1. Forward Modeling for Time-Distance Helioseismology
A C Birch
CORA, NWRA

2. Outline Introduction to time-distance
Introduction to the forward problem
Example calculations
Open questions and future work

3. SOHO/MDI dopplergram, Ni I, colors range from +/- 2 km/s

5. The basic idea of t.d. helioseismology (Duvall et al., 93)
Measure the time for a wave packet to move from x1 to x2 and also the time from x2 to x1
From the times infer subsurface conditions, e.g. sound speed, flows …

6. Time-distance helioseismology

7. Gizon (2003), f modes, hi-res SOHO/MDI

8. Zhao, Kosovichev, & Duvall (2002)
13 hours hi-res SOHO/MDI
Depth 0-3 Mm, red is down flow, longest arrow is 1 km/s

9. Linear forward and inverse problems
Linear sensitivity to small changes in the solar model
In principle have to consider all possible types of perturbations: e.g. flows, sound speed, density, source properties, magnetic field, damping properties …
Could use other quantities in addition to the travel time, for example the amplitude or central frequency of the cross-correlation

10. A recipe for computing kernels (Gizon & Birch, 2002 ApJ)
Previous work was based on ray approximation or single-source approximation, need something more general:
+ Motivated by Woodard (1997, ApJ): wavefield, excited by stochastic sources (convection)
+ Compute the expectation value of the cross-covariance function
+ Linearize (Born Approximation) the dependence
of the cross-covariance to changes in the solar interior (e.g. sound speed)

11. Damping, source strength, f-mode kernels
Gizon, Birch (Apj,2002)
Damping, source strength, f-mode kernels
How big are these effects for reasonable sunspot models ?

12. Example: mtf + phase speed (Birch, Kosovichev, & Duvall, ApJ 2004)

13. Future work and open questions
Quiet Sun: have many of the pieces, need to put them together
Sunspots: need realistic forward modeling
Radiative transfer ?
Still some very hard questions: everything is correlated … wave excitation, damping, flows, sound speed etc.