S E I S C O P E Using the Ensemble Transform Kalman Filter to - - PowerPoint PPT Presentation

▶
s e i s c o p e
SMART_READER_LITE
LIVE PREVIEW

S E I S C O P E Using the Ensemble Transform Kalman Filter to - - PowerPoint PPT Presentation

Dec 01, 2023 157 likes •714 views

S E I S C O P E Using the Ensemble Transform Kalman Filter to estimate uncertainty in Full Waveform Inversion Julien Thurin 1 , Romain Brossier 1 and Ludovic M etivier 1 , 2 Wednesday 30 th May, 2018 - 13 th International ENKF workshop 1 Univ.

slide-1
SLIDE 1

S E I S C O P E

Using the Ensemble Transform Kalman Filter to estimate uncertainty in Full Waveform Inversion

Julien Thurin1, Romain Brossier1 and Ludovic M´ etivier1,2 Wednesday 30th May, 2018 - 13th International ENKF workshop

1 Univ. Grenoble Alpes, ISTerre 2 Univ. Grenoble Alpes, CNRS, LJK

slide-2
SLIDE 2

Introduction: Seismic Tomography

S E I S C O P E

Goal of seismic tomography: find physical parameters of the subsurface from seismic wavefield data. The recorded data are directly linked to the subsurface physical properties. Comparison of observed wavefield with synthetic wavefield allows formulating an inverse problem to find the model that gives the best data-fit.

Illustration of the tomographic problem.

ETKF-FWI 1

slide-3
SLIDE 3

Introduction: Full Waveform Inversion

S E I S C O P E

In Full Waveform Inversion (FWI), we try to match the entire recorded wavefield (dobs) at receiver locations with the synthetic waveform data computed in a starting model (dcal). FWI allows to obtain higher resolution than ”classical” tomography techniques relying only on travel time. The FWI inverse problem is more difficult (more non-linear) as it attempts to fit an entire pressure recordings.

Exemple of recorded wavefield data.

ETKF-FWI 2

slide-4
SLIDE 4

Outline

S E I S C O P E

Characteristics of FWI From FWI to ETKF-FWI Application on a Synthetic Case Conclusions

ETKF-FWI 3

slide-5
SLIDE 5

Outline

S E I S C O P E

Characteristics of FWI From FWI to ETKF-FWI Application on a Synthetic Case Conclusions

ETKF-FWI 3

slide-6
SLIDE 6

Full Waveform Inversion Workflow

S E I S C O P E

Courtesy of Isabella Masoni

ETKF-FWI 4

slide-7
SLIDE 7

Full Waveform Inversion Workflow

S E I S C O P E

Courtesy of Isabella Masoni

ETKF-FWI 4

slide-8
SLIDE 8

Full Waveform Inversion Workflow

S E I S C O P E

Courtesy of Isabella Masoni

ETKF-FWI 4

slide-9
SLIDE 9

Full Waveform Inversion Workflow - Forward modeling

S E I S C O P E

As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. The misfit of the FWI problem is : minm 1

2 dcal(m) − dobs2

ETKF-FWI 5

slide-10
SLIDE 10

Full Waveform Inversion Workflow - Forward modeling

S E I S C O P E

As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. The misfit of the FWI problem is : minm 1

2 dcal(m) − dobs2

ETKF-FWI 5

slide-11
SLIDE 11

Full Waveform Inversion Workflow - Forward modeling

S E I S C O P E

As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. The misfit of the FWI problem is : minm 1

2 dcal(m) − dobs2

ETKF-FWI 5

slide-12
SLIDE 12

Full Waveform Inversion Workflow - Forward modeling

S E I S C O P E

As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. The misfit of the FWI problem is : minm 1

2 dcal(m) − dobs2

ETKF-FWI 5

slide-13
SLIDE 13

Full Waveform Inversion Workflow - Forward modeling

S E I S C O P E

As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. The misfit of the FWI problem is : minm 1

2 dcal(m) − dobs2

ETKF-FWI 5

slide-14
SLIDE 14

Full Waveform Inversion Workflow - Forward modeling

S E I S C O P E

As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. The misfit of the FWI problem is : minm 1

2 dcal(m) − dobs2

ETKF-FWI 5

slide-15
SLIDE 15

Full Waveform Inversion Workflow - Forward modeling

S E I S C O P E

As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. The misfit of the FWI problem is : minm 1

2 dcal(m) − dobs2

ETKF-FWI 5

slide-16
SLIDE 16

Full Waveform Inversion Workflow - Forward modeling

S E I S C O P E

As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. The misfit of the FWI problem is : minm 1

2 dcal(m) − dobs2

ETKF-FWI 5

slide-17
SLIDE 17

Full Waveform Inversion Workflow

S E I S C O P E

Courtesy of Isabella Masoni

ETKF-FWI 5

slide-18
SLIDE 18

Full Waveform Inversion Workflow

S E I S C O P E

Courtesy of Isabella Masoni

ETKF-FWI 5

slide-19
SLIDE 19

Full Waveform Inversion Workflow

S E I S C O P E

Courtesy of Isabella Masoni

ETKF-FWI 5

slide-20
SLIDE 20

Full Waveform Inversion Workflow

S E I S C O P E

Courtesy of Isabella Masoni

ETKF-FWI 5

slide-21
SLIDE 21

Full Waveform Inversion Workflow

S E I S C O P E

Courtesy of Isabella Masoni

ETKF-FWI 5

slide-22
SLIDE 22

Introduction : Why using FIW ?

S E I S C O P E

Traveltime tomography in the Valhall Oil Field.

ETKF-FWI 6

slide-23
SLIDE 23

Introduction : Why using FIW ?

S E I S C O P E

Full Waveform Inversion in the Valhall Oil Field.

ETKF-FWI 7

slide-24
SLIDE 24

Motivations

S E I S C O P E

  • FWI is generally applied in a deterministic fashion from a starting model
  • FWI relies on local optimization (quasi-Newton)
  • FWI results are generally difficult to assess

Only a few recent papers propose to tackle the uncertainy problem in FWI : still no systematic applications. We propose an approach relying on a mixed-method based on an Ensemble Transform Kalman Filter and the classic quasi-Newton optimization scheme to evaluate uncertainty in the FWI results.

ETKF-FWI 8

slide-25
SLIDE 25

Outline

S E I S C O P E

Characteristics of FWI From FWI to ETKF-FWI Application on a Synthetic Case Conclusions

ETKF-FWI 8

slide-26
SLIDE 26

Adapting the FWI problem to the EnKF

S E I S C O P E

We define the FWI problem as a non-linear operator F F(m) = min

m

1 2 ||dcal(m) − dobs||2

  • m is the model containing the n physical parameters
  • dcal(m) the synthetic wavefield data computed in m
  • dobs the observed data
  • ||.|| the Euclidean distance in the data space

Applying F on an initial model m0 → unique solution with a local optimization scheme. But how do we apply an EnKF to this static problem?

ETKF-FWI 9

slide-27
SLIDE 27

Multi Scale Approach

S E I S C O P E

Full Waveform data fitting is ill-posed by nature. Non-unique, its cost function can be strongly non-convex. The cost function convexity is primarily dominated by the data frequency content (due to the nature of cycle skipping problem) We can use the multi-scale frequency strategy as a proxy for evolution in the frequency domain.

1D waveform cost function, at different frequency content from (Bunks et al., 1995)

ETKF-FWI 10

slide-28
SLIDE 28

The ensemble approach and dynamic axis

S E I S C O P E

We can recast our problem as an ensemble representation. Our ensemble m is a collection of Ne models mi, with i = 1, 2, . . . , Ne. In place of the typical DA forecast forward modeling problem we have : mf

k i = F(mi k−1)

(1) We decompose the dobs in K frequency bands → solve FWI independently on each of Ne models, at a given frequency k. Allowing to consider a dynamic axis in frequency, with k = 1, . . . , K instead of temporal evolution

ETKF-FWI 11

slide-29
SLIDE 29

Scheme specificities

S E I S C O P E

The EnKF scheme we follow : the Ensemble Transform Kalman Filter (ETKF) (Bishop et al., 2001). We apply FWI in the frequency domain → complex wavefield data. We consider all measurements as uncorrelated → measurement noise operator is diagonal whose values are calibrated

  • n the data noise level.

ETKF-FWI 12

slide-30
SLIDE 30

ETKF-FWI Scheme

S E I S C O P E step (modeling frequency) k − 1 k k + 1 mk−1 ×

  • dobs,k

⋆ Forecast (FWI) mf k ×

  • Observation

(modeling) df k × ⋆ ⋆ ⋆ ⋆ ⋆ ma k ×

  • Forecast

(FWI) mf k+1 ×

  • dobs,k+1

⋆ Observation (modeling) df k+1 × ⋆ ⋆ ⋆ ⋆ ⋆ ma k+1 ×

  • ma

k+1 ×

  • ETKF-FWI

13

slide-31
SLIDE 31

Outline

S E I S C O P E

Characteristics of FWI From FWI to ETKF-FWI Application on a Synthetic Case Conclusions

ETKF-FWI 13

slide-32
SLIDE 32

Applying ETKF to FWI

S E I S C O P E

Application on 2D Marmousi model :

  • Fixed spread surface acquisition (144 sources, 660 receivers)
  • Noisy signal (SNR = 5)
  • 15 ETKF-FWI cycles from 3 to 10Hz.
  • Initial gaussian repartition

ETKF-FWI 14

slide-33
SLIDE 33

Applying ETKF to FWI

S E I S C O P E 500 1000 1500 2000 2500 3000 3500 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 500 1000 1500 2000 2500 3000 3500 Depth (m) 1500 2000 2500 3000 3500 4000 4500 Velocity (m/s)

Numerical test setting. Top : True model, bottom : Initial model

ETKF-FWI 15

slide-34
SLIDE 34

Generating the initial ensemble

S E I S C O P E 500 1000 1500 2000 2500

Depth (m)

2500 5000 7500 10000 12500

Offset (m)

500 1000 1500 2000 2500 2500 5000 7500 10000 12500 −100 −50 50 100

Velocity (m/s)

Exemple of random perturbation selected to build the initial gaussian repartition.

ETKF-FWI 16

slide-35
SLIDE 35

Applying ETKF to FWI

S E I S C O P E 500 1000 1500 2000 2500 3000 3500 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 500 1000 1500 2000 2500 3000 3500 Depth (m) 1500 2000 2500 3000 3500 4000 4500 Velocity (m/s) 1000 2000 3000 4000 5000 6000 7000 8000 Variance (m^2/s^2)

Results for 200 ensemble members. Top : Final ensemble mean, bottom : Final ensemble variance

ETKF-FWI 17

slide-36
SLIDE 36

Undersampling sensitivity - Ensemble mean

S E I S C O P E 1000 2000 3000 1000 2000 3000 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 1000 2000 3000 Depth (m) 1500 2000 2500 3000 3500 4000 Variance (m^2/s^2)

Numerical test. Ensemble mean for Ne = 2000, 200 and 20

ETKF-FWI 18

slide-37
SLIDE 37

Undersampling sensitivity - Ensemble variance

S E I S C O P E 1000 2000 3000 1000 2000 3000 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 1000 2000 3000 Depth (m) 1000 2000 3000 4000 5000 6000 7000 8000 Variance (m^2/s^2)

Numerical test. Variance for Ne = 2000, 200 and 20

ETKF-FWI 19

slide-38
SLIDE 38

A wide range of variance values

S E I S C O P E

That high degree of variability makes qualitative comparison of off-diagonal terms difficult. We propose to use the Correlation matrix Ca

k instead of the Covariance matrix to read the off-diagonal terms.

Ca

k,e = (diag(Pa k,e))−1/2Pa k,e(diag(Pa k,e))−1/2

(2) This provides dimensionless and normalized correlation maps, regardless of Ne and location in the medium.

ETKF-FWI 20

slide-39
SLIDE 39

Estimating local correlation maps

S E I S C O P E

2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 1000 2000 3000 Depth (m)

(A)

1000 2000 3000 4000 5000 6000 7000 8000 Variance (m^2/s^2)

(B) (C) (D)

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 Correlation

(E) (F) (G)

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 Correlation Ne = 20

(H)

Ne = 200

(I)

Ne = 2000

(J)

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 Correlation ETKF-FWI 21

slide-40
SLIDE 40

Variance convergence test

S E I S C O P E 250 500 750 1000 1250 1500 1750 2000 Number of members 500 1000 1500 2000 2500 Variance (m^2/s^2)

Influence of Ne on variance estimation

ETKF-FWI 22

slide-41
SLIDE 41

Velocity Log - Comparing ETKF-FWI with FWI

S E I S C O P E 500 1000 1500 2000 2500 3000 3500

ETKF-FWI

2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 500 1000 1500 2000 2500 3000 3500 Depth (m)

FWI

1500 2000 2500 3000 3500 4000 4500 Velocity (m/s) 1500 2000 2500 3000 3500 4000 4500 Velocity (m/s)

Top : Ensemble Mean. Bottom : FWI result

ETKF-FWI 23

slide-42
SLIDE 42

Velocity Log - Comparing ETKF-FWI with FWI

S E I S C O P E 500 1000 1500 2000 2500 3000 3500 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 500 1000 1500 2000 2500 3000 3500 Depth (m) 2000 3000 4000 Velocity (m/s) 500 1000 1500 2000 2500 3000 3500 Depth (m) ETKF-FWI FWI True 1500 2000 2500 3000 3500 4000 4500 Velocity (m/s) 1000 2000 3000 4000 5000 6000 7000 8000 Variance (m^2/s^2)

Velocity log through the Ne = 2000 ensemble.

ETKF-FWI 24

slide-43
SLIDE 43

Velocity Log - Comparing ETKF-FWI with FWI

S E I S C O P E 500 1000 1500 2000 2500 3000 3500 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 500 1000 1500 2000 2500 3000 3500 Depth (m) 2000 3000 4000 Velocity (m/s) 500 1000 1500 2000 2500 3000 3500 Depth (m) ETKF-FWI FWI True 1500 2000 2500 3000 3500 4000 4500 Velocity (m/s) 1000 2000 3000 4000 5000 6000 7000 8000 Variance (m^2/s^2)

Velocity log through the Ne = 2000 ensemble.

ETKF-FWI 25

slide-44
SLIDE 44

Velocity Log - Comparing ETKF-FWI with FWI

S E I S C O P E 500 1000 1500 2000 2500 3000 3500 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 500 1000 1500 2000 2500 3000 3500 Depth (m) 1000 2000 3000 4000 Velocity (m/s) 500 1000 1500 2000 2500 3000 3500 Depth (m) ETKF-FWI FWI True 1500 2000 2500 3000 3500 4000 4500 Velocity (m/s) 1000 2000 3000 4000 5000 6000 7000 8000 Variance (m^2/s^2)

Velocity log through the Ne = 2000 ensemble.

ETKF-FWI 26

slide-45
SLIDE 45

Velocity Log - Comparing ETKF-FWI with FWI

S E I S C O P E 500 1000 1500 2000 2500 3000 3500 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 500 1000 1500 2000 2500 3000 3500 Depth (m) 1000 2000 3000 4000 Velocity (m/s) 500 1000 1500 2000 2500 3000 3500 Depth (m) ETKF-FWI FWI True 1500 2000 2500 3000 3500 4000 4500 Velocity (m/s) 1000 2000 3000 4000 5000 6000 7000 8000 Variance (m^2/s^2)

Velocity log through the Ne = 2000 ensemble.

ETKF-FWI 27

slide-46
SLIDE 46

Velocity Log - Comparing ETKF-FWI with FWI

S E I S C O P E 500 1000 1500 2000 2500 3000 3500 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 500 1000 1500 2000 2500 3000 3500 Depth (m) 2000 3000 4000 Velocity (m/s) 500 1000 1500 2000 2500 3000 3500 Depth (m) ETKF-FWI FWI True 1500 2000 2500 3000 3500 4000 4500 Velocity (m/s) 1000 2000 3000 4000 5000 6000 7000 8000 Variance (m^2/s^2)

Velocity log through the Ne = 2000 ensemble.

ETKF-FWI 28

slide-47
SLIDE 47

Velocity Log - Comparing ETKF-FWI with FWI

S E I S C O P E 500 1000 1500 2000 2500 3000 3500 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 500 1000 1500 2000 2500 3000 3500 Depth (m) 2000 3000 4000 Velocity (m/s) 500 1000 1500 2000 2500 3000 3500 Depth (m) ETKF-FWI FWI True 1500 2000 2500 3000 3500 4000 4500 Velocity (m/s) 1000 2000 3000 4000 5000 6000 7000 8000 Variance (m^2/s^2)

Velocity log through the Ne = 2000 ensemble.

ETKF-FWI 29

slide-48
SLIDE 48

Velocity Log - Comparing ETKF-FWI with FWI

S E I S C O P E 500 1000 1500 2000 2500 3000 3500 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) 500 1000 1500 2000 2500 3000 3500 Depth (m) 2000 3000 4000 Velocity (m/s) 500 1000 1500 2000 2500 3000 3500 Depth (m) ETKF-FWI FWI True 1500 2000 2500 3000 3500 4000 4500 Velocity (m/s) 1000 2000 3000 4000 5000 6000 7000 8000 Variance (m^2/s^2)

Velocity log through the Ne = 2000 ensemble.

ETKF-FWI 30

slide-49
SLIDE 49

Outline

S E I S C O P E

Characteristics of FWI From FWI to ETKF-FWI Application on a Synthetic Case Conclusions

ETKF-FWI 30

slide-50
SLIDE 50

Conclusions

S E I S C O P E

Uncertainty estimation is possible with ETKF-FWI. Numerical experiments show:

  • Very low Ne = dramatic underestimation of Pa

e

  • Higher Ne = stable approximation
  • Variance underestimation = power-low trend
  • Mean is preserved
  • Possible local ”collapse” : strong undersapling in shallow zones

Uncertainty estimation not absolute uncertainty.

ETKF-FWI 31

slide-51
SLIDE 51

Perspectives

S E I S C O P E

Short term :

  • Undersampling mitigation (inflation tests in progress).
  • Initial ensemble building.

Medium term :

  • Real data application : Valhall 2D.
  • Comparison with other methodologies.

Long term prospective work:

  • Go beyond 2D frequency acoustic (3D, time domain, multiparameter...)
  • Inversion parameters influence over uncertainty
  • Sensor fusion (well log data, geophysical methods)

ETKF-FWI 32

slide-52
SLIDE 52

Acknowledgments

S E I S C O P E

Thanks for your attention

  • 2018 SEISCOPE sponsors (http://seiscope2.osug.fr):

AKERBP, CGG, CHEVRON, EXXON-MOBIL, JGI, PETROBRAS, SCHLUMBERGER, SHELL, SINOPEC, STATOIL and TOTAL.

  • CIMENT (Froggy) computing center https://ciment.ujf-grenoble.fr
  • CINES/IDRIS/TGCC computing center (allocation 046091 made by GENCI)

ETKF-FWI 33

slide-53
SLIDE 53

Questions?

ETKF-FWI 33

slide-54
SLIDE 54

Bishop, C. H., Etherton, B. J., and Majumdar, S. J. (2001). Adaptive sampling with the ensemble transform kalman

  • filter. part i: Theoretical aspects. Monthly weather review, 129(3):420–436.

Bunks, C., Salek, F. M., Zaleski, S., and Chavent, G. (1995). Multiscale seismic waveform inversion. Geophysics, 60(5):1457–1473. Ott, E., Hunt, B. R., Szunyogh, I., Zimin, A. V., Kostelich, E. J., Corazza, M., Kalnay, E., Patil, D., and Yorke, J. A. (2004). A local Ensemble Kalman filter for atmospheric data assimilation. Tellus A, 56:415–428. Wang, X., Bishop, C. H., and Julier, S. J. (2004). Which is better, an ensemble of positive-negative pairs or a centered spherical simplex ensemble? Monthly Weather Review, 132(7):1590–1605.

ETKF-FWI 34