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


➢ 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


➢ 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


receiver

16


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


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


➢ 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


sweep 2 - 19 Hz


Initialization


GSA-FWIConventional

FWIGSA-FWI-

CFWI

Conventional FWI

Conventional FWI

CFWI

27

Conventional FWIConventional FWI

initialized with GSA-FWI

00 2-5 Hz

28



100100 2-5 Hz

29



200200 2-5 Hz

30



2-7 Hz

31



2-11 Hz

32



2-15 Hz

33



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


➢ 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

Forward Backward

41

Forward Backward

42

Gradient calculated

using the reference


Reference gradient

43

Gradient calculated

using the sampled


Sampled gradient

44

Data misfit Model misfit

Conventional FWI

GSA-FWI

Conventional FWI

GSA-FWI

45

Workflow


sweep 2 - 19 Hz


Initialization


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


2-5 Hz 2-5 Hz

48


Low frequency

2-5 Hz

Low frequency

2-5 Hz

49


2-5 Hz 2-5 Hz

50


Low frequency

2-5 Hz

Low frequency

2-5 Hz

51


2-5 Hz 2-5 Hz

52


Low frequency

2-5 Hz

Low frequency

2-5 Hz

53


2-5 Hz 2-5 Hz

54


2-5 Hz 2-5 Hz

55


2-5 Hz 2-5 Hz

56



Low frequency

2-5 Hz

Low frequency

2-5 Hz

57



2-5 Hz 2-5 Hz

58



Low frequency

2-5 Hz

Low frequency

2-5 Hz

59



Low frequency

2-5 Hz

Low frequency

2-5 Hz

60



2-5 Hz 2-5 Hz

61



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


All frequency

2-19 Hz

Full-waveform inversion based on gradient sampling...

Documents

Transcript of Full-waveform inversion based on gradient sampling...