Transcript Extended Diffraction-Slice Theorem for Wavepath Traveltime

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
• Motivation
• Theory
• Numerical Results
• 2D SEG/EAGE Salt Model
• Gulf of Mexico Data
• Conclusions
2
Motivation
True Model
Depth (km)
0
Velocity (m/s)
4500
4
Waveform Tomogram
Depth (km)
0
4
1500
0
X (km)
16
3
Theory of Waveform Inversion
An acoustic wave equation:
 1

1  2 P(r, t | rs )

P(r, t | rs )  s(r, t | rs )
2
 (r )
t
  (r )

 (r )
where c(r ) 
 (r )
The waveform misfit function is
1
f    P 2 (rg , t | rs ) dt
2 s g
4
Theory of Waveform Inversion
The waveform residual is defined by
P(rg , t | rs )  P(rg , t | rs )obs  P(rg , t | rs )calc
The steepest descent method can be used to
minimize the misfit function:
ck 1 (r)  ck (r)  k g k (r)
5
Theory of Waveform Inversion
The gradient is calculated by
2
 (r, t | r ) P ' (r, t | r )
g (r ) 
dt
P

s
s

c(r ) s
where
P' (r, t | rs )   dr ' G(r,t | r ' ,0)  s(r ' , t | rs )
s(r, t | rs )    (r  rg )P(rg , t | rs )
g
6
Part I: Outline
Application of waveform inversion to marine
data
• Motivation
• Theory
• Numerical Results
• 2D SEG/EAGE Salt Model
• Gulf of Mexico Data
• Conclusions
7
2D SEG/EAGE Salt Model
4500
Depth (km)
Velocity (m/s)
0
4
0
X (km)
16 1500
8
Initial Velocity Models
Depth (km)
0
Velocity (m/s)
4500
Traveltime Tomogram
4
Depth (km)
0
1500
v(z) Model
4
0
X (km)
16
9
Waveform Inversion Results
Using Traveltime Tomogram
Depth (km)
0
Velocity (m/s)
4500
4
Using v(z) Model + Flooding
Depth (km)
0
4
1500
0
X (km)
16
10
Waveform Inversion Results
True Model
Depth (km)
0
Velocity (m/s)
4500
4
1500
0
X (km)
16
10
Flooding Technique
Using v(z) Model w/o Flooding
Depth (km)
0
Velocity (m/s)
4500
4
Waveform Tomogram after Salt Flood
Depth (km)
0
4
1500
0
X (km)
16
11
Flooding Technique
Waveform Tomogram after Sediment Flood
Depth (km)
4500
4
Waveform Tomogram using v(z) and Flooding Technique
Depth (km)
0
4
Velocity (m/s)
0
1500
0
X (km)
16
12
Part I: Outline
Application of waveform inversion to marine
data
• Motivation
• Theory
• Numerical Results
• 2D SEG/EAGE Salt Model
• Gulf of Mexico Data
• Conclusions
13
Gulf of Mexico Data
480 Hydrophones
515 Shots
a) Virtu
b) Original CSG 1
0
0
0.5
0.5
12.5 m
dt = 2 ms
Tmax = 10 s
1
Time (s)
1
1.5
1.5
2
2
2.5
2.5
3
1
2
1.5
Offset (km)
2.5
3
1
1.5
Offs
14
Data Preprocessing
(b)0-15
5-HzHz
CSG
(b)
CSG
(c)0-25
10-Hz
(c)
HzCSG
CSG
0
0
0.5
0.5
0.5
1
1
1
1.5
1.5
1.5
2
2.5
Time (s)
0
Time (s)
Time (s)
(a) Original CSG
2
2.5
2
2.5
3
3
3
3.5
3.5
3.5
4
0
2
4
Offset (km)
4
0
2
4
Offset (km)
4
0
2
4
Offset (km)
15
Adaptive Early-Arrival Muting Window
Window = 1 s
CSG(a) Original CSG(b) 5-Hz CSG
0
0
0
5
0.5
0.5
1
1
1
1
1.5
1.5
1.5
1.5
2
2.5
2
2.5
2
2.5
Time (s)
5
2
0.5
Time (s)
1
(b) 5-Hz CSG (c) 10-Hz CSG (c) 10-Hz CSG
0
0
0-15 Hz CSG
Time (s)
0.5
Time (s)
5
Window = 1 s
2
2.5
3
3
3
3
3
5
3.5
3.5
3.5
3.5
4
40
m)
4
0
2
4
Offset (km)
4
2
40
Offset (km)
4
04
2
Offset (km)
0-25 Hz CSG
4
2
40
Offset (km)
2
4
Offset (km)
16
Adaptive Early-Arrival Muting Window
Window = 2 s
CSG(a) Original CSG(b) 5-Hz CSG
0
0
0
5
0.5
0.5
1
1
1
1
1.5
1.5
1.5
1.5
2
2.5
2
2.5
2
2.5
Time (s)
5
2
0.5
Time (s)
1
(b) 5-Hz CSG (c) 10-Hz CSG (c) 10-Hz CSG
0
0
0-15 Hz CSG
Time (s)
0.5
Time (s)
5
Window = 2 s
2
2.5
3
3
3
3
3
5
3.5
3.5
3.5
3.5
4
40
m)
4
0
2
4
Offset (km)
4
2
40
Offset (km)
4
04
2
Offset (km)
0-25 Hz CSG
4
2
40
Offset (km)
2
4
Offset (km)
17
Traveltime Tomogram
Results
Depth (km)
0
Velocity (m/s)
3000
2.5
Waveform Tomogram
Depth (km)
0
2.5
1500
0
X (km)
20
18
3000
Waveform Tomogram
Depth (km)
Velocity (m/s)
0
1500
2.5
Vertical Derivative of Waveform Tomogram
Depth (km)
0
2.5
0
X (km)
20
19
Kirchhoff Migration Images
20
Kirchhoff Migration Images
20
Comparing CIGs
21
Comparing CIGs
CIG from Traveltime Tomogram
CIG from Waveform Tomogram
22
Comparing CIGs
23
Comparing CIGs
CIG from Traveltime Tomogram
CIG from Waveform Tomogram
24
Comparing CIGs
25
Comparing CIGs
CIG from Traveltime Tomogram
CIG from Waveform Tomogram
26
Part I: Outline
Application of waveform inversion to marine
data
• Motivation
• Theory
• Numerical Results
• 2D SEG/EAGE Salt Model
• Gulf of Mexico Data
• Conclusions
27
Conclusions
• Acoustic waveform inversion was applied to
both 2D synthetic and field data
• Using the traveltime tomogram, waveform
inversion failed to converge to an accurate
solution due to high velocity contrast
• Using v(z) model with the flooding technique,
an accurate result was obtained
28
Conclusions
• Acoustic waveform inversion with a dynamic
early-arrival muting window can invert the
Gulf of Mexico data to obtain a velocity model
that is more accurate than the traveltime
tomogram.
• The accuracy of waveform tomogram was
assessed by comparing the migration images
and common image gathers.
29
Part II: Outline
Spatial Resolution Analysis using Beylkin’s
Formula
• Motivation
• Theory
• Numerical Results
• Homogeneous Model
• Smoothed 2D SEG/EAGE Salt Model
• Conclusions
1
Motivation
Model with Resolution Limits
RTM Image
2
Part II: Outline
Spatial Resolution Analysis using Beylkin’s
Formula
• Motivation
• Theory
• Numerical Results
• Homogeneous Model
• Smoothed 2D SEG/EAGE Salt Model
• Conclusions
3
2D Spatial Resolution Formulas
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 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 Formulas
Source/receiver configuration can be described
by coordinate 
2D common-shot gather can be described by
rs  ( X ,0) and rg  ( ,0)
for fixed X
Beylkin’s resolution mapping:
f : ( ,  )  (k x , k z )
data
image
6
2D Spatial Resolution Formulas
Beylkin et al. (1985) derived the following
resolution formulas:
k   (r,  )
where
k  (k x , k z )
is the wavenumber vector in the
image domain
 (r,  )
is the traveltime surface of a
diffractor at r for shot/receiver
pairs described by 
7
2D Spatial Resolution Formulas
A diffractor traveltime  (r,  ) can be described as
 (r,  )   (r, rs )   (r, rg )   s   g
where
 (x, y ) is the traveltime from surface position y
to subsurface position x.
Similarly, k can be written as vectorial sum
k  ks  kg
 ( s   g )
8
Wavenumber Illumination
k  ks  kg
r
9
2D Spatial Resolution Formulas
Horizontal and vertical resolution limits:
2
x 
max | k x (r,  ,  ) |
2
z 
max | k z (r,  ,  ) |
10
Part II: Outline
Spatial Resolution Analysis using Beylkin’s
Formula
• Motivation
• Theory
• Numerical Results
• Homogeneous Model
• Smoothed 2D SEG/EAGE Salt Model
• Conclusions
11
Homogeneous Model
Depth (km)
Homogeneous Model with Resolution Limits
0
1
0
X (km)
1
12
Wavenumber
Illumination
13
Homogeneous Model
Reverse-Time Migration Image
Depth (km)
0
1
0
X (km)
1
14
Part II: Outline
Spatial Resolution Analysis using Beylkin’s
Formula
• Motivation
• Theory
• Numerical Results
• Homogeneous Model
• Smoothed 2D SEG/EAGE Salt Model
• Conclusions
15
Smoothed SEG/EAGE Salt Model
Smoothed Salt Model with Resolution Limits
0
Depth (km)
Velocity (m/s)
6000
3.5
1500
0
X (km)
16
16
Wavenumber
Illumination
17
Smoothed SEG/EAGE Salt Model
Reverse-Time Migration Image
Depth (km)
0
3.5
0
X (km)
16
18
RTM Image Line at x = 8 km
1
0.2
0.15
Amplitude
Amplitude
z = 2 km
Amplitude
z = 1 km
0
0
1.9 Depth (km) 2.1
0
0.9 Depth (km) 1.1
z = 3 km
2.9 Depth (km) 3.1
19
Smoothed SEG/EAGE Salt Model
Reverse-Time Migration Image
Depth (km)
0
3.5
0
X (km)
16
20
RTM Image Line at z = 2 km
x = 8 km
0.2
Amplitude
Amplitude
0.3
0.5
Amplitude
x = 2 km
0
0
7.8
0
13.8
1.8
X (km)
2.2
X (km)
8.2
x = 14 km
X (km) 14.2
21
Part II: Outline
Spatial Resolution Analysis using Beylkin’s
Formula
• Motivation
• Theory
• Numerical Results
• Homogeneous Model
• Smoothed 2D SEG/EAGE Salt Model
• Conclusions
22
Conclusions
• 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.
23
Acknowledgments
• My advisor: Dr. Gerard Schuster
• Mentors: Paul Valasek, Partha Routh, and Peng Chen
• Committee members: Dr. Ron Brunh and
Dr. Richard Jarrard
• 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
24