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Deliverable eSDO-1121: Mode Asymmetry Analysis

T. Toutain, Y. Elsworth, W. Chaplin

28 June 2005

The algorithm developed here is based on a work done internally by the eSDO Birmingham group and which is submitted to the MNRAS journal. It is based on a modification of the usual formula describing an asymmetrical p-mode profile accounting for *c* the so-called correlated-noise coefficient. This coefficient describes to which extend the excitation of a mode can be described by the solar background noise. We assume that the excitation function of a p-mode is the same as the background noise component having the same spatial pattern as the mode. In that case, the p-mode line profile in the power spectrum is no longer modeled with a Lorentzian profile but instead with the following formula :

where L is the usual Lorentzian profile :

H,ν_{0},Γ and n(ν) are the mode height, frequency, linewidth and the background noise, respectively. *c* is related to *a* the usual asymmetry as defined by Nigam and Kosovichev (see ref. 1 here below) with :

Note: the asymmetry or the coefficient of correlated-noise are parameters which define the shape of the p-mode profile as do also the usual parameters (frequency, linewidth...). Therefore their determination requests to simultaneaously determine the others parameters. Hence "mode asymmetry analysis" is somehow a part of "mode frequency analysis" which means that both have similar Inputs, Outputs... as described here below.

- HMI dopplergrams.

- FITS file: table containing frequency, mode asymmetry for each p mode of low degree and their associated error bars.

- Artificial helioseismic time series.
- MDI helioseismic time series.

- commandline; commandline: input HMI dopplergrams of AIA images, output of FITS files containing images and statistical data.
- AstroGrid CEA web service: this algorithm will be deployed as a CEA service hosted in the UK that users can call the web service to process datasets on the grid. Due to computational intensity, access to this service may be restricted to registered solar users of AstroGrid.
- SolarSoft routine: the C module will be wrapped in IDL and distributed through the MSSL SolarSoft gateway. Users will access to a SolarSoft installation can call the routine from the commandline or GUI to process locally held data.
- JSOC module: the C module will be installed in the JSOC pipeline. Users can access the routine through pipeline execution to operate on data local to the JSOC data centre.

- The user identifies a period of observation during which p-mode parameters will be calculated
- The user obtains HMI dopplergrams covering this period of observation and constructs time series using optimal masks.
- Next, the mode frequency analysis algorithm is applied to sht time series.
- The algorithm runs and returns a list of p-modes with the frequency and asymmetry for each mode (inlcuding error bars for each parameter).

- Apply the Optimal Mask technique to an HMI dopplergram:
- First, divide the dopplergram into a number of bins (for example the binning of the LOI-proxy as defined for MDI data).
- Next, model the signal of each image bin produced by a specific mode; average the signals coming from
*N*CCD pixels in each bin._{k} - Finally, choose the optimal mask vector to maximize the target mode's signal while minimizing signals from other nearby modes by using the singular value decomposition (SVD)
^{2}method to minimize contamination or leakage of modes onto the target mode. It consists of the following steps:- Construct a local optical mask: identify a window around the target mode, and filter out modes whose frequencies fall in this window.
- Add a regularization term to model the noise contribution.

- Apply the preceeding method to all the Dopplergrams in the period of observations as defined by the user obtaining a time series for the target mode.
- A discrete Fourier transform is applied to convert the time series to a power spectrum. * The resulting power spectrum should reflect the frequency of the selected mode while minimizing frequencies from nearby modes.
- The mode frequency analysis algorithm is called with the target mode time series data as input.

- the mode parameter determination is based on a standard likelihood-minimization technique using a Lorentzian profile as a model for the mode profile in the power spectrum.

- Return the frequency and asymmetry of the target mode.

- Artificial time series (fits format) These artificial time series are made using the following steps:
- Make independent complex spectra for each simulated mode. Modes of degrees
*l*=0-20 and azimuthal order*|m|*≤ *l*are modelled using observed solar p-mode frequencies. Their linewidths and amplitudes are obtained fitting a curve to exisiting measurements of these parameters. Asymmetry is included via a correlation between background noise and excitation function of the mode. - Inverse Fourier transform each spectrum obtaining a time series of complex amplitude for each mode.
- Multiply at each instant
*t*the complex amplitude of each mode with its corresponding spherical harmonics pattern of projected velocity onto CCD pixels and sum-up for all modes. - To obtain an artificial time series for a given mode (
*l*,*m*),multiply the signal on each CCD pixel with the value of the complex conjuguate of the corresponding spherical harmonics and sum-up for all pixels.

- Make independent complex spectra for each simulated mode. Modes of degrees

- Nigam, R.; Kosovichev, A. G., "Measuring the Sun's Eigenfrequencies from Velocity and Intensity Helioseismic Spectra: Asymmetrical Line Profile-fitting Formula ", 1998, Astrophysical Journal Letters v.505, p.L51

-- ElizabethAuden - 29 Jun 2005

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