The art of fitting p-mode spectra: Part II. Leakage and noise covariance matrices

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 $(m,\nu)$ 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.Comment: 17 pages with tex and ps files, submitted to A&A, [email protected]

The art of fitting p-mode spectra: Part I. Maximum Likelihood Estimation

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 $\chi^{2}$ with 2 degrees of freedom. Results for $l=1$ 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.Comment: 13 pages, ps file gzipped. Submitted to A&

On the choice of parameters in solar structure inversion

The observed solar p-mode frequencies provide a powerful diagnostic of the internal structure of the Sun and permit us to test in considerable detail the physics used in the theory of stellar structure. Amongst the most commonly used techniques for inverting such helioseismic data are two implementations of the optimally localized averages (OLA) method, namely the Subtractive Optimally Localized Averages (SOLA) and Multiplicative Optimally Localized Averages (MOLA). Both are controlled by a number of parameters, the proper choice of which is very important for a reliable inference of the solar internal structure. Here we make a detailed analysis of the influence of each parameter on the solution and indicate how to arrive at an optimal set of parameters for a given data set.Comment: 14 pages, 15 figures. Accepted for publication on MNRA

On The Determination of MDI High-Degree Mode Frequencies

The characteristic of the solar acoustic spectrum is such that mode lifetimes get shorter and spatial leaks get closer in frequency as the degree of a mode increases for a given order. A direct consequence of this property is that individual p-modes are only resolved at low and intermediate degrees, and that at high degrees, individual modes blend into ridges. Once modes have blended into ridges, the power distribution of the ridge defines the ridge central frequency and it will mask the true underlying mode frequency. An accurate model of the amplitude of the peaks that contribute to the ridge power distribution is needed to recover the underlying mode frequency from fitting the ridge. We present the results of fitting high degree power ridges (up to l = 900) computed from several two to three-month-long time-series of full-disk observations taken with the Michelson Doppler Imager (MDI) on-board the Solar and Heliospheric Observatory between 1996 and 1999. We also present a detailed discussion of the modeling of the ridge power distribution, and the contribution of the various observational and instrumental effects on the spatial leakage, in the context of the MDI instrument. We have constructed a physically motivated model (rather than some ad hoc correction scheme) resulting in a methodology that can produce an unbiased determination of high-degree modes, once the instrumental characteristics are well understood. Finally, we present changes in high degree mode parameters with epoch and thus solar activity level and discuss their significance.Comment: 59 pages, 38 figures -- High-resolution version at http://www-sgk.harvard.edu:1080/~sylvain/preprints/ -- Manuscript submitted to Ap

Blackbody temperature of 200+ stellar flares observed with the CoRoT satellite

We estimated blackbody temperature for 209 flares observed at 69 F-K stars, significantly increasing the number of flare temperature determinations. We used the Blue and Red channels obtained by the 27 cm telescope of the CoRoT satellite at high cadence and long duration. The wavelength limits of the channels were estimated using spectra from the Pickles library for the spectral type and luminosity class of each star, provided by the Exodat Database. The temperatures were obtained from the flare energy Blue-to-Red ratio, using the flare equivalent duration and stellar flux in both channels. The expected value of the analyzed flares is equal to 6,400 K with a standard deviation of 2,800 K, where the mean stellar spectral type, weighted by the number of flares in each spectral subclass, is equal to G6. Contrary to our results, a stellar white-light flare is often assumed to emit as a blackbody with a temperature of 9,000 K or 10,000 K. Our estimates agree, however, with values obtained for solar flares. The GAIA G-band transmissivity is comparable to that of the CoRoT White channel, which allows us to calibrate the flares to the Gaia photometric system. The energy in the G band of the analyzed flares varies between $10^{32}$ and $10^{37}$ erg and the flare area ranges from 30$\mu$sh to 3 sh (solar hemisphere). The energy release per area in a flare is proportional to $T_{\rm flare}^{2.6}$, at least up to 10,000 K.Comment: Accepted Astronomical Journa

Subsurface structure evolution associated with the rise and fall of intensely active regions

In order to study the sub-surface structure variations associated with the formation and evolution of major active regions producing large, long-lived sunspot groups and intense flare activity, we analyze the localized power spectra at the sites of selected active regions from well before their emergence through their disappearance, comparing the helioseismic data with those for quiet regions of the same size and at the same latitude during similar time ranges. Because of the need for continuous data through disc passages of the selected regions over several Carrington rotations, we have analyzed GONG Doppler data, for which nearly continuous observations are always available, whereas continuous MDI data are usually limited to two rotations or less. The studied active regions were selected from among those that attained the greatest sunspot group area during the years for which there is reasonably complete GONG+ data coverage, from the middle of 2001 through 2005

Image Quality of the Helioseismic and Magnetic Imager (HMI) Onboard the Solar Dynamics Observatory (SDO)

We describe the imaging quality of the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO) as measured during the ground calibration of the instrument. We describe the calibration techniques and report our results for the final configuration of HMI. We present the distortion, modulation transfer function, stray light,image shifts introduced by moving parts of the instrument, best focus, field curvature, and the relative alignment of the two cameras. We investigate the gain and linearity of the cameras, and present the measured flat field