Multisource Least-squares Migration and Prism Wave Reverse Time Migration
description
Transcript of Multisource Least-squares Migration and Prism Wave Reverse Time Migration
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Multisource Least-squares Migration and Prism Wave
Reverse Time Migration
Wei Dai
Oct. 31, 2012
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Outline• Introduction and Overview
• Chapter 2: Multisource least-squares migration
• Chapter 3: Plane-wave least-squares reverse time
migration
• Chapter 4: Prism wave reverse time migration
• Summary
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Introduction: Least-squares Migration
• Seismic migration: Given: Observed data
modelling operator
Goal: find a reflectivity model to explain by solving
the equation
Direct solution: expensive
Conventional migration:
Iterative solution:
Migration velocity
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0 X (km) 60 X (km) 6
30
Z (k
m)
• Problems in conventional migration image
Introduction: Motivation for LSM
migration artifacts
imbalanced amplitude
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• Least-squares migration has been shown to
produce high quality images, but it is considered
too expensive for practical imaging.
• Solution: combine multisource technique and
least-squares migration (MLSM).
Problem of LSM
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Motivation for Multisource
Multisource LSMTo: Increase efficiency Remove artifacts Suppress crosstalk
• Problem: LSM is too slow
• Solution: multisource phase-encoding techniqueMany (i.e. 20) times slower than standard migration
Multisource Migration Image
Multisource Crosstalk
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Overview• Chapter 2 : MLSM is implemented with Kirchhoff migration
method and the performance is analysed with signal-to-
noise ratio measurements.
• Chapter 3: MLSM is implemented with reverse time
migration and plane-wave encoding.
• Chapter 4: A new method is proposed to migrate prism
waves with reverse time migration.
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Outline• Introduction and Overview
• Chapter 2: Multisource least-squares migration
• Chapter 3: Plane-wave least-squares reverse time
migration
• Chapter 4: Prism wave reverse time migration
• Summary
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Random Time Shift𝑳𝟏𝒎=𝒅𝟏
O(1/S) cost!
Encoding Matrix
Supergather
Random source time shifts
𝑳𝟐𝒎=𝒅𝟐
𝒅=𝑵𝟏𝒅𝟏+𝑵𝟐𝒅𝟐
Encoded supergather modeler
𝑳𝒎=[𝑵 ¿¿𝟏𝑳𝟏+𝑵𝟐𝑳𝟐]𝒎¿
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Given: Supergather modeller
Multisource Migration
shots are encoded in the supergather
Define: Supergather migration
)
𝑵 𝒊𝑻 𝑵 𝒊= 𝑰
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)
1 Signal term S-1 noise terms
SNR
Repeat for all the shotsSNR
The signal-to-noise ratio of the migration image from one supergather is 1, when .
If there are more supergathersSNR is the number of stacks.
Multisource Migration
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Numerical VerificationTrue Model
0 X (km) 5
0Z
(km
)1.
5
𝑺=𝟑𝟐𝟎
0 X (km) 5
Conventional Image 𝑺=𝟏𝟔𝟎𝑺=𝟖𝟎𝑺=𝟒𝟎Image of One supergather
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Numerical VerificationTrue Model
0 X (km) 5
0Z
(km
)1.
5
𝑰=𝟏
0 X (km) 5
𝑰=𝟓𝑰=𝟏𝟎𝑰=𝟐𝟎Conventional ImageImage of I supergathers
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Numerical Verification
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Multisource LSMOne supergather, static encoding
True Model
0 X (km) 5
0Z
(km
)1.
5
Iteration: 1
0 X (km) 5
Iteration: 10Iteration: 30Iteration: 60
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Multisource LSMOne supergather, dynamic encoding
True Model
0 X (km) 5
0Z
(km
)1.
5
Iteration: 1
0 X (km) 5
Iteration: 10Iteration: 30Iteration: 60
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Static vs Dynamic
0 X (km) 5
0Z
(km
)1.
5
Iteration: 1
0 X (km) 5
Iteration: 1Static dynamic
Iteration: 10Iteration: 10 Iteration: 30Iteration: 30 Iteration: 60Iteration: 60
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SNR vs Iteration
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Chapter 2: Conclusions• MLSM can produce high quality images efficiently.
LSM produces high quality image.
Multisource technique increases computational
efficiency.
SNR analysis suggests that not too many iterations
are needed.
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Chapter 2: Limitations• MLSM implemented with Kirchhoff migration can
only reduce I/O cost.
• Random encoding method requires fixed spread
acquisition geometry.
need to be implemented with reverse time
migration.
Plane-wave encoding.
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Limitation of Random Encoding
• It is not applicable to marine streamer data.Fixed spread geometry (synthetic) Marine streamer geometry (observed)
6 traces 4 traces
Mismatch between acquisition geometries will dominate the misfit.
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Outline• Introduction and Overview
• Chapter 2: Multisource least-squares migration
• Chapter 3: Plane-wave least-squares reverse time
migration
• Chapter 4: Prism wave reverse time migration
• Summary
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Chapter 3: Plane-wave LSRTM
• Implemented with wave-equation based method
Significant computation saving.
• Instead of inverting for one stacked image, image from
each shot is separated.
Common image gathers are available.
Good convergence even with bulk velocity error.
• Plane-wave encoding
Applicable to marine-streamer data.
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Plane Wave Encoding
0 xs
Δt=pxs
θ
d(p,g,t)=
p=
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0 X (km) 12
0Ti
me
(s)
12
A common shot gather
0 X (km) 12
A supergather (p=0 μs/m)
Plane Wave Encoding
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Least-squares Migration with Prestack Image
𝒅=𝑳𝒎• Equation:
• Equations with
stacked image:
= m
• Equations with
prestatck image:
=
• Misfit:𝒇 (𝒎 )=𝟏
𝟐‖𝑳𝒎−𝒅‖𝟐Solution:
𝒎𝟏=(𝑳𝟏𝑻 𝑳𝟏 )−𝟏𝑳𝟏𝒅𝟏
𝒎𝟐=(𝑳𝟐𝑻 𝑳𝟐 )−𝟏𝑳𝟐𝒅𝟐
𝒎𝟑=(𝑳𝟑𝑻 𝑳𝟑 )−𝟏𝑳𝟑𝒅𝟑
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• Gradient:-d)-λ
Theory: Least-squares Migration+
• Misfit:
Penalty on image difference
of nearby angles
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Prestack Images𝒎=𝒎(𝒙 ,𝒑 )• Prestack image:
stack
extract
Z
p
X
Z
X
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The Marmousi2 Model
0 X (km) 8
0Z
(km
)3.
5
4.5
1.5
km/s
• Model size: 801 x 351 • Source freq: 20 hz• shots: 801 • geophones: 801• Plane-wave gathers: 31
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0 X (km) 8
0Z
(km
)3.
50
Z (k
m)
3.5
Smooth Migration Velocity
Conventional RTM Image
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0 X (km) 8
0Z
(km
)3.
50
Z (k
m)
3.5
Plane-wave RTM image
Plane-wave LSRTM image (30 iterations)
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0 X (km) 8
0Z
(km
)3.
50
Z (k
m)
3.5
Common Image Gathers from RTM Image
Common Image Gathers from LSRTM Image
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0 X (km) 8
0Z
(km
)3.
50
Z (k
m)
3.5
RTM Image /w 5% Velocity Error
LSRTM Image /w 5% Velocity Error
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0 X (km) 8
0Z
(km
)3.
50
Z (k
m)
3.5
CIGs from RTM Image /w 5% Velocity Error
CIGs from LSRTM Image /w 5% Velocity Error
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Convergence Curves
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Plane-wave LSRTM of 2D Marine Data
0 X (km) 16
0Z
(km
)2.
5
2.1
1.5
km/s
• Model size: 16 x 2.5 km • Source freq: 25 hz• Shots: 515 • Cable: 6km• Receivers: 480
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WorkflowRaw data
Transform into CDP domain
Apply Normal Moveout to flat reflections
2D spline interpolation
Shift all the events back
Tau-p transform in CRG domain to generate
plane waves
Transform into CRG domain
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0 X (km) 16
0Z
(km
)2.
5Conventional RTM (cost: 1)
0Z
(km
)2.
5
Plane-wave RTM (cost: 0.2)
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X (km) 16
Plane-wave LSRTM (cost: 12)
0
0Z
(km
)2.
50
Z (k
m)
2.5
Plane-wave LSRTM /w One Angle per Iteration (cost: 0.4)
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Zoom ViewsConventional RTM
Plane-wave RTM
Plane-wave LSRTM
Plane-wave LSRTM (one angle)
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Zoom ViewsConventional RTM
Plane-wave RTM
Plane-wave LSRTM
Plane-wave LSRTM (one angle)
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Convergence Curves
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X (km) 3.750
0Ti
me
(s)
3
Observed Data
Observed Data vs Predicted Data(Plane Waves)
X (km) 3.750
Predicted Data
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Time (s) 30
Am
plitu
de
Observed Data (Red lines) vs Predicted Data (Black lines)
Plane waves are fitted perfectly
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Chapter 3: Conclusions• Plane-wave LSRTM can efficiently produce high quality
images.
LSM produces high quality image.
Plane-wave encoding applicable to marine data.
Prestack image incorporated to produce common
image gathers and enhance robustness.
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Limitations• Prestack images need to be stored during iterations.
Large memory cost..
• Plane wave encoding.
Regular sampling in shot axis is required (interpolation).
Sufficient amount of angles to reduce aliasing artifacts
(i.e. 31).
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Outline• Introduction and Overview
• Chapter 2: Multisource least-squares migration
• Chapter 3: Plane-wave least-squares reverse time
migration
• Chapter 4: Prism wave reverse time migration
• Summary
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Chapter 4: Introduction• Problem: Vertical boundaries (salt flanks) are
difficult to image because they are usually not illuminated by primary reflections.
• Solution: Prism waves contain valuable information.
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Conventional Method• When the known boundaries are embedded in
the velocity model, conventional RTM can migrate prism waves correctly.
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Recorded Trace
Time (s) 20
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Horizontal Reflector Embedded in the Velocity0
Z (k
m)
3
0 X (km) 6
0Z
(km
)3
Conventional RTM Image
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Reverse Time Migration Formula
𝒎𝒎𝒊𝒈(𝒙)=∑𝝎𝝎𝟐𝑾 ∗(𝝎 )𝑮∗ (𝒙|𝒔 )𝑮∗ (𝒙|𝒈 )𝒅 (𝒈|𝒔 )
Angular Freq. Source SpectrumGreen’s functions
Input Data
0Z
(km
)3
𝒙
𝑮 (𝒙|𝒔 )=𝑮𝒐 (𝒙|𝒔 )+𝑮𝟏(𝒙∨𝒔)𝑮 (𝒙|𝒈 )=𝑮𝒐 (𝒙|𝒈 )+𝑮𝟏(𝒙∨𝒈 )
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+ 𝑮𝒐∗ (𝒙|𝒔 )𝑮𝒐
∗ (𝒙|𝒈 ) 𝒅𝟐 (𝒈|𝒔 )
+ +
+ + + Other terms.]
0 X (km) 6
0Z
(km
)3
Ellipses
Rabbit Ears
Prism Wave Kernels
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𝒎𝒎𝒊𝒈=∑𝝎𝝎𝟐𝑾 ∗ (𝝎 )𝑮𝟏
∗ (𝒙|𝒔 )𝑮𝒐∗ (𝒙|𝒈 )𝒅𝟐 (𝒈|𝒔 )
𝑮𝟏❑ (𝒙|𝒔 )=∫𝝎𝟐𝒎 (𝒙 ′)𝑮𝒐 (𝒙 ′|𝒔 )𝑮𝒐 ( 𝒙′|𝒙 )𝒅𝒙 ′
Born Modeling
0Z
(km
)3
0 X (km) 6
Migration of Prism Waves
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0 X (km) 6
0Z
(km
)3
0Z
(km
)3
Migration Image of Prism Waves
RTM Image /w Smooth Velocity
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The Salt Model
• Model size: 601 x 601
• Source freq: 20 hz
• shots: 601
• geophones: 601
0 X (km) 6
0Z
(km
)6
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0 X (km) 6
0Z
(km
)6
0 X (km) 6
Migration Velocity
RTM with Smooth VelocityRTM Image
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RTM ImageFinal Image
0 X (km) 6
0Z
(km
)6
0 X (km) 6
Migration Velocity
If the Horizontal Reflectors are embedded in the velocity
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Chapter 4: Conclusions• I propose a new method to migrate prism waves
separately.
Limitations• Computational cost is doubled.
Avoid the modification of migration velocity.
Reduce cross interference between different waves.
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Outline• Introduction and Overview
• Chapter 2: Multisource least-squares migration
• Chapter 3: Plane-wave least-squares reverse time
migration
• Chapter 4: Prism wave reverse time migration
• Summary
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Summary• Chapter 2 : MLSM is proposed and tested with Kirchhoff
migration.
True Model
0 X (km)
5
0Z
(km
)1.
5
0 X (km)
5
Iteration: 60
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Summary• Chapter 3: MLSM is implemented with reverse time
migration and plane-wave encoding and tested with field
data example.
Conventional RTM Plane-wave LSRTM
6 X (km)
8
1Z
(km
)1.
5
6 X (km)
8
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Summary• Chapter 4: A new method is proposed to migrate prism
waves with reverse time migration for salt flank
delineation.Old Method
0 X (km) 6
0Z
(km
)6
New Method
0 X (km) 6
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Acknowledgements
I thank the sponsors of UTAH consortium for their financial support.
I thank my committee members for the supervision over my program of study.
I thank my fellow graduate students for the collaborations and help over last 4 years.