Full-waveform inversion based on
gradient sampling algorithmJizhong Yang1, Yunyue Elita Li1,
Yanwen Wei2, Haohuan Fu2, and Yuzhu Liu3
1 National University of Singapore2 Tsinghua University
3 Tongji University
Outline
⚫ Introduction
⚫ Methodology
⚫ Numerical Examples
➢ 2004 BP model
⚫ Conclusions
2
Outline
⚫ Introduction
⚫ Methodology
⚫ Numerical Examples
➢ 2004 BP model
⚫ Conclusions
3
Overview of FWI
Iterative
Process
Starting Model
Final Model
Real Data
( ) ( )2
2
1
2J m u m d= −
d: recorded seismic data
u(m): modeled seismic data
m: seismic velocity
Surface acquisition geometry
Time-domain implementation
Acoustic regime Workflow of FWI
4
FWI Challenges
▪ Nonlinear inversion
▪ Iterative methods
▪ Nonconvex
▪ Local minimum
5
FWI Challenges
▪ Low frequency data
▪ Missing from
previous acquisitions
▪ Expensive to acquire
▪ Very slow convergence
at large velocity error
6
FWI challenges: slow convergence
• Small gradient at the lowest frequency
• Takes too many iterations to update
200 iterations
7
1000 iterations
fn=1Hz
FWI challenges: slow convergence
200 iterations
200 iterations
Inversion may still get stuck in local minima
at high frequencies!
8
fn=1Hz
fn=4Hz
• Computing descent directions as a weighted sum
over all sampled gradients
• Calculating gradients for each sample
• Updating the model using the gradient
sampling direction
• Sampling N+1 model vectors in a region close to the current model
Gradient sampling algorithm
(Burke et al., 2005; Curtis and Que, 2013)
Direction at m
Gradient sampling direction
9
• Use the weighted gradient to update model
• Better gradients at low frequency and faster convergence
FWI with GSA: global optimal solution
• Sample the vicinity of current model
200 iterations
10
fn=1Hz
FWI with GSA: global optimal solution
200 iterations
200 iterations
Inversion arrives at global minima at
high frequencies!
11
fn=1Hz
fn=4Hz
Outline
⚫ Introduction
⚫ Methodology
⚫ Numerical Examples
➢ 2004 BP model
⚫ Conclusions
12
Methodology
Gradient Computational Cost
Conventional
FWI2Ns
GSA-FWI 2Ns*(N+1)
( ) ( ) ( ) ( ); 2 , ; , ;g U t V t dt= x m m x x m x m
( ) ( ) ( ) ( ); 2 , ; , ;i i i ig U t V t dt= x m m x x m x m
0
N
i mi
i
g g=
=0
0, 1N
i i
i
=
=
( ), ;U tx m
Ns is the number of sources, N+1 is the number of sampled vectors in the vicinity of the current model
: Forward propagated wavefield ( ), ;V tx m : Backward propagated wavefield
: Weighting coefficient
13
How can we approximate the sampled
gradient in an efficient way?
14
v(iz)=2000+grad*iz, vmax=5000 source
receiver
15
v(iz)=2000+grad*iz, vmax=3400 source
receiver
16
v(iz)=2000+grad*iz, vmax=4600 source
receiver
17
Forward Backward Gradient* =
Reference model
Sampled model
18
Gradient calculated
using the reference
velocity model at t=0.5s
Reference gradient
19
Gradient calculated
using the sampled
velocity model at t=0.5s
Sampled gradient
20
How can we approximate the sampled gradient in an
efficient way?
( ) ( ); ;i i −g x m g x h m
Randomly sample one h within the First Fresnel Zone at each time step!
( ) ( ) ( ) ( ) ( )2 ; , ;i i i iU t V t dt= − − − g x h m x h x h , m x h m
( ) ( ) ( ) ( )2 , ; , ;t t tU t V t dt − − −g x m x h x h m x h m
( ) ( ) ( )2 , ; , ;i i iU t V t dt= − − −m x h x h m x h m
0
0, 1N
i i
i
=
=
21
Gradient Computational Cost
Conventional
FWI2Ns
GSA-FWI 2Ns*(N+1)
GSA-FWI
(Our method)2Ns
( ) ( ) ( ) ( ); 2 , ; , ;g U t V t dt= x m m x x m x m
( ) ( ) ( ) ( ); 2 , ; , ;i i i ig U t V t dt= x m m x x m x m
0
N
i mi
i
g g=
=0
0, 1N
i i
i
=
=
22
( ) ( ) ( ) ( )2 , ; , ;t t tU t V t dt − − −g x m x h x h m x h m
Outline
⚫ Introduction
⚫ Methodology
⚫ Numerical Examples
➢ 2004 BP model
⚫ Conclusions
23
Workflow
Final solutionFrequency
sweep 2 - 19 Hz
Low frequency 2 - 5 Hz
Initialization
Half-space initial velocity
GSA-FWIConventional
FWIGSA-FWI-
CFWI
Conventional FWI
Conventional FWI
CFWI
24
True velocity Initial velocity
25
Conventional FWI GSA-FWI
Re-initialize conventional FWI using both models
26
2-5 Hz
Workflow
Final solutionFrequency
sweep 2 - 19 Hz
Low frequency 2 - 5 Hz
Initialization
Half-space initial velocity
GSA-FWIConventional
FWIGSA-FWI-
CFWI
Conventional FWI
Conventional FWI
CFWI
27
Conventional FWIConventional FWI
initialized with GSA-FWI
00 2-5 Hz
28
Conventional FWIConventional FWI
initialized with GSA-FWI
100100 2-5 Hz
29
Conventional FWIConventional FWI
initialized with GSA-FWI
200200 2-5 Hz
30
Conventional FWIConventional FWI
initialized with GSA-FWI
2-7 Hz
31
Conventional FWIConventional FWI
initialized with GSA-FWI
2-11 Hz
32
Conventional FWIConventional FWI
initialized with GSA-FWI
2-15 Hz
33
Conventional FWIConventional FWI
initialized with GSA-FWI
2-19 Hz
34
True velocity
All frequency
2-19 Hz
Conventional FWI
35
Conventional FWI
initialized with GSA-FWITrue velocity
All frequency
2-19 Hz
36
Outline
⚫ Introduction
⚫ Methodology
⚫ Numerical Examples
➢ 2004 BP model
⚫ Conclusions
37
Conclusions
⚫ GSA-based FWI speeds up the convergence, and is hence less sensitive
to the cycle-skipping problem
⚫ We proposed an approximated sampling scheme, which makes the
resulting GSA-based FWI as efficient as conventional FWI
⚫ The proposed GSA-based FWI is robust when the seismic data contains
abundant diving waves and refraction waves
38
Discussion
⚫ GSA-based FWI did not overcome the non-convexity of FWI
➢ Numerical tests starting from 5Hz did not yield significant results
⚫ Further study is needed on an optimal strategy of space shift
➢ Numerical tests with different shift strategies show similar, yet different
results
⚫ Applicability of GSA-based FWI to reflection-dominated seismic data
➢ Numerical tests on transmission-dominated data show better results
39
40
Forward Backward
41
Forward Backward
42
Gradient calculated
using the reference
velocity model at t=0.5s
Reference gradient
43
Gradient calculated
using the sampled
velocity model at t=0.5s
Sampled gradient
44
Data misfit Model misfit
Conventional FWI
GSA-FWI
Conventional FWI
GSA-FWI
45
Workflow
Final solutionFrequency
sweep 2 - 19 Hz
Low frequency 2 - 5 Hz
Initialization
Half-space initial velocity
GSA-FWIConventional
FWIRobust FWI
results
Conventional FWI
Conventional FWI
FWI stuck in local minima
46
Conventional FWI
2-5 Hz 2-5 Hz
GSA-FWI
47
Conventional FWI GSA-FWI
2-5 Hz 2-5 Hz
48
Conventional FWI GSA-FWI
Low frequency
2-5 Hz
Low frequency
2-5 Hz
49
Conventional FWI GSA-FWI
2-5 Hz 2-5 Hz
50
Conventional FWI GSA-FWI
Low frequency
2-5 Hz
Low frequency
2-5 Hz
51
Conventional FWI GSA-FWI
2-5 Hz 2-5 Hz
52
Conventional FWI GSA-FWI
Low frequency
2-5 Hz
Low frequency
2-5 Hz
53
Conventional FWI GSA-FWI
2-5 Hz 2-5 Hz
54
Conventional FWI GSA-FWI
2-5 Hz 2-5 Hz
55
Conventional FWI GSA-FWI
2-5 Hz 2-5 Hz
56
Conventional FWIConventional FWI
initialized with GSA-FWI
Low frequency
2-5 Hz
Low frequency
2-5 Hz
57
Conventional FWIConventional FWI
initialized with GSA-FWI
2-5 Hz 2-5 Hz
58
Conventional FWIConventional FWI
initialized with GSA-FWI
Low frequency
2-5 Hz
Low frequency
2-5 Hz
59
Conventional FWIConventional FWI
initialized with GSA-FWI
Low frequency
2-5 Hz
Low frequency
2-5 Hz
60
Conventional FWIConventional FWI
initialized with GSA-FWI
2-5 Hz 2-5 Hz
61
Conventional FWIConventional FWI
initialized with GSA-FWI
2-5 Hz 2-5 Hz
62
True velocityInitial velocity
63
True velocity
All frequency
2-19 Hz
Conventional FWI
64
65
True velocityConventional FWI
initialized with GSA-FWI
All frequency
2-19 Hz