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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 :

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triple correlation helioseismology

Triple correlation Helioseismology

Frank P. Pijpers

Imperial College London

with thanks to HELAS for financial support

multiple correlations
multiple 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)

slide3

The three arc-averaging masks used

Use the standard phase-speed filter for this

separation

3

2

1

what to expect
What to expect ?
  • The travel time over each of the three sides should be (almost) equal so that : τ1=τ2=τgrp
  • Power in Fourier domain concentrated around ridge(s) with ω1+ω2= 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
slide5

Analogous to what

is done in speckle

masking.

Figure from

Lohmann, Weigelt,

Wirnitzer, (1983)

App. Opt. 22,4028

average

triple correlation

ratio

slide6

The average triple correlation for a cube of 512 128 x 128 images

with an averaging mask of arcs on an equilateral triangle (8 by 8 sets)

Fourier modulus (logarithmic)

Fourier phase

slide7

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 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
The cost ?
  • Memory : for long time series the storage requirement goes up as N2. 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