Thierry Corbard, LoHCo Tucson Feb 10 2004 Inversion for ring diagram analysis in GONG++ Pipeline...

Thierry Corbard, LoHCo Tucson Feb 10 2004

Inversion for ring diagram analysis in GONG++ Pipeline

• What are we currently doing ?RLS inversion and its details

• What could be done?Tests with OLA

Thierry Corbard, LoHCo Tucson Feb 10 2004

The Inverse problem in Ring Diagram Analysis

Ring Fitting => (n,ע) ≡ i {ux, σx, uy, σy } i=1…N

iyxyx

iiyx drrrKu ,,, )( v)(

Mode SelectionKernel Interpolation Choice of

functional form

Data Statistic

),0( ,, yxi

yx N

)( 2 diagB

Thierry Corbard, LoHCo Tucson Feb 10 2004

Mode Selection

:ifkept is ),,,,,( yxyx uun

1. n- ≤ n ≤ n+ n-=0 n+=6

2. |ux| < umax & |uy|< umax umax= 500m/s

3. There exists 2 modes in kernel file such thata)

=> we exclude the extreme frequencies for each n

b) ℓ- > ℓmin & ℓ+ > ℓmin ℓmin=175

),min(),max( maxmin =1000µHz

Thierry Corbard, LoHCo Tucson Feb 10 2004

Kernel Interpolation

We use:bc

),(

),(

C, b

)(,2

KKKKc

b

Thierry Corbard, LoHCo Tucson Feb 10 2004

RLS Inversion

dr

drrrKu

i i

ii 2

2

22

2 r

v(r))(v)(min

•Choice of regularization: second derivative

•Choice of functional form for v(r): step function

0

v)(v)(vv(r) 1

1j

jjjm

jj

rrrrr

radius accoustic in spacedequally 11 mjjr

r1=0.9

m=50

Thierry Corbard, LoHCo Tucson Feb 10 2004

j

K

r

r

ij

i

mj

Niij

j

j

drrKdrrrK

1

1

1

)(v)( v)(

1-m2j1jj1-j2

2

vv2vr

v

j

•Central differences for second derivatives

2221 ||v||||v)(||minargv LKuB

000

12100

01210

00121

L

Thierry Corbard, LoHCo Tucson Feb 10 2004

Diagnostic Tools vKu

uBKLLKBKA

uTT

uT

111v

AvvRe

matrixsolution

KA

(r) dr

r

rKA

j

i

iji v

)(

)(v j

Resolution Kernel

mN

KuB

221

2 ||)v(||

Chi-square value

TAABB v

Covariance Matrix

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

What could be done?

• Exploring the choice (automatic?) of the regularization parameter for different CMD

• Use other functional forms: spline, related to theories …

• Use GSVD for RLS – Fast and more robust than solving normal equations– Solve quasi-simultaneously for a set of regularization parameters

=> automatic choices (L-curve, GCV) easier to implement

• Use OLA type of method– Allow different regularization for different depths– Easier to interpret in terms of resolution

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Thierry Corbard, LoHCo Tucson Feb 10 2004

Correlation Matrix

21v

21v

BBBC

iiki

iiji

iikiji

kj

AA

AACor

22

2

)v,v(