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

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


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


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


Kernel Interpolation

We use:bc

),(

),(

C, b

)(,2

KKKKc

b


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


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


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








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











Correlation Matrix

21v

21v

BBBC

iiki

iiji

iikiji

kj

AA

AACor

22

2

)v,v(

Documents

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