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-Covariance Use 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 Examples Holography Time-Distance

Ring Kernels

read Woodard 2006 ApJ, coming out soon Advertisement:

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)(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

6 mHz True velocity VxVx VyVy 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.