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Deliverable eSDO-1121: Subsurface Flow Analysis

S. Zharkov, M. Thompson

28 June 2005

The purpose of this algorithm is to measure and interpret the travel times of the acoustic waves between any two locations on the solar surface in terms of bulk subsurface flows. An anomaly in the travel-time difference for waves propagating in opposite directions contains the seismic signature of the subsurface flow within the proximity of the ray path. The subsurface flow is obtained by building a forward model using the Rytov approximation and then solving the inverse problem for a particular set of observed travel-time differences.

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

Travel-time difference maps for different skip distances and orientations; Subsurface flow maps under the tracked region obtained by inversion; Combinations of flow maps under tracked regions organised into synoptic maps.

command line

The aim of the package is to measure subsurface flows 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 travel-time differences are then obtained from one-way travel times. These contain information about subsurface flow 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: travel-time differences, sensitivity kernels and inversion results. Travel times and sensitivity kernels could be used as input for Local Helioseismology Inversion package to refine the inversion. Also it may be possible to use user generated sensitivity kernels.

The problem consists of three stages: data interpretation via filtering stages and cross-correlation, and estimation of travel times and travel-time differences; 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 subsurface flow.

**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-distances.

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 differences 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 flow 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 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: Subsurface flow perturbation sensitivity kernels, 3D data cube.

**Inversion: **

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

Input: Travel time differences for various skip distances, corresponding sensitivity kernels, Solar model, data error covariance matrix

1. Perform 2D Fourier transforms of the input travel time difference 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 subsurface flow estimate.

Output: subsurface flow as a function of depth and position, covariance matrix of the estimated model

Other methods considered for inversion: Regularised Least Squares; 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

Topic revision: r17 - 2005-09-29 - SergeiZharkov

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