Angle-domain parameters computed via weighted slant-stack

Claudio Guerra SEP-131

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

Outline

• Introduction

• Weighted OFF2ANG

• Results

• Conclusions

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||LTm– dobs||2

2S(m)/m2 = T*L*LT

H(x,; x’,’) = T*(,h) H(x,h; x’,h’) T(,h)

H(x,; x’,’) – angle-domain Hessian H(x,h; x’,h’) – offset-domain Hessian m – ADCIG mh – SODCIGT(,h) – angle-to-offset transformation T*(,h) – offset-to-angle transformationL – modeling operator L* - migration

angle-10 60

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

dept

h

offset-1200 1200

angle-10 60

dept

h

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

t

z

M(x,z)

x – *

N(*,t)

Weighted OFF2ANG – phase behavior

Slant – stack

Q – ADCIG P – SODCIG – stacking linef (z) – wavelet zr – reflector A – amplitude h – subsurface offset – reflection angle – rho filter

Weighted OFF2ANG – phase behavior

Slant – stack

Q – ADCIG – phase functionf (z) – wavelet A – amplitude h* – stationary offset – reflection angle

Weighted OFF2ANG

Weighted Slant – stack

– ADCIG – phase functionf (z) – wavelet A – amplitude h* – stationary offset – reflection angle

Results

Sigsbee2b

dep

th

cmp

Results – Input dataoffset

-1200 1200de

pth

offset-1200 1200

SODCIG Diagonal of the Hessian

Results –ADCIGsangle

-10 60angle

-10 60angle

-10 60

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th

angle-10 60

angle-10 60

angle-10 60

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th

dep

th

angle-10 60

angle-10 60

angle-10 60

Maindiagonal

Results – Angle sections15o 30o 40o

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th

cmp cmp

dep

th

cmp cmp

dep

th

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dep

th

cmp

dep

th

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Maindiagonal

Results – Amplitude correction

angle-10 60

angle-10 60

angle-10 60

dep

th

angle-10 60

angle-10 60

angle-10 60

dep

th

Maindiagonal

Results – Amplitude correction

15º angle section

dept

h

cmp cmp cmp

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h

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30º angle section

dept

h

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45º angle section

Maindiagonal

Results – Amplitude correctiond

epth

cmp cmp

Angle stack

Main diagonal 5th off-diagonal

Results – 0o Off-diagonals d

epth

cmp cmpcmp cmp

15th off-diagonal

Main diagonal 5th off-diagonal

Results – 15º Off-diagonals d

epth

cmp cmpcmp cmp

15th off-diagonal

Conclusions

• Alternative approach to transform the Hessian to the angle domain

• Well balanced ADCIGs– Better angle-stack

• Off-diagonal terms– Still no direct application