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Deliverable eSDO-1121: Mode Frequency Analysis

T. Toutain, Y. Elsworth, W. Chaplin

28 June 2005

The aim of the algorithm is to apply a helioseismic data analysis technique developed by T. Toutain and A. Kosovichev(member of MDI and HMI teams) the so-called optimal-mask technique^{1} to the HMI data. This technique allows to clean a p-mode power spectrum around a given target mode making the determination of its parameters (frequency, linewidth, power amplitude) more reliable. Usual techniques based on spherical-harmonic masks are known to produce "mode leakage" around the target mode making determination of the parameters more difficult.

Once a cleaned power spectrum is obtained around the target mode a standard likelihood-minimization fitting method^{3} with a Lorentzian profile model is applied to extract the parameters of the target mode. The Lorentzian profile writes:

The parameters are : ν_{0} the mode central frequency, Γ the mode linewidth and H the power height. The central frequency is given by the position of the Lorentzian profile in the Fourier power spectrum, the typical unit is mHz. The linewidth is given by the width of the Lorentzian profile at half-height in the Fourier power spectrum, its unit is μHz. The power height is given by the height of the Lorentzian profile in the Fourier power spectrum, its typical unit for velocity observations is cm^{2}/μHz.

The frequency range in the power spectrum for which mode central frequencies will be calculated is 1.0 - 5.0 mHz.

The algorithm increases in numerical complexity and therefore stability as the degree of the p-modes increases. At first, only low-degree modes will be targeted therefore
the range of degrees for which mode parameters will be calculated is between 0 and 5.
This range can be extended to higher degrees as the algorithm proves stable and fast.

In real data the mode parameters are not known very precisely so it might be difficult to quantify how well this method performs compared to the existing method implemented in the MDI peak-bagging pipeline. It is therefore useful as a first step to implement an algorithm to produce artificial helioseismic time series. It will then be possible to check how well the outputs both from the existing fitting routine and the new one compare to the parameters put in the artificial time series. We have now finished the development of such an algorithm and artificial timeseries for modes of degree up to l=5 have been produced. The time series will be tested in the MDI peak-bagging pipeline.

Because of the similarities between MDI and HMI data the algorithm will be "validated" on the existing MDI data.

- HMI dopplergrams.

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

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

- commandline:
- 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, line width and amplitude 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, linewidth and amplitude 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. - 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

- Toutain, T.; Kosovichev, A. G., 2000, "Optimal Masks for Low-Degree Solar Acoustic Modes", The Astrophysical Journal, Volume 534, Issue 2, pp. L211-L214.
- Kosovichev, A. G. 1986, Bull. Crimean Astrophys. Obs., 75, 19
- Anderson, E.R.; Duvall,T.L.; Jefferies, S.M., 1990, "Modeling of Solar Oscillation Spectra", The Astrophysical Journal, Volume 364, pp. 699-705

-- ElizabethAuden - 29 Jun 2005

Topic revision: r16 - 2005-10-03 - ElizabethAuden

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