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Deliverable eSDO-1121: Perturbation Map Generation

S. Zharkov, M. Thompson

28 June 2005

The purpose of this algorithm is to measure and interpret the travel times of the waves between any two locations on the solar surface in terms of subsurface wave-speed between them. An anomaly in the mean travel-time contains the seismic signature of the wave speed perturbation within the proximity of the ray path. The wave speed perturbation is obtained by solving the inverse problem.

HMI tracked and remapped Dopplergrams of rectangular regions of solar disk.

Travel-time anomaly maps for different skip-distances; Subsurface wave speed maps under the tracked region; Synoptic wave speed anomaly maps;

command line

The aim of the package is to measure sound speed perturbations in upper convection zone, for use in understanding and predicting AR evolution and evolution of atmospheric magnetic structures.

First, input tracked datacube is pre-processed and center to annulus travel-time measurements are extracted by computing the temporal cross covariance of the signal at a point on the solar surface with the signal at another point. The mean travel-times are then obtained from one-way travel times. These contain information about solar interior wave speed perturbation which is extracted by solving the equation

This is done by building wave speed sensitivity kernels in Rytov's approximation

and solving the first equation using the Multichannel Deconvolution method. The package will provide three main outputs: mean travel times, sensitivity kernels and inversion results. Travel times and sensitivity kernels can be used as input for Local Helioseismology Inversion package to refine the inversion. Also it may The travel time data could also be possible to use user generated used on its own with sensitivity kernels. kernels and inversion routines generated or provided by the user.

The problem consists of three stages: data interpretation via filtering stages and cross-correlation and estimation of travel time and travel-time means; building a forward model of the Sun to tie the surface data and subsurface features; the solution of the resulting inversion problem to recover the soundspeed perturbation.

**Data Interpretation:**

1. Input Doppler tracked and remapped datacube is Fourier transformed and filtered by applying a high-pass filter to remove convective motions, f-mode filter (removing f-mode ridge) and then a phase speed filter to select the waves that travel similar skip-distance.

2. From the filtered signal compute the cross-covariance function, suitably averaging to increase the signal-to-noise ratio.

3. Travel-times of the waves travelling in each direction are obtained by fitting the averaged cross-covariance function with a smooth cross-covariance function computed from a solar model or from quiet Sun data. Travel-time means are then computed.

4. The noise covariance matrix is estimated by measuring the rms travel time within a quiet Sun region.

**Forward Model:** building travel time sensitivity kernels for sound speed perturbation using Rytov approximation

Input: Solar model, spatial resolution, skip-distance

1. For every pair of points in the output data cube, calculate ray paths and theoretical travel times for rays of the given frequency travelling to and from surface points via depth point. The horizontal translational invariance of the background model greatly reduces the amount of computing required.

2. Using the ray travel times and ray path length calculate approximate sensitivity kernels for each of the skip-distances in the Rytov approximation.

Output: Sound speed perturbation sensitivity kernels, 3D data cube.

**Inversion: **

To infer the soundspeed perturbation from the observation we invert the travel-time differences using the travel-time sensitivity kernels and multi-channel deconvolution algorithm.

Input: Mean travel time for various skip distances, corresponding sensitivity kernels, Solar model, mean travel time error covariance matrix

1. Perform 2D Fourier transforms of the input mean travel time perturbations and sensitivity kernels

2. Calculate weight matrices for model vector using error covariance matrix and chosen trade-off parameter

3. Calculate the Fourier transform of the estimated soundspeed perturbation

4. Apply layer by layer inverse Fourier transform to obtain soundspeed perturbation estimate.

Output: sound speed perturbation as a function of depth and position, covariance matrix of the estimated model

Other methods considered Regularised Least Square; Optimally Localised Averages; LSQR; Singular Value Decomposition;

Gizon, L., Birch, A.C., Local helioseismology, Living Reviews of Solar Physics, 2005

Giles, P.M., Time-distance Measurements of Large Scale Flows in the Solar Convection Zone (Ph.D. Thesis)

J.M. Jensen and F.P. Pijpers, Sensitivity kernels for time-distance inversion based on the Rytov approximation, Astronomy & Astrophysics, 412, 257-265 (2003)

J.M. Jensen, Helioseismic Time-Distance Inversion, (Ph.D. thesis), 2001

-- ElizabethAuden - 29 Jun 2005

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