1. Travel Time Measurements Markus Roth 1, Laurent Gizon 1, John G. Beck 2 1 Max-Planck-Institut für Sonnensystemforschung 2 Stanford University HELAS Workshop „Roadmap for Local Helioseismology“, September 25, 2006 Markus Roth 1, Laurent Gizon 1, John G. Beck 2 1 Max-Planck-Institut für Sonnensystemforschung 2 Stanford University HELAS Workshop „Roadmap for Local Helioseismology“, September 25, 2006

2. Travel Time Measurements Time-Distance Helioseismology IntroductionMethodsNoiseResultsConclusions Δ Source Observer Calculation of the Cross Correlation of the observed signals as function of travel-distance and time lag: Observation of oscillation signal at two locations on the Sun  16.5° Duvall et al (1993)

3. Travel Time Measurements Cross-Correlation as function of distance  and time lag  Positive time-lag: outgoing-wave Negative time-lag: incoming wave Difference allows to conclude on flows, mean sound speed, etc... 1st ridge: one bounce 2nd ridge: two bounces 3rd ridge: three bounces... IntroductionMethodsNoiseResultsConclusions Observation at the equator, 90 day average Time-Distance Diagram

4. Travel Time Measurements Measuring Travel Times Used Methods: 1)Fitting Garbor Wavelet (five parameters) ! phase travel time t ph 2)Extra Cross-Correlation (Zhao & Jordan 1998, Gizon & Birch 2004 for stochastic sources), minimization of “badness of fit”: yields travel-time measurement Reference cross-correlation: symmetrized long-term average Correlation coefficient: 0.76 IntroductionMethodsNoiseResultsConclusions

5. Travel Time Measurements Measured Travel Times IntroductionMethodsNoiseResultsConclusions Measuring the travel time difference between waves travelling East – West ! sensitive to differential rotation North – South ! sensitive to meridional flow

6. Travel Time Measurements Wavelet Fits Measuring travel times for 1st & 2nd bounce IntroductionMethodsNoiseResultsConclusions 1st bounce measurements are smooth 2nd bound measurements need more averaging clear due to signal/noise

7. Travel Time Measurements Minimization Method IntroductionMethodsNoiseResultsConclusions Advantages: Works on noisy data, too. Fast computation Stable results

8. Travel Time Measurements Comparison of Performance IntroductionMethodsNoiseResultsConclusions Measured signal in the order of 1 sec (peak near 0 sec) Wavelet fit: often no convergence outliers not Gaussian Extra Cross-Correlation: No outliers Gaussian distribution --- Garbor --- Extra CC

9. Travel Time Measurements Variance Garbor Wavelets IntroductionMethodsNoiseResultsConclusions Garbor Wavelets: Small errors at distances ¼ 10-20° & at latitutdes ¼ 0° Observational errors are largest for short and long distances, and for high latitudes (consistent with P. Giles 1999, 90 day average) North-South East-West

10. Travel Time Measurements Variance Extra Cross-Correl. IntroductionMethodsNoiseResultsConclusions North-South East-West Extra Cross-Correlation: As function of distance: dip at 10°, but more flat than with the Garbor wavelets As function of latitude: minimal error at the equator, appears more flat

11. Travel Time Measurements Noise Sources IntroductionMethodsNoiseResultsConclusions Main Source: stochastic nature of solar oscillations (excitation, damping) Additional sources (actually the main sources!): systematic noise sources (important for long averages, e.g. for precisions to measure meridional circulation, P-Angle, foreshortening, focus position of telescope) Understanding of noise necessary for: correct understanding of measurements solving the inverse problem, kernels (speed-up calculations)

12. Travel Time Measurements Estimating the Covariance Matrix IntroductionMethodsNoiseResultsConclusions Time-distance helioseismology: Information on noise is given by the covariance matrix of the travel time measurements x1x1 x2x2 x1´x1´x2´x2´ Estimation: directly by averaging over many samples ´´ 

13. Travel Time Measurements Covariance Matrix Wavelet Fit IntroductionMethodsNoiseResultsConclusions

14. Travel Time Measurements Covariance Matrix Extra Cross-Correlation IntroductionMethodsNoiseResultsConclusions

15. Travel Time Measurements Covariance Matrix Wavelet Fit IntroductionMethodsNoiseResultsConclusions

16. Travel Time Measurements Covariance Matrix Extra Cross-Correlation IntroductionMethodsNoiseResultsConclusions

17. Travel Time Measurements Conclusions IntroductionMethodsNoiseResultsConclusions The two methods are somehow different in their noise behaviour (1st & 2nd bounce) ! needs to be worked out in detail ! information gained about solar interior Common: Covariance matrix is confined to a few degrees around the target distance confined to a few degrees around the target latitude (+ few oscillations) Differences: between East-West and North-South measurments due to used points for averaging ! data dependence in covariance matrix