Triple correlation Helioseismology Frank P. Pijpers Imperial College London with thanks to HELAS for financial support.

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Presentation transcript:

Triple correlation Helioseismology Frank P. Pijpers Imperial College London with thanks to HELAS for financial support

multiple correlations definition : c(τ 1,τ 2, …,τ n-1 ) = ∫ f 1 (t) f 2 (t+τ 1 )…f n (t+τ n-1 ) dt Triple correlation in the Fourier domain : C(ω 1,ω 2 ) = F 1 (ω 1 ) F 2 (ω 2 ) F 3 * (ω 1 +ω 2 )

The three arc-averaging masks used Use the standard phase-speed filter for this separation

What to expect ? The travel time over each of the three sides should be (almost) equal so that : τ 1 =τ 2 =τ grp Power in Fourier domain concentrated around ridge(s) with ω 1 +ω 2 = cst. but with a lot of structure in each ridge. Eliminate structure by dividing by mean triple correlation. If the wavelet is merely displaced one should find a clear signature in the Fourier phase

Analogous to what is done in speckle masking. Figure from Lohmann, Weigelt, Wirnitzer, (1983) App. Opt. 22,4028 triple correlation average ratio

The average triple correlation for a cube of x 128 images with an averaging mask of arcs on an equilateral triangle (8 by 8 sets) Fourier modulus (logarithmic)Fourier phase

Ratio of triple correlation of arc-set (4,4) and the average triple correlation Fourier modulusFourier phase

What is gained ? the ridges are visible in the Fourier modulus : wavelet changes over field. This is indicative of dispersion changes Differential travel times are directly determined from the triple correlation ratios. Fast and robust extraction of the quantity of interest

The cost ? Memory : for long time series the storage requirement goes up as N 2. Working with large fields and long time series may require large cache/swap space Time : on 8 cpu sparc machine the examples shown here took 11 minutes