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.