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Acoustic Waveform Inversion of 2D Gulf of Mexico Data. Chaiwoot Boonyasiriwat April 13, 2009. Outline. Part I: Application of waveform inversion to marine data Part II: Resolution analysis using Beylkin’s formula. 1. Part I: Outline. Application of waveform inversion to marine data - PowerPoint PPT Presentation
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Acoustic Waveform Inversion ofAcoustic Waveform Inversion of2D Gulf of Mexico Data2D Gulf of Mexico Data
Chaiwoot BoonyasiriwatChaiwoot Boonyasiriwat
April 13, 2009April 13, 2009
OutlineOutline
1
Part I: Application of waveform inversion to Part I: Application of waveform inversion to marine datamarine data
Part II: Resolution analysis using Beylkin’s Part II: Resolution analysis using Beylkin’s formulaformula
Part I: OutlinePart I: Outline
2
Application of waveform inversion to marine Application of waveform inversion to marine datadata
• MotivationMotivation
• TheoryTheory
• Numerical ResultsNumerical Results
• 2D SEG/EAGE Salt Model2D SEG/EAGE Salt Model
• Gulf of Mexico DataGulf of Mexico Data
• ConclusionsConclusions
MotivationMotivation
300 161644
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Dep
th (
km)
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th (
km)
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th (
km)
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th (
km)
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X (km)X (km)
Waveform TomogramWaveform Tomogram
15001500
45004500
Vel
ocit
y (m
/s)
Vel
ocit
y (m
/s)
True ModelTrue Model
An acoustic wave equation:An acoustic wave equation:
)|,()|,()(
1)|,(
)(
12
2
sss rtstP
t
tPrrr
r
rr
r
The waveform misfit function isThe waveform misfit function is
dttPfs g
sg )|,(2
1 2 rr
4
Theory of Waveform InversionTheory of Waveform Inversion
wherewhere)(
)()(
r
rr
c
Theory of Waveform InversionTheory of Waveform Inversion
The waveform residual is defined byThe waveform residual is defined by
calcsgobssgsg tPtPtP )|,()|,()|,( rrrrrr
The steepest descent method can be used to The steepest descent method can be used to minimize the misfit function:minimize the misfit function:
)()()(1 rrr kkkk gcc
5
Theory of Waveform InversionTheory of Waveform Inversion
The gradient is calculated byThe gradient is calculated by
s
ss tPtPdtc
g )|,(')|,( )(
2)( rrrr
rr
wherewhere
)|,'()0,'|,(')|,(' ss tstGdtP rrrrrrr
)|,()()|,( sggg
s tPts rrrrrr
6
Part I: OutlinePart I: Outline
7
Application of waveform inversion to marine Application of waveform inversion to marine datadata
• MotivationMotivation
• TheoryTheory
• Numerical ResultsNumerical Results
• 2D SEG/EAGE Salt Model2D SEG/EAGE Salt Model
• Gulf of Mexico DataGulf of Mexico Data
• ConclusionsConclusions
8
2D SEG/EAGE Salt Model2D SEG/EAGE Salt Model
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/s)
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Initial Velocity ModelsInitial Velocity Models
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v(z) Modelv(z) Model
Traveltime TomogramTraveltime Tomogram
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45004500
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y (m
/s)
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ocit
y (m
/s)
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Waveform Inversion ResultsWaveform Inversion Results
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Using v(z) Model + FloodingUsing v(z) Model + Flooding
Using Traveltime TomogramUsing Traveltime Tomogram
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45004500
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/s)
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/s)
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Waveform Inversion ResultsWaveform Inversion Results
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km)
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X (km)X (km)
True ModelTrue Model
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45004500
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/s)
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ocit
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/s)
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Flooding TechniqueFlooding Technique
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Waveform Tomogram after Salt FloodWaveform Tomogram after Salt Flood
Using v(z) Model w/o FloodingUsing v(z) Model w/o Flooding
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ocit
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/s)
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ocit
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Flooding TechniqueFlooding Technique
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km)
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km)
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Waveform Tomogram using v(z) and Flooding TechniqueWaveform Tomogram using v(z) and Flooding Technique
Waveform Tomogram after Sediment FloodWaveform Tomogram after Sediment Flood
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45004500
Vel
ocit
y (m
/s)
Vel
ocit
y (m
/s)
Part I: OutlinePart I: Outline
13
Application of waveform inversion to marine Application of waveform inversion to marine datadata
• MotivationMotivation
• TheoryTheory
• Numerical ResultsNumerical Results
• 2D SEG/EAGE Salt Model2D SEG/EAGE Salt Model
• Gulf of Mexico DataGulf of Mexico Data
• ConclusionsConclusions
515 Shots480 Hydrophones
12.5 mdt = 2 msTmax = 10 s
1 1.5 2 2.5
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Offset (km)
Tim
e (s)
b) Original CSG 1
1 1.5 2 2.5
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Offset (km)Tim
e (s)
a) Virtual CSG 1
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Gulf of Mexico DataGulf of Mexico Data
Data PreprocessingData Preprocessing
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Offset (km)
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e (s)
(a) Original CSG
0 2 4
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Adaptive Early-Arrival Muting WindowAdaptive Early-Arrival Muting Window
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Window = 1 s Window = 1 s
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(c) 10-Hz CSG
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Window = 2 s Window = 2 s
Adaptive Early-Arrival Muting WindowAdaptive Early-Arrival Muting Window
1800 20202.52.5
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Traveltime TomogramTraveltime Tomogram
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ocit
y (m
/s)
Vel
ocit
y (m
/s)
Waveform TomogramWaveform Tomogram2.52.5
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ResultsResults
1900 2020
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th (
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Waveform TomogramWaveform Tomogram
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30003000
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ocit
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/s)
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ocit
y (m
/s)
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th (
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th (
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Vertical Derivative of Waveform TomogramVertical Derivative of Waveform Tomogram
Kirchhoff Migration ImagesKirchhoff Migration Images
20
Kirchhoff Migration ImagesKirchhoff Migration Images
20
Comparing CIGsComparing CIGs
21
Comparing CIGsComparing CIGs
22
CIG from Traveltime Tomogram CIG from Waveform Tomogram
Comparing CIGsComparing CIGs
23
Comparing CIGsComparing CIGs
24
CIG from Traveltime Tomogram CIG from Waveform Tomogram
Comparing CIGsComparing CIGs
25
Comparing CIGsComparing CIGs
26
CIG from Traveltime Tomogram CIG from Waveform Tomogram
Part I: OutlinePart I: Outline
27
Application of waveform inversion to marine Application of waveform inversion to marine datadata
• MotivationMotivation
• TheoryTheory
• Numerical ResultsNumerical Results
• 2D SEG/EAGE Salt Model2D SEG/EAGE Salt Model
• Gulf of Mexico DataGulf of Mexico Data
• ConclusionsConclusions
ConclusionsConclusions
28
• Acoustic waveform inversion was applied to Acoustic waveform inversion was applied to both 2D synthetic and field databoth 2D synthetic and field data
• Using the traveltime tomogram, waveform Using the traveltime tomogram, waveform inversion failed to converge to an accurate inversion failed to converge to an accurate solution due to high velocity contrastsolution due to high velocity contrast
• Using v(z) model with the flooding technique, Using v(z) model with the flooding technique, an accurate result was obtainedan accurate result was obtained
ConclusionsConclusions
29
• Acoustic waveform inversion with a dynamic Acoustic waveform inversion with a dynamic early-arrival muting window can invert the early-arrival muting window can invert the Gulf of Mexico data to obtain a velocity model Gulf of Mexico data to obtain a velocity model that is more accurate than the traveltime that is more accurate than the traveltime tomogram.tomogram.
• The accuracy of waveform tomogram was The accuracy of waveform tomogram was assessed by comparing the migration images assessed by comparing the migration images and common image gathers.and common image gathers.
Part II: OutlinePart II: Outline
1
Spatial Resolution Analysis using Beylkin’s Spatial Resolution Analysis using Beylkin’s FormulaFormula
• MotivationMotivation
• TheoryTheory
• Numerical ResultsNumerical Results
• Homogeneous ModelHomogeneous Model
• Smoothed 2D SEG/EAGE Salt ModelSmoothed 2D SEG/EAGE Salt Model
• ConclusionsConclusions
MotivationMotivation
2
Model with Resolution LimitsModel with Resolution Limits RTM ImageRTM Image
Part II: OutlinePart II: Outline
3
Spatial Resolution Analysis using Beylkin’s Spatial Resolution Analysis using Beylkin’s FormulaFormula
• MotivationMotivation
• TheoryTheory
• Numerical ResultsNumerical Results
• Homogeneous ModelHomogeneous Model
• Smoothed 2D SEG/EAGE Salt ModelSmoothed 2D SEG/EAGE Salt Model
• ConclusionsConclusions
Given:
source/receiver configuration and source frequency
Find:
spatial resolution limits
Method:
use a mapping function that maps the given information to resolution limits
4
2D Spatial Resolution Formulas2D Spatial Resolution Formulas
Why Beylkin’s resolution analysis?
• It is simple
• It can be used for heterogeneous media
• It is fast -- ray-based method
5
2D Spatial Resolution Formulas2D Spatial Resolution Formulas
Source/receiver configuration can be described by coordinate
6
2D Spatial Resolution Formulas2D Spatial Resolution Formulas
XX gs fixedfor )0,( and )0,( rr
2D common-shot gather can be described by
),(),(: zx kkf
Beylkin’s resolution mapping:
data image
Beylkin et al. (1985) derived the following resolution formulas:
7
2D Spatial Resolution Formulas2D Spatial Resolution Formulas
),( rk
),( zx kkk
whereis the wavenumber vector in the
image domain
),( r is the traveltime surface of a diffractor at r for shot/receiver pairs described by
A diffractor traveltime can be described as
8
2D Spatial Resolution Formulas2D Spatial Resolution Formulas
where
is the traveltime from surface position y to subsurface position x.
),( r
gsgs ),(),(),( rrrrr
),( yx
Similarly, k can be written as vectorial sum
gs kkk
)( gs
9
Wavenumber IlluminationWavenumber Illumination
gs kkk
rr
Horizontal and vertical resolution limits:Horizontal and vertical resolution limits:
|),,(|max
2
rxk
x
10
2D Spatial Resolution Formulas2D Spatial Resolution Formulas
|),,(|max
2
rzk
z
Part II: OutlinePart II: Outline
11
Spatial Resolution Analysis using Beylkin’s Spatial Resolution Analysis using Beylkin’s FormulaFormula
• MotivationMotivation
• TheoryTheory
• Numerical ResultsNumerical Results
• Homogeneous ModelHomogeneous Model
• Smoothed 2D SEG/EAGE Salt ModelSmoothed 2D SEG/EAGE Salt Model
• ConclusionsConclusions
Homogeneous ModelHomogeneous Model
1200 11X (km)X (km)
11
00D
epth
(km
)D
epth
(km
)
Homogeneous Model with Resolution LimitsHomogeneous Model with Resolution Limits
Wavenumber Wavenumber IlluminationIllumination
13
Homogeneous ModelHomogeneous Model
1400 11X (km)X (km)
11
00D
epth
(km
)D
epth
(km
)
Reverse-Time Migration ImageReverse-Time Migration Image
Part II: OutlinePart II: Outline
15
Spatial Resolution Analysis using Beylkin’s Spatial Resolution Analysis using Beylkin’s FormulaFormula
• MotivationMotivation
• TheoryTheory
• Numerical ResultsNumerical Results
• Homogeneous ModelHomogeneous Model
• Smoothed 2D SEG/EAGE Salt ModelSmoothed 2D SEG/EAGE Salt Model
• ConclusionsConclusions
Smoothed SEG/EAGE Salt ModelSmoothed SEG/EAGE Salt Model
16
00 1616X (km)X (km)
3.53.5
00
Dep
th (
km)
Dep
th (
km)
Smoothed Salt Model with Resolution LimitsSmoothed Salt Model with Resolution Limits
15001500
60006000
Vel
ocit
y (m
/s)
Vel
ocit
y (m
/s)
Wavenumber Wavenumber IlluminationIllumination
17
Smoothed SEG/EAGE Salt ModelSmoothed SEG/EAGE Salt Model
18
00 1616X (km)X (km)
3.53.5
00
Dep
th (
km)
Dep
th (
km)
Reverse-Time Migration ImageReverse-Time Migration Image
RTM Image Line at x = 8 kmRTM Image Line at x = 8 km
19
0.90.9 1.11.1Depth (km)Depth (km)00
11
Am
plit
ude
Am
plit
ude
z = 1 kmz = 1 km
1.91.9 2.12.1Depth (km)Depth (km)00
0.20.2
Am
plit
ude
Am
plit
ude
z = 2 kmz = 2 km
00
0.150.15
Am
plit
ude
Am
plit
ude
2.92.9 3.13.1Depth (km)Depth (km)
z = 3 kmz = 3 km
Smoothed SEG/EAGE Salt ModelSmoothed SEG/EAGE Salt Model
20
00 1616X (km)X (km)
3.53.5
00
Dep
th (
km)
Dep
th (
km)
Reverse-Time Migration ImageReverse-Time Migration Image
RTM Image Line at z = 2 kmRTM Image Line at z = 2 km
21
1.81.8 2.22.2X (km)X (km)00
0.50.5
Am
plit
ude
Am
plit
ude
x = 2 kmx = 2 km
7.87.8 8.28.2X (km)X (km)00
0.20.2
Am
plit
ude
Am
plit
ude
x = 8 kmx = 8 km
00
0.30.3
Am
plit
ude
Am
plit
ude
13.813.8 14.214.2X (km)X (km)
x = 14 kmx = 14 km
Part II: OutlinePart II: Outline
22
Spatial Resolution Analysis using Beylkin’s Spatial Resolution Analysis using Beylkin’s FormulaFormula
• MotivationMotivation
• TheoryTheory
• Numerical ResultsNumerical Results
• Homogeneous ModelHomogeneous Model
• Smoothed 2D SEG/EAGE Salt ModelSmoothed 2D SEG/EAGE Salt Model
• ConclusionsConclusions
ConclusionsConclusions
23
• Beylkin’s ray-based spatial resolution analysis can provide accurate resolution limits for both homogeneous and smooth heterogeneous models in this study.
• For a homogeneous model, vertical resolution is constant while horizontal resolution degrades with depth.
• For a heterogeneous model, both vertical and horizontal resolutions generally degrade with depth since the velocity value is getting higher.
• My advisor: Dr. Gerard Schuster• Mentors: Paul Valasek, Partha Routh, and Peng Chen• Committee members: Dr. Ron Brunh and
Dr. Richard Jarrard
AcknowledgmentsAcknowledgments
24
• UTAM alumni: Jianming Sheng, Min Zhou,
Ruiqing He, Xiang Xiao, George Jiang
• UTAM colleagues: Weiping Cao, Ge Zhan, Sherif Hanafy, Sam Brown, etc.
• UTAM sponsors• Development and Promotion of Science and
Technology (DPST) Project of Thailand
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