Forward Modeling for Time-Distance Helioseismology

A C Birch

CORA, NWRA

Outline

• Introduction to time-distance

• Introduction to the forward problem

• Example calculations

• Open questions and future work

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

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 …

Time-distance helioseismology

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

Zhao, Kosovichev, & Duvall (2002)13 hours hi-res SOHO/MDI

Depth 0-3 Mm, red is down flow, longest arrow is 1 km/s

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

A recipe for computing kernels(Gizon & Birch, 2002 ApJ)

+ Motivated by Woodard (1997, ApJ): wavefield, excited by stochastic sources (convection)+ Compute the expectation value of the cross-covariance function+ Linearize (Born Approximation) the dependenceof the cross-covariance to changes in the solar interior (e.g. sound speed)

Previous work was based on ray approximation or single-source approximation,need something more general:

Damping, source strength, f-mode kernelsGizon, Birch (Apj,2002)

How big are these effects for reasonable sunspot models ?

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

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.