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Sensitivity Kernels for Local Helioseismology. Aaron Birch NWRA, CoRA Division. Outline. Introduction to the forward problem Examples of kernels Artificial data Open questions. The Forward Problem. Given a model, e.g. sound-speed, flows want to know what to expect for measurements. - PowerPoint PPT Presentation
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Sensitivity Kernels for Local Helioseismology
Aaron Birch
NWRA, CoRA Division
Outline
• Introduction to the forward problem
• Examples of kernels
• Artificial data
• Open questions
The Forward Problem• Given a model, e.g. sound-speed, flows
want to know what to expect for measurements.
• Want linear forward problem; necessary for linear inversions
Linear Forward Problems
Two Steps:
1) Linearize the dependence of the measurements (e.g. travel times) on first order changes in wavefield covariance:
2) Linearize dependence of wavefield covariance on changes in the interior of the model (e.g. change in sound speed)
First-Order Change in Cross-CovarianceUse Born approximation to get the sensitivity of the cross-covariance to perturbations to the background model:
Gizon & Birch 2002 ApJ
…
Example Kernels
J. Jackiewicz and L. Gizon
Line of Sight
Phase-Speed Filtering
Birch, Duvall & Kosovichev 2004 ApJ
Two More ExamplesHolographyTime-Distance
Ring Kernels
read Woodard 2006 ApJ,coming out soon
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Software• Currently implemented in matlab
• Now with web interface, very nice !
• This code can do– Sound-speed kernels for time-distance– Unperturbed power spectrum and cross-
covariance – Coming Soon: flow kernels for holography and
time-distance
• Requires a background model, source parameters, and damping model
• Pretty flexible code: (example input file)
Artificial Data
Artificial Data
• Use wave propagation simulations to generate artificial data (many groups are now working on this !)
• Once the machinery is in place will provide a very efficient means for solving complicated non-linear forward problems
• Explore bias, signal-to-noise ratios, optimize measurements, study cross-talk, validation of forward and inverse methods
Holography @ 6 mHz True velocity
Vx
Vy
w/D. Braun. Simulations from Stein & Nordlund also help from Dali Georgobiani.
Some Open Questions• For details: Gizon & Birch 2005 Living Reviews Article • Wave propagation through magnetic regions !• Source effects ? Parchevsky• Range of validity of linearizations ?• Linearization around something other than quiet Sun
models ? Will this help with Born approx for magnetic fields ?
• Currently unknown which kernels are needed only for small corrections and which are crucial. Simulations will help.
• Line of sight & foreshortening. In principle need kernels for each position on the disk. Likely important for small-scale flows and sound-speed variations.