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Deliverable eSDO-1121: Local Helioseismology Inversion

S. Zharkov, M. Thompson

28 June 2005

The aim of this package is to provide front-end interface for specifying and launching inversions of helioseismic data, on local or remote machines; maintain queueing system and indices of data and inversions results; retrieve inversions results and allow display and local storage.

Travel times (mean/difference), corresponding travel time sensitivity kernels, noise covariance matrix; regularisation option, trade-off parameter;

Solar interior inversion, error estimates, resolution estimates

command line

In general, the inverse problem in local helioseismology is ill posed and thus requires some "a priori" knowledge which is used to select the proper solution from the set of possible solutions given by the measurements. This normally consists of choosing an additional constraint which causes the chosen solution to be "smooth" in some way. The relative influence of such regularization is controlled by "regularization trade-off parameter". In general, model solutions are computed for a range of different values, and then the "best" model is chosen.

In addition, the degree to which any inversion method succeeds depends on the accuracy of the forward model, noise level in travel times, the depth and spatial scales of real variations in the Sun, the number and type of travel times that are available as input to inversion, the accuracy of the travel time covariance matrix. This package will provide the user with tools to apply different inversion methods with varying trade-off parameters to the data enabling the user to choose "best" results.

For Perturbation Map Generation and Subsurface Flow Analysis algorithms during the inversion part the choice of regularisation and trade-off parameter will be made automatically as a first approximation. User will be able to take travel-times, error covariance matrix and travel time sensitivity kernels available as additional output in the above algorithms and use those as input for A Local Helioseismology Inversion Workbench to refine the inversion results.

Inversion methods considered for implementation: Multi channel deconvolution, Regularized Least Squares, Singular Value Decomposition, Subtractive Optimally Localized Averages.

** Option 1: Multi Channel Deconvolution (Priority for implementation)**

Input: Travel times, corresponding sensitivity kernels, Solar model, noise covariance matrix, trade-off parameter, regularization operator

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

2. Calculate weight matrices for model vector using chosen regularization and trade-off parameter.

3. Calculate the Fourier transform of the inverse operator using weight matrices, Fourier transformed model data.

4. Calculate the Fourier transform of the estimated soundspeed perturbation

5. The soundspeed perturbation estimate is found by inverse Fourier transforming of 3 layer by layer.

6. Calculate the covariance matrix of the estimated model using inverse operator and data error covariance matrix

7. Calculate the resolution matrix using inverse operator and Fourier transformed data.

Output: Solar interior inversion, error estimates, resolution estimates

** Option 2: Subtractive Optimally Localized Averages **

Input: travel times, corresponding sensitivity kernels, noise covariance matrix, trade-off parameter

1. From sensitivity kernels, noise covariance matrix and trade-off parameter calculate the the mode kernel cross-correlation matrix and calculate its inverse.

2. Choose Gaussian target function parameters, ie spatial location in 3D, spatial horizontal and vertical extent.

3. For each data point calculate the cross-correlation vector of the mode kernels with the target function with an extra column used for constraint.

4. From 1. and 3. calculate weight coefficients at each data point.

5. Calculate the solar interior parameters using input data and weight coefficients.

6. Calculate error bars and spatial resolution attained.

** Option 3: Generalized Singular Value Decomposition **

Here we are essentially minimizing the expression

where Lambda is the trade-off parameter, **L** is the regularization operator, and ||.|| denotes L^{2} norm.

Input: travel times, corresponding sensitivity kernels, Solar model, noise covariance matrix, trade-off parameter

1. Using sensitivity kernels and top hat functions as basis calculate matrix **A**.

2. Solve generalized SVD for **A** and regularization operator **L** obtaining generalized singular values of (**A**, **L**), rank of **A** and decomposition matrices **U**, **V**, **W ^{-1}**.

3. Calculate **W** and filter factors.

4. Using **U**, **V**, **W**, filter factors and non-zero singular values calculate the inversion from input data at each data point.

5. Calculate resolution matrix and error estimates.

** Regularization operator choice ** The choice of the regularization operator will be the zero-th, first or second derivative of the model.

1. B.H. Jacobsen, I. Moller, J.M. Jensen and F.Efferso, Multichannel Deconvolution, MCD, in Geophysics and Helioseismology, Phys. Chen. Earth(A), Vol. 24, No. 3, 215-220, 1999

- B.H. Jacobsen, I. Moller, J.M. Jensen and F.Efferso, Multichannel Deconvolution, MCD, in Geophysics and Helioseismology, Phys. Chen. Earth(A), Vol. 24, No. 3, 215-220, 1999
- Pijpers, F. P., Thompson, M. J., The SOLA method for helioseismic inversion, A&A, 1994,
- Christensen-Dalsgaard, J., Hansen, P. C., Thompson, M. J., Generalized Singular Value Decomposition Analysis of Helioseismic Inversions, 1993
- Tikhonov A N and Arsenin V Ya, Solution of Ill-Posed Problems, 1977

2. Pijpers, F. P., Thompson, M. J., The SOLA method for helioseismic inversion, A&A, 1994,

3. Christensen-Dalsgaard, J., Hansen, P. C., Thompson, M. J., Generalized Singular Value Decomposition Analysis of Helioseismic Inversions, 1993

4. Tikhonov A N and Arsenin V Ya, Solution of Ill-Posed Problems, 1977

-- ElizabethAuden - 29 Jun 2005

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