Triple correlation Triple correlation HelioseismologyHelioseismology

Frank P. PijpersFrank P. Pijpers

Imperial College LondonImperial College London

with thanks to HELAS for financial support

multiple correlationsmultiple correlations

definition :

c(τ1 ,τ2 , …,τn-1) = ∫ f1(t) f2(t+τ1)…fn(t+τn-1) dt

Triple correlation in the Fourier domain :

C(ω1,ω2) = F1(ω1 ) F2(ω2) F3*(ω1+ω2)

The three arc-averaging masks used

1 2

3

Use the standard phase-speed filter for this separation

What to expect ?What to expect ?The travel time over each of the three sides The travel time over each of the three sides should be (almost) equal so that : should be (almost) equal so that : ττ11==ττ22==ττgrpgrp

Power in Fourier domain concentrated around Power in Fourier domain concentrated around ridge(s) with ridge(s) with ωω11++ωω22= = cst.cst. but with a lot of but with a lot of

structure in each ridge.structure in each ridge.

Eliminate structure by dividing by mean triple Eliminate structure by dividing by mean triple correlation.correlation. If the wavelet is merely displaced If the wavelet is merely displaced one should find a clear signature in the one should find a clear signature in the Fourier phaseFourier phase

Analogous to whatis done in specklemasking.Figure fromLohmann, Weigelt,Wirnitzer, (1983)App. Opt. 22,4028

triple correlationaverage

ratio

The average triple correlation for a cube of 512 128 x 128 imageswith 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 modulus Fourier phase

What is gained ?What is gained ?

the ridges are visible in the Fourier the ridges are visible in the Fourier modulus : wavelet changes over field. modulus : wavelet changes over field. This is indicative of dispersion changesThis is indicative of dispersion changes

Differential travel times are directly Differential travel times are directly determined from the triple correlation determined from the triple correlation ratios. ratios. Fast and robust extraction of the Fast and robust extraction of the quantity of interest quantity of interest

The cost ?The cost ?

Memory :Memory : for long time series the storage for long time series the storage requirement goes up as Nrequirement goes up as N22. Working with . Working with large fields and long time series may large fields and long time series may require large cache/swap spacerequire large cache/swap space

Time :Time : on 8 cpu sparc machine the on 8 cpu sparc machine the examples shown here took 11 minutesexamples shown here took 11 minutes