Angledomain parameters computed via weighted slantstack Claudio Guerra

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

Motivation • Post migration processes in the reflection-angle domain – – residual migration-velocity analysis residual multiple attenuation AVA regularization of the least-squares inverse imaging • Compensate for illumination problems in ADCIGs – kinematics is not corrected • Map properties from the subsurface-offset domain into the reflection-angle domain – diagonals of the Hessian

Outline • • Introduction Weighted OFF 2 ANG Results Conclusions

Introduction • SEP 125 - Valenciano and Biondi – “The Hessian. . . in the angle dimension lacks of resolution to be able to interpret which angles get more illumination. ” S(m) = ½||Lmh – dobs||2 = ½||LTmg – dobs||2 2 S(m)/ m 2 = 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 mg – ADCIG T(g, h) – angle-to-offset transformation L – modeling operator H(x, h; x’, h’) – offset-domain Hessian mh – SODCIG T*(g, h) – offset-to-angle transformation L* - migration

Introduction • SEP 125 - 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 -10 depth -1200 no compact support angle 60 -10 angle 60

Weighted OFF 2 ANG • Stationary-phase 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

Weighted OFF 2 ANG – phase behavior Slant – stack mg (z, g) = mh(z, h) dh z=z W z(g, h) = z 0 + h tan(g) mh(z, h) = A(h) f (z – zr(h)) mg – ADCIG f (z) – wavelet h – subsurface offset mh – SODCIG zr(h) – reflector g – reflection angle z – stacking line A(h) – amplitude

Weighted OFF 2 ANG – phase behavior Slant – stack Mg (kz, g) = F(kz) A(h) e-ikz(z(g, h) – zr(h)) dh W Mg (kz, g) ~ r(kz)-1 A(h*) F(kz) e-ikz. F (g, h*) |F’’(g, h*)|0. 5 mg (z, g) ~ A(h*) f (F (g, h*)) |F’’(g, h*)|0. 5 mg – ADCIG f (z) – wavelet h – subsurface offset Mg – FT [ADCIG] zr(h) – reflector h* – stationary offset F – phase function A(h) – amplitude g – reflection angle

Weighted OFF 2 ANG Weighted Slant – stack mg (z, g) = w(z, h) mh(z, h) dh z=z W mg (z, g) ~ wg (z, h*) mg (z, g) wg (z, h*) ~ mg (z, g) mg – ADCIG h – subsurface offset h* – stationary offset mg – weighted ADCIG g – reflection angle wg (z, h*) – angle domain weight mh – SODCIG w (z, h)– weight

Results Sigsbee 2 b depth cmp

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

Results – ADCIGs 60 -10 angle 60 -10 depth angle depth -10 angle 60 -10 angle 60

Results – Angle sections 30 o 40 o cmp depth cmp depth 15 o depth cmp

Results – Amplitude correction depth -10 angle 60

Results – Amplitude correction depth -10 angle 60

Results – Amplitude correction 15º angle section depth cmp cmp

Results – Amplitude correction 30º angle section depth cmp cmp

Results – Amplitude correction 45º angle section depth cmp cmp

Results – Amplitude correction depth cmp

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

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

Conclusions • Alternative approach to transform the Hessian to the angle domain – Need of a compact support to slant-stack – General framework to transform any • Well balanced ADCIGs – Better angle-stack • Off-diagonal terms – Still no direct application