Bayesian approach for g‐mode detection,or how to restrict our imagination

Nowadays, g‐mode detection is based upon a priori theoretical knowledge. By doing so, detection becomes more restricted to what we can imagine. De facto, the universe of possibilities ismade narrower. Such an approach is pertinent for Bayesian statisticians. Examples of how Bayesian inferences can be applied to spectral analysis and helioseismic power spectra are given. Our intention is not to give the full statistical framework (much too ambitious) but to provide an appetizer for going further in the direction of a proper Bayesian inference, especially for detecting gravity modes. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Brightness fluctuations of the Sun and p‐mode oscillations: The inverse problem and nonadiabatic waves in the photosphere

Using photometric observations of the Sun as a star (DIFOS, SoHO) we were able to solve the inverse heloiseismic problem and determine the global time‐dependent relative temperature fluctuations as functions of the geometric height. This was done under the adiabatic assumption. A mathematical tool was developed to solve the inverse problem, which is ill‐posed. The calculations were done using the numerical software Matlab 7. The adiabatic solution shows signs of temperature waves in the lower photosphere, which agrees with calculations done by Rodríguez Hidalgo et al. (2001) and Stodilka (2011). (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variations of the low‐degree solar p‐mode frequencies for 10 years of GOLF observations

We experiment with a method of measuring the frequency of solar p modes, intended to extend the passband for the variations of the frequency spectrum as high as possible. So far this passband is limited to a fraction of μ Hz for the classical analysis based on numerical fits of a theoretical line profile to a power spectrum averaged over periods lasting at least several weeks. This limit for the present analysis can be shifted to the mHz range, corresponding to some of the “5 min” oscillations, but in this range we use a lower resolution which allows us to separate odd and even p modes. We show an example of the results for long term variations and apply this analysis to search for a modulation of the p‐mode frequency spectrum by asymptotic series of solar g modes. A faint signal is found in the analysis of 10 years of GOLF data. This very preliminary result possibly indicates the detection of a small number of g modes of degree l = 1. A tentative determination of an observational value of the parameter P₀ follows. P₀ is the scaling factor of the asymptotic series of g modes and is a key data for solar core physics. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

p‐mode power variation with solar atmosphere as observed in the Na D1 and K spectral lines

In this work we investigate p‐mode power variation with solar atmosphere. To this aim, we use THÉMIS observations of the Na D1 (λ 5896 Å) and K (λ 7699 Å) spectral lines. While the formation heights of the K spectral line are essentially located in the photospheric layer, the formation heights of the Na D1 line span a much wider region: from photosphere up to chromosphere. Hence, we had the opportunity to infer p‐mode power variation up to the chromospheric layer. By analyzing power spectra obtained by temporal series at different points of the Na D1 and K spectral lines, we confirm and quantify the increase in p‐mode power towards higher atmospheric layers. Furthermore, the large span in formation heights of the Na D1 line induces a larger enhancement of p‐mode power with solar atmosphere compared to the K spectral line. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anomalous variations in low-degree helioseismic mode frequencies

We compare changes in the frequencies of solar acoustic modes with degree between 0 and 2, as derived from Global Oscillation Network Group (GONG), Birmingham Solar Oscillations Network (BiSON) and Michelson Doppler Imager (MDI) spectra obtained between 1995 and 2003. We find that, after the solar-activity dependence has been removed from the frequencies, there remain variations that appear to be significant, and are often well correlated between the different data sets. We consider possible explanations for these fluctuations, and conclude that they are likely to be related to the stochastic excitation of the modes. The existence of such fluctuations has possible relevance to the analysis of other low-degree acoustic mode spectra such as those from solar-type stars.     

YAOPBM – yet another peak bagging method

I present and discuss the fitting methodology I developed for very‐long time series (2088‐day‐long). This new method was first used to fit low degree modes, 𝓁 ≤ 25. That time series was also sub‐divided in somewhat shorter segments (728‐daylong) and also fitted for these low degrees, in order to measure changes with the solar activity level. I have recently extended the fitting in several “directions”: 1) to substantially higher degrees (𝓁 ≤ 125), 2) to shorter time series (364‐ and 182‐day‐long), and, 3) to additional 728‐day‐long segments, covering now some 10 years of observations. I present and discuss issues related to this expansion, namely problems at low frequencies affecting the f and p₁ modes, and the inadequacy of the leakage matrix at higher degrees. I also present some of the characteristics of the observed temporal changes in the resulting frequencies. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The visibility of low‐frequency solar acoustic modes

We make predictions of the detectability of low‐frequency p modes. Estimates of the powers and damping times of these low‐frequency modes are found by extrapolating the observed powers and widths of higher‐frequency modes with large observed signal‐to‐noise ratios. The extrapolations predict that the low‐frequency modes will have small signal‐to‐noise ratios and narrow widths in a frequency‐power spectrum. Monte Carlo simulations were then performed where timeseries containing mode signals and normally distributed Gaussian noise were produced. The mode signals were simulated to have the powers and damping times predicted by the extrapolations. Various statistical tests were then performed on the frequency‐amplitude spectra formed from these timeseries to investigate the fraction of spectra in which the modes could be detected. The results of these simulations were then compared to the number of p‐modes candidates observed in real Sun‐as‐a‐star data at low frequencies. The fraction of simulated spectra in which modes were detected decreases rapidly as the frequency of modes decreases and so the fraction of simulations in which the low‐frequency modes were detected was very small. However, increasing the signal‐to‐noise (S/N) ratio of the low‐frequency modes by a factor of 2 above the extrapolated values led to significantly more detections. Therefore efforts should continue to further improve the quality of solar data that is currently available. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the effect of error correlation on linear inversions

We have examined the effect on linear helioseismic inversions of correlations in data errors, taking an example from one-dimensional rotational splitting inversion. Artificial data with correlated errors were generated and then inverted with or without using the proper covariance matrix. The effects of using incorrect covariance matrices, on solutions as well as on trade-offs, are discussed. It is found that improper account of the correlations can be deleterious to the faithfulness of the inversions, and yields incorrect error estimates, which under some circumstances can lead to misleading inferences.     

Detection and temporal coherence of p‐modes below 1.4 mHz

Data collected recently by the helioseismic experiments aboard the SOHO spacecraft have allowed the detection of low degree p‐modes with increasingly lower order n. In particular, the GOLF experiment is currently able to unambiguously identify low degree modes with frequencies as low as 1.3 mHz. The detection of p‐modes with very low frequency (i.e., low n), is difficult due to the low signal‐to‐noise ratio in this spectral region and its contamination by solar signals that are not of acoustic origin. To address this problem without using any theoretical a priory, we propose a methodology that relies only on the inversion of observed values to define a spectral window for the expected locations of these low frequency modes. The application of this method to 2920‐day‐long GOLF observations is presented and its results discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Meridional flow profile measurements with SOHO/MDI

We present meridional flow measurements of the Sun using a novel helioseismic approach for analyzing SOHO/MDI data in order to push the current limits in radial depth. Analyzing three consecutive months of data during solar minimum, we find that the meridional flow is as expected poleward in the upper convection zone, turns equatorward at a depth of around 40 Mm (∼ 0.95 R_⊙), and possibly changes direction again in the lower convection zone. This may indicate two meridional circulation cells in each hemisphere, one beneath the other. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling solar atmospheric gravity oscillation modes

Solar oscillations are investigated in a one‐dimensional hydrodynamic plane‐parallel model with an atmosphere. Besides the acoustic pressure (p) modes, the fundamental (f) and Lamb mode, another set of eigenmodes, a group of atmospheric gravity (g) modes, is found in the low‐frequency region of the spectrum. Their frequencies and spatial behaviour are studied. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar-cycle variation of the sound-speed asphericity from GONG and MDI data 1995–2000

We study the variation of the frequency splitting coefficients describing the solar asphericity in both GONG and MDI data, and use these data to investigate temporal sound-speed variations as a function of both depth and latitude during the period 1995–2000 and a little beyond. The temporal variations in even splitting coefficients are found to be correlated to the corresponding component of magnetic flux at the solar surface. We confirm that the sound-speed variations associated with the surface magnetic field are superficial. Temporally averaged results show a significant excess in sound speed around     and latitude of 60°.     

The phase function for stellar acoustic oscillations — IV. Solar-like stars

In recent years there has been some progress towards detecting solar-like oscillations in stars. The goal of this challenging project is to analyse frequency spectra similar to that observed for the Sun in integrated light. In this context it is important to investigate what can be learned about the structure and evolution of the stars from such future observations. Here we concentrate on the structure of the upper layers, as reflected in the phase function. We show that it is possible to obtain this function from low-degree p modes, at least for stars on the main sequence. We analyse its dependence on several uncertainties in the structure of the uppermost layers. We also investigate a filtered phase function, which has properties that depend on the layers around the second helium ionization zone.     

Helioseismology program for the PICARD satellite

The PICARD mission is a CNES micro‐satellite to be launched in 2009. Its goal is to better understand the Sun and the potential impact of its activity on earth climate by measuring simultaneously the solar total and spectral irradiance, diameter, shape and oscillations. We present the scientific objectives, instrumental requirements and data products of the helioseismology program of PICARD which aims to observe the low to medium l p‐mode oscillations in intensity and search for g‐mode oscillation signatures at the limb. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A modified peak‐bagging technique for fitting low‐𝓁 solar p‐modes

We introduce a modified version of a standard power spectrum ‘peak‐bagging’ technique which is designed to gain some of the advantages that fitting the entire low‐degree p‐mode power spectrum simultaneously would bring, but without the problems involved in fitting a model incorporating many hundreds of parameters. Employing Monte‐Carlo simulations we show that by using this modified fitting code it is possible to determine the true background level in the vicinity of the p‐mode peaks. In addition to this we show how small biases in other mode parameters, which are related to inaccurate estimates of the true background, are also consequently removed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New aspects of Doppler imaging in Sun-as-a-star observations

Birmingham Solar Oscillations Network (BiSON) instruments use resonant scattering spectrometers to make unresolved Doppler velocity observations of the Sun. Unresolved measurements are not homogenous across the solar disc and so the observed data do not represent a uniform average over the entire surface. The influence on the inhomogeneity of the solar rotation and limb darkening has been considered previously and is well understood. Here, we consider a further effect that originates from the instrumentation itself. The intensity of light observed from a particular region on the solar disc is dependent on the distance between that region on the image of the solar disc formed in the instrument and the detector. The majority of BiSON instruments have two detectors positioned on opposite sides of the image of the solar disc and the observations made by each detector are weighted towards differing regions of the disc. Therefore, the visibility and amplitudes of the solar oscillations and the realization of the solar noise observed by each detector will differ. We find that the modelled bias is sensitive to many different parameters such as the width of solar absorption lines, the strength of the magnetic field in the resonant scattering spectrometer, the orientation of the Sun's rotation axis, the size of the image observed by the instrument and the optical depth in the vapour cell. We find that the modelled results best match the observations when the optical depth at the centre of the vapour cell is 0.55. The inhomogeneous weighting means that a 'velocity offset' is introduced into unresolved Doppler velocity observations of the Sun, which varies with time, and so has an impact on the long-term stability of the observations.