full-waveform inversion based on gradient sampling...

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