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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.
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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: • Linearize the dependence of the measurements (e.g. travel times) on first order changes in wavefield covariance: • Linearize dependence of wavefield covariance on changes in the interior of the model (e.g. change in sound speed)
First-Order Change in Cross-Covariance Use Born approximation to get the sensitivity of the cross-covariance to perturbations to the background model: … Gizon & Birch 2002 ApJ
Line of Sight J. Jackiewicz and L. Gizon
Phase-Speed Filtering Birch, Duvall & Kosovichev 2004 ApJ
Two More Examples Holography Time-Distance
Advertisement: read Woodard 2006 ApJ, coming out soon
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 • 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
True velocity Holography @ 6 mHz 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.