Travel Time Measurements

Travel Time Measurements

Markus Roth1, Laurent Gizon1, John G. Beck2

1Max-Planck-Institut für Sonnensystemforschung2Stanford University

HELAS Workshop „Roadmap for Local Helioseismology“, September 25, 2006

Travel Time MeasurementsTime-Distance

Helioseismology

Introduction Methods Noise Results Conclusions

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)

Travel Time Measurements

Cross-Correlation as function ofdistance and time lag

Positive time-lag: outgoing-wave

Negative time-lag: incoming wave

Difference allows to conclude onflows, mean sound speed, etc...

1st ridge: one bounce2nd ridge: two bounces3rd ridge: three bounces...

Introduction Methods Noise Results Conclusions

Observation at the equator, 90 day average

Time-Distance Diagram

Travel Time Measurements

Measuring Travel Times

Used Methods:

1) Fitting Garbor Wavelet (five parameters) ! phase travel time tph

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

Introduction Methods Noise Results Conclusions

Travel Time Measurements

Measured Travel Times

Introduction Methods Noise Results Conclusions

Measuring the travel time difference between waves travelling

East – West

! sensitive to differential rotation

North – South

! sensitive to meridional flow

Travel Time Measurements

Wavelet Fits

Measuring travel times for 1st & 2nd bounce

Introduction Methods Noise Results Conclusions

1st bounce measurements are smooth

2nd bound measurements need more averagingclear due to signal/noise

Travel Time Measurements

Minimization Method

Introduction Methods Noise Results Conclusions

Advantages:

•Works on noisy data, too.

•Fast computation

•Stable results

Travel Time MeasurementsComparison of

PerformanceIntroduction Methods Noise Results Conclusions

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

Travel Time Measurements

Variance Garbor Wavelets

Introduction Methods Noise Results Conclusions

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

Travel Time Measurements

Variance Extra Cross-Correl.

Introduction Methods Noise Results Conclusions

North-South East-West

Extra Cross-Correlation:

As function of distance: dip at 10°, but more flat than with the Garbor waveletsAs function of latitude: minimal error at the equator, appears more flat

Travel Time Measurements

Noise Sources

Introduction Methods Noise Results Conclusions

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)

Travel Time MeasurementsEstimating the

Covariance MatrixIntroduction Methods Noise Results Conclusions

Time-distance helioseismology:

Information on noise is given by the covariance matrix of the travel time measurements

x1

x2

x1´ x2´

Estimation: directly by averaging over many samples

´

Travel Time MeasurementsCovariance Matrix

Wavelet FitIntroduction Methods Noise Results Conclusions

Travel Time MeasurementsCovariance Matrix

Extra Cross-CorrelationIntroduction Methods Noise Results Conclusions

Travel Time MeasurementsCovariance Matrix

Wavelet FitIntroduction Methods Noise Results Conclusions

Travel Time MeasurementsCovariance Matrix

Extra Cross-CorrelationIntroduction Methods Noise Results Conclusions

Travel Time Measurements

Conclusions

Introduction Methods Noise Results Conclusions

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