Angle-domain parameters computed via weighted slant-stack

1. Angle-domain parameters computed via weighted slant-stack Claudio Guerra SEP-131

2. Motivation • Post migration processes in the reflection-angle domain • migration-velocity analysis • residual multiple attenuation • AVA • regularization of the least-squares inverse imaging • Compensate for illumination problems in ADCIGs

3. Outline • Introduction • Weighted OFF2ANG • Results • Conclusions

4. Introduction • SEP125 - Valenciano and Biondi • Compute the Hessian in the angle domain by chaining operators T*, H and T. S(m) = ½||Lmh – dobs||2 = ½||LTmg – dobs||2 2S(m)/m2 = T*L*LT H(x,g; x’,g’) = T*(g,h) H(x,h; x’,h’)T(g,h) H(x,g; x’,g’) – angle-domain Hessian H(x,h; x’,h’)– offset-domain Hessian mg– ADCIG mh– SODCIG T(g,h)– angle-to-offset transformation T*(g,h)– offset-to-angle transformation L – modeling operator L* - migration

5. Introduction • SEP125 - Valenciano and Biondi • “The Hessian ... in the angle dimension lacks of resolution to be able to interpret which angles get more illumination.” offset -1200 1200 offset -1200 1200 angle -10 60 angle -10 60 depth depth

6. Weighted OFF2ANG • Assymptotic approximation of Kirchhoff Migration • Main contribution comes from the vicinity of the stationary point • Bleistein(1987) and Tygel et.al(1993) • migration with two different weights • division of the migrated images x – x* t N(x*,t) M(x,z) z

7. Q– ADCIG P– SODCIG z – stacking line f (z) – wavelet zr – reflector A – amplitude h – subsurface offset g – reflection angle – rho filter Weighted OFF2ANG – phase behavior Slant – stack

8. Weighted OFF2ANG – phase behavior Slant – stack Q– ADCIG F – phase function f (z) – wavelet A – amplitude h* – stationary offsetg – reflection angle

9. Weighted OFF2ANG Weighted Slant – stack – ADCIG F – phase function f (z) – wavelet A – amplitude h* – stationary offsetg – reflection angle

10. Results Sigsbee2b cmp depth

11. offset -1200 1200 offset -1200 1200 depth SODCIG Diagonal of the Hessian Results – Input data

12. Results –ADCIGs angle -10 60 angle -10 60 angle -10 60 angle -10 60 angle -10 60 angle -10 60 depth depth angle -10 60 angle -10 60 angle -10 60 depth Main diagonal

13. Results – Angle sections 30o 40o 15o cmp cmp cmp cmp cmp cmp depth depth depth cmp cmp cmp depth depth depth Main diagonal

14. Results – Amplitude correction angle -10 60 angle -10 60 angle -10 60 angle -10 60 angle -10 60 angle -10 60 depth depth Main diagonal

15. Results – Amplitude correction cmp cmp cmp cmp cmp cmp depth depth 30º angle section 15º angle section cmp cmp cmp depth Main diagonal 45º angle section

16. Results – Amplitude correction Angle stack cmp cmp depth

17. Results – 0o Off-diagonals Main diagonal 5th off-diagonal 15th off-diagonal cmp cmp cmp cmp depth

18. Results – 15º Off-diagonals Main diagonal 5th off-diagonal 15th off-diagonal cmp cmp cmp cmp depth

19. Conclusions • Alternative approach to transform the Hessian to the angle domain • Well balanced ADCIGs • Better angle-stack • Off-diagonal terms • Still no direct application