Transcript Slide 1

Multisource Least-squares
Reverse Time Migration
Wei Dai
Outline
• Introduction and Overview
• Chapter 2: Multisource least-squares reverse time
migration
• Chapter 3: Frequency-selection encoding LSRTM
• Chapter 4: Super-virtual inteferometric diffractions
• Summary
Introduction: Least-squares Migration
• Seismic migration:
Given: Observed data 𝒅
Migration velocity  modelling operator 𝑳
Goal: find a reflectivity model 𝒎 to explain 𝒅 by solving
the equation 𝑳𝒎 = 𝒅
Direct solution: 𝒎𝒎𝒊𝒈 = (𝑳𝑻 𝑳)−𝟏 𝑳𝑻 𝒅  expensive
Conventional migration:
𝑳𝑻 𝑳 ≈ 𝑰
 𝒎𝒎𝒊𝒈 = 𝑳𝑻 𝒅
Iterative solution: 𝒎(𝒌+𝟏) = 𝒎(𝒌) − 𝜶𝑳𝑻 (𝑳𝒎
𝒌
− 𝒅)
Introduction: Motivation for LSM
Z (km)
0
• Problems in conventional migration image
3
migration
artifacts
0
X (km)
imbalanced amplitude
6
0
X (km)
6
Problem of LSM
• 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).
Motivation for Multisource
• Problem: LSM is too slow
Many (i.e. 20) times slower than standard migration
• Solution: multisource phase-encoding technique
Multisource  Crosstalk
Multisource Migration Image
Multisource LSM
To:
 Increase efficiency
 Remove artifacts
 Suppress crosstalk
Overview
• Chapter 2 : Multisource least-squares reverse time
migration is implemented with random time-shift and
source-polarity encoding functions.
• Chapter 3: Multisource LSRTM is implemented with
frequency-selection encoding for marine data.
• Chapter 4: An interferometric method is proposed to
extract diffractions from seismic data and enhance its
signal-to-noise ratio.
Outline
• Introduction and Overview
• Chapter 2: Multisource least-squares reverse time
migration
• Chapter 3: Frequency-selection encoding LSRTM
• Chapter 4: Super-virtual inteferometric diffractions
• Summary
Random Time Shift
𝑳𝟏 𝒎 = 𝒅𝟏
𝑳𝟐 𝒎 = 𝒅𝟐
Random source
time shifts
O(1/S) cost!
Supergather
Encoding
𝒅 = 𝑵𝟏 𝒅𝟏 + 𝑵𝟐 𝒅𝟐 Matrix
Encoded supergather modeler
𝑳𝒎 = [𝑵𝟏 𝑳𝟏 + 𝑵𝟐 𝑳𝟐 ]𝒎
Random Time Shift
𝑳𝟏 𝒎 = 𝒅𝟏
𝑳𝟐 𝒎 = 𝒅𝟐
× (-1)
× (+1)
Supergather
Encoding
𝒅 = 𝑵𝟏 𝒅𝟏 + 𝑵𝟐 𝒅𝟐 Matrix
Encoded supergather modeler
𝑳𝒎 = [𝑵𝟏 𝑳𝟏 + 𝑵𝟐 𝑳𝟐 ]𝒎
Conventional Least-squares
Given:
𝑳𝟏
𝑳=
𝑳𝟐
𝒅𝟏
& 𝒅=
𝒅𝟐
Find: an 𝒎 s.t. min 𝑳𝒎 − 𝒅
Direct solution: 𝒎 = 𝐿𝑇 𝐿
𝟐
In general, huge
dimension matrix
−1 𝐿𝑇 𝑑
If 𝑳 is too big
Iterative solution: 𝒎(𝒌+𝟏) = 𝒎(𝒌) − 𝜶𝑳𝑻 (𝑳𝒎
= 𝒎(𝒌) − 𝜶[𝑳𝑻𝟏 𝑳𝟏 𝒎
𝒌
− 𝒅𝟏 + 𝑳𝑻𝟐 𝑳𝟐 𝒎
𝒌
𝒌
− 𝒅)
− 𝒅𝟐 ]
Note: subscripts agree
Conventional Least-squares
𝒎(𝒌+𝟏) = 𝒎(𝒌)
−𝜶[𝑳𝑻𝟏 𝑳𝟏 𝒎
𝒌
− 𝒅𝟏 + 𝑳𝑻𝟐 𝑳𝟐 𝒎
𝒌
Problem: Each prediction is a FD solve
Solution: Multisource technique
− 𝒅𝟐 ]
Multisource Least-squares
Given:
𝑳 = 𝑵𝟏 𝑳𝟏 + 𝑵𝟐 𝑳𝟐 & 𝒅 = 𝑵𝟏 𝒅𝟏 + 𝑵𝟐 𝒅𝟐
Find: an 𝒎 s.t. min 𝑳𝒎 − 𝒅
Direct solution: 𝒎 =
𝟐
In general, small
dimension matrix
−𝟏 𝑻
𝑻
𝑳 𝑳 𝑳 𝒅
If 𝑳 is too big
Iterative solution: 𝒎(𝒌+𝟏) = 𝒎(𝒌) − 𝜶𝑳𝑻 (𝑳𝒎
= 𝒎(𝒌) − 𝜶[𝑳𝑻𝟏 𝑳𝟏 𝒎
+𝑳𝑻𝟐
𝒌
− 𝒅)
− 𝒅𝟏 + 𝑳𝑻𝟐 𝑳𝟐 𝒎
𝒌
− 𝒅𝟐
+ crosstalk
𝑳𝟏 𝒎 𝒌 − 𝒅𝟏 + 𝑳𝑻𝟏 𝑳𝟐 𝒎
𝒌
− 𝒅𝟐 ]
𝒌
HESS VTI Model
Size: 1800 x 750
Grid interval: 10 m
Source number: 1800
Receiver number:
1800
FD kernel: 2-4 staggered grid
Source: 15 Hz
7.5
Z (km)
0
km/s
4.5
1.5
0
X (km)
18
HESS VTI Model
Delta and Epsilon Models
1.5
7.5
Z (km)
0
Delta
0
2.5
7.5
Z (km)
0
Epsilon
0
X (km)
18
0
km/s
Migration Velocity
4.5
7.5
Z (km)
0
Migration Velocity and
Reflectivity
1.5
0.2
7.5
Z (km)
0
Reflectivity
-0.4
0
X (km)
18
Standard RTM
Z (km)
0
RTM VS Multisource LSRTM
Resolution
Enhanced
7.5
Artifacts
removed
X (km)
Multisource
LSRTM,
MultisourceLSRTM,
LSRTM, 418Supergather
Supergather
Multisource
Supergather
18
Z (km)
0
0
7.5
8 supergather
30 iterations
Speedup:
3.75
0
X (km)
18
Signal-to-noise Ratio
SNR ∞ 𝑁𝐼
3D SEG/EAGE Model
400 Shots Evenly Distributed
Size: 676 x 676 x 201
Grid interval: 20 m
Receiver: 114244
Source: 5.0 hz
13.5 km
4.0 km
13.5 km
Smooth Migration Velocity
Obtained by 3D boxcar smoothing
13.5 km
4.0 km
13.5 km
20
Conventional RTM
400 Shots, Migrated One by One
13.5 km
4.0 km
13.5 km
LSRTM
400 Shots, 25 Shots/Supergather
13.5 km
4.0 km
13.5 km
Conventional RTM
100 Shots
13.5 km
4.0 km
13.5 km
LSRTM
100 Shots, 10 Shots/Supergather
13.5 km
4.0 km
13.5 km
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
• Random encoding is not applicable to marine
streamer data.
Fixed spread geometry (synthetic)
6 traces
Marine streamer geometry (observed)
4 traces
Mismatch between acquisition geometries will dominate the
misfit.
Outline
• Introduction and Overview
• Chapter 2: Multisource least-squares reverse time
migration
• Chapter 3: Frequency-selection encoding LSRTM
• Chapter 4: Super-virtual inteferometric diffractions
• Summary
Problem with Marine Data
misfit =
observed
data
erroneous
misfit
simulated
data
28
Solution
•
Every source is encoded with a unique
signature.
• Every receiver acknowledge the contribution
from the ‘correct’ sources.
observed
simulated
29
Frequency Selection
Nw frequency bands of source spectrum:
R(w)
Accommodate up to Nw shots
w
4 shots/group
Group 1
2 km
Single Frequency Modeling
Helmholtz Equation
𝟐
𝝎
𝜵𝟐 + 𝟐 𝑷 = −𝐖 𝝎 𝛅(𝒙 − 𝒔)
𝒗
Acoustic Wave Equation
𝟐
𝟏
𝝏
𝜵𝟐 − 𝟐 𝟐 𝐏 = −𝐑𝐞{𝐖 𝝎 𝐞𝐱𝐩(−𝒊𝝎𝒕)}𝛅(𝒙 − 𝒔)
𝒗 𝝏 𝒕
Harmonic wave source
• Advantages:
 Lower complexity in 3D case.
 Applicable with multisource technique.
Single Frequency Modeling
Amplitude
𝟐
𝟏
𝝏
𝜵𝟐 − 𝟐 𝟐 𝐏 = −𝐑𝐞{𝐖 𝝎 𝐞𝐱𝐩(−𝒊𝝎𝒕)}𝛅(𝒙 − 𝒔)
𝒗 𝝏 𝒕
T
T
Amplitude
Single Frequency Modeling
Freqency (Hz)
50
Amplitude
0
20
Freqency (Hz)
30
Marmousi2
• Model size: 8 x 3.5 km • Freq.: 400 (0~50 hz)
• Shots: 301
• Receivers: 201
• Cable: 2km
3.5
Z (km)
0
km/s
4.5
1.5
0
X (km)
8
3.5
Z (km)
0
Conventional RTM
X (km)
8
LSRTMImage
Image(iteration=20)
(iteration=1)
LSRTM
(iteration=80)
Cost: 2.4
X (km)
8
3.5
Z (km)
0
0
0
Frequency-selection LSRTM of 2D
Marine Data
• Model size: 18.7 x 2.5 km • Freq: 625 (0-62.5 Hz)
• Shots: 496
• Cable: 6km
• Receivers: 480
2.5
Z (km)
0
km/s
2.1
1.5
0
X (km)
18.7
2.5
Z (km)
0
Conventional RTM
2.5
Z (km)
0
Frequency-selection LSRTM
0
X (km)
18.7
Zoom Views
Conventional RTM
Conventional RTM
Freq. Select LSRTM
Freq. Select LSRTM
Chapter 3: Conclusions
• MLSM can produce high quality images efficiently.
 LSM produces high quality image.
 Frequency-selection encoding applicable to marine
data.
• Limitation:
 High frequency noises are present.
Outline
• Introduction and Overview
• Chapter 2: Multisource least-squares reverse time
migration
• Chapter 3: Frequency-selection encoding LSRTM
• Chapter 4: Super-virtual inteferometric diffractions
• Summary
Chapter 4: Super-virtual
inteferometric diffractions
• Diffracted energy contains valuable
information about the subsurface structure.
• Goal: extract diffractions from seismic data
and enhance its SNR.
Guide Stars
Rotate
Super-virtual stacking theory
Step 1: Virtual Diffraction Moveout + Stacking

w
dt
dt

dt
=
dt
w3 w2 w1
y
z
y
y’
z
y
y’
z
Super-virtual stacking theory
Step 2: Redatum virtual refraction to known
surface position
x
y
z
y
z
x
y
z
=
*
y’
x
y
i.e.
z
x
=
y’
y
z
Super-virtual stacking theory
Step 3: Repeat Steps 1&2 for a Different Master
Trace
x
y
z
y
z
x
y
z
=
*
y’
x
y
z
i.e.
x
=
y’
y
z
Super-virtual stacking theory
Stacking Over Master Trace Location
Desired shot/
receiver combination
Common raypaths
x
z
Super-virtual Diffraction Algorithm
1. Crosscorrelate and stack to generate virtual diffractions

w
z

w
z
w
z
=
w
Virtual src
excited at -tzz’
z’
2. Convolve to generate super-virtual diffractions
w
w
z
*
z
=
3. Stack super-virtual diffractions to increase SNR
w
z
w
+
z
w
+
z
Synthetic Results: Fault Model
0
km/s
Z (km)
3.4
3
1.8
0
X (km)
6
0
Synthetic Shot Gather: Fault
Model
3
time (s)
Diffraction
0
Offset
(km)
6
1.5
time (s) 0.5 1.5
Our
Method
Median Filter
0
Offset (km) 6
1.5
time (s) 0.5
Raw Data
time (s) 0.5
Synthetic Shot Gather: Fault Model
0
Offset (km) 6
0.5
Estimation of Statics
Picked
Traveltimes
time (s)
Predicted
Traveltimes
1.0
Estimate
statics
0
Offset (km)
6
180
280
Depth (m)
Picked Moveout
0.9
1.0
time (s)
0.6
time (s)
0.9
time (s)
0.3
0.6
Experimental Cross-well Data
0
Depth (m)
300
180
Depth (m)
280
Time Windowed
0.9
time (s)
0.6
Experimental Cross-well Data
180
Depth (m)
280
180
time (s)
0.6
Super-virtual Diffractions
Depth (m)
0.9
0.9
time (s)
0.6
Median Filter
280
180
Depth (m)
280
Experimental Cross-well Data
0.9
time (s)
0.3
0.6
Median Filtered
Depth (m)
280
Super-virtual Diffraction
0.9
1.0
time (s)
0.6
time (s)
180
0
Depth (m)
300
180
Depth (m)
280
Chapter 4: Conclusions
• Super-virtual diffraction algorithm can greatly improve
the SNR of diffracted waves..
Limitation
• Dependence on median filtering when there are other coherent
events.
• Wavelet is distorted (solution: deconvolution or match filter).
Outline
• Introduction and Overview
• Chapter 2: Multisource least-squares reverse time
migration
• Chapter 3: Frequency-selection encoding LSRTM
• Chapter 4: Super-virtual inteferometric diffractions
• Summary
Chapter 2: Multisource LSRTM
• Multisource LSRTM is implemented with random encoding
functions.
 LSM produces high quality image.
 Multisource technique increases computational
efficiency.
Multisource LSRTM, 8 Supergather
Chapter 2: Frequency-selection
LSRTM
• Multisource LSRTM is implemented with frequencyselection encoding functions.
 Applicable to marine data.
Frequency-selection LSRTM
Chapter 4: Super-virtual
inteferometric diffractions
• Super-virtual diffraction algorithm can extract diffraction
waves and greatly improve its SNR.
Before
Before
After
Acknowledgements
I thank my advisor Prof. Gerard T. Schuster and
other committee members for the supervision
over my program of study.
I thank the sponsors of CSIM consortium for their
financial support.
I thank my fellow graduate students for the
collaborations and help over last 4 years.
Workflow
Raw data
Pick a master trace
Cross-correlate all the traces
with the master trace
Repeat for all the shots and stack
the result to give virtual diffractions
Convolve the virtual diffractions with the
master trace to restore original traveltime
Stack to generate Supervirtual Diffractions
Diffraction Waveform Modeling
Born
0
Modeling
4.0
1.2
Depth (km) 0
Reflectivity
time (s)
1.2
Depth (km) 0
0
Velocity
Distance (km)
3.8
0
Distance (km)
3.8
Diffraction Waveform Inversion
1.2
Distance (km)
3.8
True Velocity
1.2
1.2
Depth (km) 0
Depth (km) 0
Estimated Reflectivity
0
Inverted Velocity
1.2
Depth (km) 0
Depth (km) 0
Initial Velocity
0
Distance (km)
3.8