Multisource Least-squares Migration and Prism Wave

Reverse Time Migration

Wei Dai

Oct. 31, 2012

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

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

0 X (km) 60 X (km) 6

30

Z (k

m)

• Problems in conventional migration image

Introduction: Motivation for LSM

migration artifacts

imbalanced amplitude

• 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

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

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.

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

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¿

Given: Supergather modeller

Multisource Migration

shots are encoded in the supergather

Define: Supergather migration

)

𝑵 𝒊𝑻 𝑵 𝒊= 𝑰

)

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

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

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

Numerical Verification

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

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

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

SNR vs Iteration

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.

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.

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.

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

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.

Plane Wave Encoding

0 xs

Δt=pxs

θ

d(p,g,t)=

p=

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

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

• Gradient:-d)-λ

Theory: Least-squares Migration+

• Misfit:

Penalty on image difference

of nearby angles

Prestack Images𝒎=𝒎(𝒙 ,𝒑 )• Prestack image:

stack

extract

Z

p

X

Z

X

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

0 X (km) 8

0Z

(km

)3.

50

Z (k

m)

3.5

Smooth Migration Velocity

Conventional RTM Image

0 X (km) 8

0Z

(km

)3.

50

Z (k

m)

3.5

Plane-wave RTM image

Plane-wave LSRTM image (30 iterations)

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

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

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

Convergence Curves

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

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

0 X (km) 16

0Z

(km

)2.

5Conventional RTM (cost: 1)

0Z

(km

)2.

5

Plane-wave RTM (cost: 0.2)

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)

Zoom ViewsConventional RTM

Plane-wave RTM

Plane-wave LSRTM

Plane-wave LSRTM (one angle)

Zoom ViewsConventional RTM

Plane-wave RTM

Plane-wave LSRTM

Plane-wave LSRTM (one angle)

Convergence Curves

X (km) 3.750

0Ti

me

(s)

3

Observed Data

Observed Data vs Predicted Data(Plane Waves)

X (km) 3.750

Predicted Data

Time (s) 30

Am

plitu

de

Observed Data (Red lines) vs Predicted Data (Black lines)

Plane waves are fitted perfectly

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.

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).

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

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.

Conventional Method• When the known boundaries are embedded in

the velocity model, conventional RTM can migrate prism waves correctly.

Recorded Trace

Time (s) 20

Horizontal Reflector Embedded in the Velocity0

Z (k

m)

3

0 X (km) 6

0Z

(km

)3

Conventional RTM Image

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 )

+ 𝑮𝒐∗ (𝒙|𝒔 )𝑮𝒐

∗ (𝒙|𝒈 ) 𝒅𝟐 (𝒈|𝒔 )

+ +

+ + + Other terms.]

0 X (km) 6

0Z

(km

)3

Ellipses

Rabbit Ears

Prism Wave Kernels

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

0 X (km) 6

0Z

(km

)3

0Z

(km

)3

Migration Image of Prism Waves

RTM Image /w Smooth Velocity

The Salt Model

• Model size: 601 x 601

• Source freq: 20 hz

• shots: 601

• geophones: 601

0 X (km) 6

0Z

(km

)6

0 X (km) 6

0Z

(km

)6

0 X (km) 6

Migration Velocity

RTM with Smooth VelocityRTM Image

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

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.

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

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

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

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

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.