1. 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

2. Thierry Corbard, LoHCo Tucson Feb 10 2004 The Inverse problem in Ring Diagram Analysis Ring Fitting => (n,ע) ≡ i {u x, σ x, u y, σ y } i=1…N Mode Selection Kernel Interpolation Choice of functional form Data Statistic

3. Thierry Corbard, LoHCo Tucson Feb 10 2004 Mode Selection 1. n - ≤ n ≤ n + n - =0 n + =6 2.|u x | < u max & |u y |< u max u max = 500m/s 3.There exists 2 modes in kernel file such that a) => we exclude the extreme frequencies for each n b)ℓ - > ℓ min & ℓ + > ℓ min ℓ min =175 =1000µHz

4. Thierry Corbard, LoHCo Tucson Feb 10 2004 Kernel Interpolation We use: C, b

5. Thierry Corbard, LoHCo Tucson Feb 10 2004 RLS Inversion Choice of regularization: second derivative Choice of functional form for v(r): step function r 1 =0.9 m=50

6. Thierry Corbard, LoHCo Tucson Feb 10 2004 Central differences for second derivatives

7. Thierry Corbard, LoHCo Tucson Feb 10 2004 Diagnostic Tools Resolution Kernel Chi-square value Covariance Matrix

8. Thierry Corbard, LoHCo Tucson Feb 10 2004

14. 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

15. Thierry Corbard, LoHCo Tucson Feb 10 2004

24. Correlation Matrix