full waveform inversion in practice - seiscope consortium · 2017. 3. 30. · some references this...

Full Waveform Inversion in Practice

Romain Brossier and Jean [email protected] [email protected]

ISTerre, Universite Joseph Fourier, Grenoble, FranceSEISCOPE Consortium

12th International Congress of the Brazilian Geophysical SocietyShort Course on Full Waveform Inversion

15 August 2011

R. Brossier & J. Virieux (Univ Joseph Fourier) FWI in practice SBGF - 15/08/2011 1 / 43

Outline

1 Introduction

2 FWI Resolution

3 Multiscale algorithms

4 Source time function estimation

5 Starting model building


Some references

This part of the course is mainly based on the following paper and presentation

Virieux J. and S. Operto, [2009] An overview of full-waveform inversion in explorationgeophysics. Geophysics, 74(6), WCC1-WCC26.

S. Operto [2006] Seismic imaging by frequency-domain full-waveform inversion Part 3 :synthetic examples. SEISCOPE SUMMER SCHOOL


Outline

1 Introduction

2 FWI Resolution





FWI principleIntuitive understanding in time domain ... the incident wavefield


FWI principleIntuitive understanding in time domain ... the adjoint wavefield from receivers


FWI principleCorrelation of the incident and adjoint wavefields


Key questions to be addressed

1 What is the FWI resolution ?I Reminders on the diffraction tomographyI Effects of the Hessian information

2 Multiscale/Hierarchical FWI algorithms

3 Source time function reconstruction ?4 Starting model definition ?

I Accuracy and cycle skippingI Possible methods for building starting models


Outline

1 Introduction

2 FWI ResolutionFWI gradient estimationFWI Hessian estimation





Outline

1 Introduction






Monochromatic gradient : The Fresnel zone

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Relation between frequency, aperture and resolution

FWI is based on diffraction tomography principle and can be seen as a spatialinverse Fourier transform (Devaney, 1982)Spatial resolution of FWI (cf. Sirgue and Pratt, 2004)

~k =2ω

c0cos (θ/2)~n, (1)

S R

ps

pr

x

q

θ

k = f q

λ= c/f

q = ps + prps = pr = 1/c

k : wavenumber for theacquisition aperture

f : frequency for the acquisitiontime series


Relation between frequency, aperture and resolution

FWI is based on diffraction tomography principle and can be seen as a spatialinverse Fourier transform (Devaney, 1982)

Spatial resolution of FWI (cf. Sirgue and Pratt, 2004)

~k =2ω

c0cos (θ/2)~n, (2)

f

kz

kz

max

kz

min

f1 f2 f3


Impact on acquisitionTransmission acquisition : single ball

7 freqs, 5 iter/freq, [4,5,7,10,13,16,20]Hz


Impact on acquisitionTransmission acquisition : single ball (2)



Impact on acquisitionTransmission + reflection acquisition : single ball



Impact on acquisitionTransmission acquisition : two balls



Impact on acquisitionRay based tomography


Impact on acquisitionTransmission + reflection acquisition : two balls



Impact on acquisitiona last example

Source in red

Receiver in blue

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Resolution and wavenumbersConclusion

To avoid local minima, the full spatial wavenumber must be filled

FWI requires wide angle acquisitionsI Large angle (long offsets) to constraint low wavenumber (long wavelengths)I Small angles (short offset) to reach the optimal resolution (λ/2)

Thanks to redundancy, the frequency spectrum could be decimated


Outline

1 Introduction






Hessian of FWI

The Hessian (related to resolutionmatrix) contains information onspatial correlation betweendiffractors

Using Hessian in optimizationintroduces this information : acts asa deconvolving operator on thegradient→ Gradient versus Gauss Newtonoptimization 4 freqs, 1 iter/freq,[4,5,7,10]Hz

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w n

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m)

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a)

b)


Hessian of FWI

The Hessian (related to resolutionmatrix) contains information onspatial correlation betweendiffractors

Using Hessian in optimizationintroduces this information : acts asa deconvolving operator on thegradient→ Gradient versus Gauss Newtonoptimization 4 freqs, 1 iter/freq,[4,5,7,10]Hz

0 1 2 30

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pth

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m)

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m)

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pth

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m)

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VP (km/s)

d)

e)


Hessian of FWI

The Hessian information improve significantly the results...

... but computationally expensive. Only few realistic applications with Newtonor Gauss-Newton methods

AlternativesI Quasi-Newton approach as L-BFGS (Brossier et al., 2009)I Incomplete Gauss-Newton (Hu et al., 2009)


Outline

1 Introduction

2 FWI Resolution





Efficient multiscale frequency-domain FWI algorithm

Efficient frequency-domain FWI algorithms use only few discrete frequencies fromlow to high frequencies for wide aperture acquisitions (Sirgue and Pratt, 2004;Brenders and Pratt, 2007a)

multiscale approach and mitigation of non-linearities

efficiency

Algorithm

1: for frequency = frequency 1 to frequency n do2: while (NOT convergence AND iter < nitermax) do3: Estimate source wavelet if required

4: Build gradient vector G(k)m

5: Build perturbation vector δm6: Update model m(k+1) = m(k) + αδm7: end while8: end for



Synthetic application fromSourbier et al. (2009) (true,starting, 3.5 Hz, 9.2 Hz,20.6 Hz)



Crustal imaging on theNankai thrust, Japan(Operto et al., 2006)



Complex frequency preconditioning allows to select arrivals from the first arrival(Shin et al., 2002; Brenders and Pratt, 2007b).

Remove complex late arrivals

An heuristics to select apertures in the data

F (ω + iγ)eγt0 =

∫ +∞

−∞f (t)e−γ(t−t0)e−iωtdt (3)



Two-levels hierarchical algorithm (Brossier et al., 2009)

1: for frequency = frequency 1 to frequency n do2: for data damping = highdamping to lowdamping do3: while (NOT convergence AND iter < nitermax) do4: Estimate source wavelet if required

5: Build gradient vector G(k)m

6: Build perturbation vector δm7: Update model m(k+1) = m(k) + αδm8: end while9: end for

10: end for


Outline

1 Introduction

2 FWI Resolution





Source time function estimation

The source time function can be estimated as an unknown of the inverseproblem (Pratt, 1999) :

C =1

2∆d†∆d (4)

I ∆d = dobs − dcalc

I dcalc = sg with g the Green function

C =1

2(dobs − sg)†(dobs − sg) (5)

s =< gcal(m0)|dobs >

< gcal(m0)|gcal(m0) >, (6)

It can be seen as the division of the cross-correlation of the computed andobserved data, by the autocorrelation of the computed data.


Source time function estimation

Analysis of the repeatability of source wavelet to qualify model accuracy(under the assumption that sources are repeatable on the field)

-20 -10 0 10 20

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b) c) d)

e) f ) g) h)

NMO model FATT modelAnisotropic model

Anisotropic FWI model3D FWI model

NMO model

(limited o!sets)

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s)

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NMO+ FWI model

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Outline

1 Introduction

2 FWI Resolution





Starting model buildingPresentation

Current multidimensional FWI based on local optimizationI High number of unknownsI Numerical cost of forward problem

For classical misfit functions (L2, L1, log,...), local method implies a “good”starting model

I No cycle skipping of the dataI Wavenumber content close to the smallest wavenumbers of FWI (to avoid gap

in the spectrum)

n n+1n-1

n

Time (s)

n+1nn-1

T/2 T/2

n+1n-1


Starting model buildingPresentation

Limitations for starting from scratchI Limited low frequencies in the data

(recent studies ≈ 1-2 Hz, Plessix et al., 2010)I Incomplete illumination

An other method is required to delineate the starting model of FWI


Possibles methods and their limitationsReflection tomography and MVA

ProsI Usage of reflected eventsI Strong experience in O&G industryI High resolution for FWI

ConsI Applicability in complex environments ?I Sensitive to NMO velocity : validity at long offset in case of anisotropy ?

(discussion later on the anisotropy issue)

Examples in Sirgue et al. (2009, 2010)

150 m 150 m

1050 m1050 m

from Sirgue et al. (2009)


Possibles methods and their limitationsFirst-arrival travel times tomography

ProsI RobustI Weak numerical cost (in particular with adjoint formulation Taillandier et al.,

2009)

ConsI Requires picking of first breakI Low resolution (requires low frequency for FWI)I Limited penetration in case of LVZI Sensitive to “horizontal” velocity

Some examples in Ravaut et al. (2004); Operto et al. (2006); Brenders andPratt (2007b,a)


Possibles methods and their limitationsStereotomography

ProsI Can use reflected and

refracted events (slope oflocally coherent events)

I Hierarchical scheme fromlarge to weak aperture(Prieux et al., 2010)

ConsI Requires picking (automatic)

and QCI Numerical cost of the

hierarchical scheme

preliminary example ofStereotomo + FWI in Prieuxet al. (2010), review ofstereotomo in Lambare (2008)

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epth

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1.6 2.0 2.4 2.8 3.2km/s

a) b)

c) d)

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(km

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Possibles methods and their limitationsLaplace domain inversion

ProsI Seems efficient on large

contrast (salt)I Affordable cost

ConsI Requires picking of the first

break and careful mute toensure robustness

I Requires long time recordingsI Instabilities (incomplete

Laplace transforms)

Example in Shin and Ha (2009)

a)

b)

c)

d)

e)

(image courtesy of C. Shin and Y. H. Cha)


Possibles methods and their limitationsEarly waveform tomography (EWT)

Avoid phase ambiguity byconsidering only first phases

Increase the size of theminimum valley

Expected better resolution thantraveltime tomography Shenget al. (2006); Ellefsen (2009)

left (FWI), middle (EWT), right (traveltime) (Sheng et al., 2006)


Starting model buildingConclusion

Currently, no generic method for starting model building...

The choice in mainly driven by the data properties (acquisition,frequency-contain...)

... still an active research field

Some alternative choice of the misfit function are proposed to avoid the thecycle-skipping limitation

I cross-correlation (Tape et al., 2009; Baumstein et al., 2011)I time-frequency criteria (Fichtner et al., 2009)


Baumstein, A., Ross, W., and Lee, S. (2011). Simultaneous source elastic inversion of surfacewaves. In Expanded Abstracts, page C040. European Association of Geoscientists andEngineers.

Brenders, A. J. and Pratt, R. G. (2007a). Efficient waveform tomography for lithosphericimaging : implications for realistic 2D acquisition geometries and low frequency data.Geophysical Journal International, 168 :152–170.

Brenders, A. J. and Pratt, R. G. (2007b). Full waveform tomography for lithospheric imaging :results from a blind test in a realistic crustal model. Geophysical Journal International,168 :133–151.

Brossier, R., Operto, S., and Virieux, J. (2009). Seismic imaging of complex onshore structuresby 2D elastic frequency-domain full-waveform inversion. Geophysics, 74(6) :WCC63–WCC76.

Devaney, A. J. (1982). A filtered backprojection algorithm for diffraction tomography. UltrasonicImaging, 4 :336–350.

Ellefsen, K. J. (2009). A comparison of phase inversion and traveltime tomography forprocessing near-surface refraction traveltimes. Geophysics, 74(6) :WCB11–WCB24.

Fichtner, A., Kennett, B. L. N., Igel, H., and Bunge, H. P. (2009). Full waveform tomographyfor upper-mantle structure in the Australasian region using adjoint methods. GeophysicalJournal International, 179(3) :1703–1725.

Hu, W., Abubakar, A., and Habashy, T. (2009). Preconditioned non-linear conjugate gradientmethod for seismic full-waveform inversion. In Expanded Abstracts. EAGE.

Lambare, G. (2008). Stereotomography. Geophysics, 73(5) :VE25–VE34.

Operto, S., Virieux, J., Dessa, J. X., and Pascal, G. (2006). Crustal imaging from multifoldocean bottom seismometers data by frequency-domain full-waveform tomography :application to the eastern Nankai trough. Journal of Geophysical Research,111(B09306) :doi :10.1029/2005JB003835.


Plessix, R.-E., Baeten, G., de Maag, J. W., Klaassen, M., Rujie, Z., and Zhifei, T. (2010).Application of acoustic full waveform inversion to a low-frequency large-offset land data set.SEG Technical Program Expanded Abstracts, 29(1) :930–934.

Pratt, R. G. (1999). Seismic waveform inversion in the frequency domain, part I : theory andverification in a physic scale model. Geophysics, 64 :888–901.

Prieux, V., Operto, S., Lambare, G., and Virieux, J. (2010). Building starting model for fullwaveform inversion from wide-aperture data by stereotomography. SEG Technical ProgramExpanded Abstracts, 29(1) :988–992.

Ravaut, C., Operto, S., Improta, L., Virieux, J., Herrero, A., and dell’Aversana, P. (2004).Multi-scale imaging of complex structures from multi-fold wide-aperture seismic data byfrequency-domain full-wavefield inversions : application to a thrust belt. Geophysical JournalInternational, 159 :1032–1056.

Sheng, J., Leeds, A., Buddensiek, M., and Schuster, G. T. (2006). Early arrival waveformtomography on near-surface refraction data. Geophysics, 71(4) :U47–U57.

Shin, C. and Ha, Y. H. (2009). Waveform inversion in the Laplace-Fourier domain. GeophysicalJournal International, 177 :1067–1079.

Shin, C., Min, D.-J., Marfurt, K. J., Lim, H. Y., Yang, D., Cha, Y., Ko, S., Yoon, K., Ha, T.,and Hong, S. (2002). Traveltime and amplitude calculations using the damped wave solution.Geophysics, 67 :1637–1647.

Sirgue, L., Barkved, O. I., Dellinger, J., Etgen, J., Albertin, U., and Kommedal, J. H. (2010).Full waveform inversion : the next leap forward in imaging at Valhall. First Break, 28 :65–70.

Sirgue, L., Barkved, O. I., Gestel, J. P. V., Askim, O. J., and Kommedal, J. H. . (2009). 3Dwaveform inversion on Valhall wide-azimuth OBC. In 71th Annual International Meeting,EAGE, Expanded Abstracts, page U038.

Sirgue, L. and Pratt, R. G. (2004). Efficient waveform inversion and imaging : a strategy forselecting temporal frequencies. Geophysics, 69(1) :231–248.


Sourbier, F., Operto, S., Virieux, J., Amestoy, P., and L’Excellent, J.-Y. (2009). Fwt2d : Amassively parallel program for frequency-domain full-waveform tomography of wide-apertureseismic data–part 1 : Algorithm. Computers & Geosciences, 35(3) :487 – 495.

Taillandier, C., Noble, M., Chauris, H., and Calandra, H. (2009). First-arrival travel timetomography based on the adjoint state method. Geophysics, 74(6) :WCB1–WCB10.

Tape, C., Liu, Q., Maggi, A., and Tromp, J. (2009). Seismic tomography of the southerncalifornia crust based on spectral-element and adjoint methods. Geophysical JournalInternational, 180 :433–462.


full waveform inversion in practice - seiscope consortium · 2017. 3. 30. · some references this...

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