Solar low-degree p-mode parameters from the GONG network (original) (raw)
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The art of fitting p-mode spectra: Part I. Maximum Likelihood Estimation
1997
In this article we present our state of the art of fitting helioseismic p-mode spectra. We give a step by step recipe for fitting the spectra: statistics of the spectra both for spatially unresolved and resolved data, the use of Maximum Likelihood estimates, the statistics of the p-mode parameters, the use of Monte-Carlo simulation and the significance of fitted parameters. The recipe is applied to synthetic low-resolution data, similar to those of the LOI, using Monte-Carlo simulations. For such spatially resolved data, the statistics of the Fourier spectrum is assumed to be a multi-normal distribution; the statistics of the power spectrum is \emph{not} a chi2\chi^{2}chi2 with 2 degrees of freedom. Results for l=1l=1l=1 shows that all parameters describing the p modes can be obtained without bias and with minimum variance provided that the leakage matrix is known. Systematic errors due to an imperfect knowledge of the leakage matrix are derived for all the p-mode parameters.
The art of fitting p-mode spectra: Part II. Leakage and noise covariance matrices
1997
In Part I we have developed a theory for fitting p-mode Fourier spectra assuming that these spectra have a multi-normal distribution. We showed, using Monte-Carlo simulations, how one can obtain p-mode parameters using 'Maximum Likelihood Estimators'. In this article, hereafter Part II, we show how to use the theory developed in Part I for fitting real data. We introduce 4 new diagnostics in helioseismology: the (m,nu)(m,\nu)(m,nu) echelle diagramme, the cross echelle diagramme, the inter echelle diagramme, and the ratio cross spectrum. These diagnostics are extremely powerful to visualize and understand the covariance matrices of the Fourier spectra, and also to find bugs in the data analysis code. These diagrammes can also be used to derive quantitative information on the mode leakage and noise covariance matrices. Numerous examples using the LOI/SOHO and GONG data are given.
The art of tting p-mode spectra I. Maximum likelihood estimation
In this article we present our state of the art of tting helioseismic p-mode spectra. We give a step by step recipe for tting the spectra: statistics of the spectra both for spatially unresolved and resolved data, the use of Maximum Likelihood estimates, the statistics of the p- mode parameters, the use of Monte-Carlo simulation and the signicance of tted parameters. The recipe is applied to synthetic low-resolution data, similar to those of the LOI, using Monte-Carlo simulations. For such spatially resolved data, the statistics of the Fourier spectrum is as- sumed to be a multi-normal distribution; the statistics of the power spectrum is not a 2 with 2 degrees of free- dom. Results for l = 1 shows that all parameters describ- ing the p modes can be obtained with negligible bias and with minimum variance provided that the leakage matrix is known. Systematic errors due to an imperfect knowl- edge of the leakage matrix are derived for all the p-mode parameters.
LOWL P-MODE FREQUENCIES AND THEIR VARIATION WITH SOLAR ACTI VITY
We present an analysis of the frequency shift and the even terms of the frequency splitting coefficients carried out using six years of LOWL data, starting in 1994. The temporal variations, and their dependences with the fre- quency and degree are addressed. The results are consis- tent with previous analysis.
Secular variations in the spectrum of solar p-modes
Solar Physics, 1994
The solar p-mode spectrum of very low I is measured with high accuracy for a long enough period of time so as to allow the search for solar cycle variations, in this paper solar cycle variations of the frequency and energy of the modes are confirmed. Moreover, a slight variation,within errors, of its rotational splitting with the solar cycle, is suggested.
MEASUREMENT OF LOW SIGNAL-TO-NOISE RATIO SOLAR p -MODES IN SPATIALLY RESOLVED HELIOSEISMIC DATA
The Astrophysical Journal, 2009
We present an adaptation of the rotation-corrected, m-averaged spectrum technique designed to observe low signal-to-noise-ratio, low-frequency solar p modes. The frequency shift of each of the 2l + 1 m spectra of a given (n, l) multiplet is chosen that maximizes the likelihood of the m-averaged spectrum. A high signal-to-noise ratio can result from combining individual low signal-to-noise-ratio, individual-m spectra, none of which would yield a strong enough peak to measure. We apply the technique to GONG and MDI data and show that it allows us to measure modes with lower frequencies than those obtained with classic peak-fitting analysis of the individual-m spectra. We measure their central frequencies, splittings, asymmetries, lifetimes, and amplitudes. The low-frequency, low-and intermediate-angular degrees rendered accessible by this new method correspond to modes that are sensitive to the deep solar interior down to the core (l ≤ 3) and to the radiative interior (4 ≤ l ≤ 35). Moreover, the low-frequency modes have deeper upper turning points, and are thus less sensitive to the turbulence and magnetic fields of the outer layers, as well as uncertainties in the nature of the external boundary condition. As a result of their longer lifetimes (narrower linewidths) at the same signal-to-noise ratio the determination of the frequencies of lower-frequency modes is more accurate, and the resulting inversions should be more precise.
solarFLAG hare and hounds: estimation of p-mode frequencies from Sun-as-star helioseismology data
Monthly Notices of the Royal Astronomical Society, 2008
We report on the results of the latest solarFLAG hare-and-hounds exercise, which was concerned with testing methods for extraction of frequencies of low-degree solar p modes from data collected by Sun-as-a-star observations. We have used the new solarFLAG simulator, which includes the effects of correlated mode excitation and correlations with background noise, to make artificial timeseries data that mimic Doppler velocity observations of the Sun as a star. The correlations give rise to asymmetry of mode peaks in the frequency power spectrum. Ten members of the group (the hounds) applied their "peak bagging" codes to a 3456-day dataset, and the estimated mode frequencies were returned to the hare (who was WJC) for comparison. Analysis of the results reveals a systematic bias in the estimated frequencies of modes above ≈ 1.8 mHz. The bias is negative, meaning the estimated frequencies systematically underestimate the input frequencies.
Journal of Physics: Conference Series, 2008
In order to take full advantage of the long time series collected by the GONG and MDI helioseismic projects, we present here an adaptation of the rotation-corrected m-averaged spectrum technique in order to observe low radial-order solar p modes. Modeled profiles of the solar rotation demonstrated the potential advantage of such a technique [1, 2, 3]. Here we develop a new analysis procedure which finds the best estimates of the shift of each m of a given (n, ℓ) multiplet, commonly expressed as an expansion in a set of orthogonal polynomials, which yield the narrowest mode in the m-averaged spectrum. We apply the technique to the GONG data for modes with 1 ≤ ℓ ≤ 25 and show that it allows us to measure lower-frequency modes than with classic peak-fitting analysis of the individual-m spectra.
Astronomy and Astrophysics
We estimate the statistical uncertainties of low-l solar p-modes parameters based on a Monte Carlo approach. Random perturbations of ideal Lorentz profiles L(a, ν i) can provide many estimations of the set of p-modes parameters a and allow one to estimate statistical error-bars σ a by modelling the parameters' distribution function. Unlike frequencies, which show symmetric distributions, amplitudes and linewidths have asymmetric probability density function similar to the distribution function for time-averaged energies of stochastically excited solar p-modes (Kumar, 1988). A comparison between σ ν and uncertainties based on Hessian's computation (Libbrecht 1992, Toutain and Appourchaux 1994) shows a nice agreement. However, our error-bars take into account more statistical effects, and rely less on the initial parameters' estimation. Such a technique has been used on the IRIS power spectra computed from gapped data, and on one GONG power spectrum computed from almost continuous data. We also present IRIS linewidths and error bars averaged over the years 1989-92 and computed with a fitting strategy using imposed frequency which improves the value of both the parameter and its uncertainty.
Astronomy and Astrophysics, 2004
Low-angular-degree solar p modes provide a sensitive probe of the radiative interior and core of the Sun. Estimates of their centroid frequencies can be used to constrain the spherically symmetric structure of these deep-lying layers. The required data can be extracted from two types of observation: one where the modes are detected in integrated sunlight, i.e., a Sun-as-astar view; and a second where the visible disc is imaged onto many pixels, and the collected images then decomposed into their constituent spherical harmonics. While the imaging strategy provides access to all of the individual components of a multiplet, the Sun-as-a-star technique is sensitive to only just over half of these, with those modes that are detected having different levels of visibility. Because the various components can have contrasting spatial structure over the solar surface, they can respond very differently to changes in activity along the solar cycle. Since the Sun-as-a-star and resolved analyses take as input a different 'subset' of modes, the extracted frequency estimates are expected to differ depending upon the phase of the cycle. Differences also arise from the types of models used to fit the modes. Here, we present expressions that allow the sizes of the differences to be predicted.