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Full waveform inversion for high resolution seismic imaging: HPC issueson recent applications and ongoing research.

L. Metivier1,2, R. Brossier2, Q. Merigot3, A. Minuissi4,S. Operto4, E. Oudet1, J. Virieux2

1LJK, CNRS & Universite Grenoble Alpes2ISTerre, Universite Grenoble Alpes

3CEREMADE, CNRS & Universite Paris-Dauphine4Geoazur, CNRS & Universite Nice Sophia Antipolis

[email protected]

http://seiscope2.osug.fr

4iemes Journees Scientifiques Equip@Meso : Sciences de l’Univers, Toulouse,CALMIP, 26 et 27 Novembre 2015

SEISCOPE

L. Metivier et al. (Univ. Grenoble, CNRS) Full waveform inversion 27/11/2015 1 / 27

Outline

1 Introduction: principle of full waveform inversion

2 Example of application: 3D acoustic frequency-domain FWI

3 Reducing the sensitivity to the initial model accuracy using an optimal transport distance


Seismic imaging of the subsurface

We aim at reconstructing the Earth subsurface mechanical properties from surfacemeasurements of seismic waves

Example of a seismic survey in a marine environment. The vessel tows cables containing series ofequidistant hydrophones. Air guns are shot at regular interval. The hydrophones record theenergy coming back from the subsurface. This signal is interpreted to reconstruction the

subsurface mechanical properties.


Seismic imaging of the Earth

Illustration: numerical modeling of seismic wave propagation


Full waveform inversion : principle

28th'September'2012'

Full(waveform(inversion((FWI)(F(Principle(

Progress'MeeMng'';''Isabella'MASONI' 7'

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Calculated'data'Observed'data'Misfit'funcMon'

'Δd = dobs − dcal






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Time-domain full waveform inversion : mathematical settings

Nonlinear optimization problem

Minimization of

minc

f (c) =12

NsX

s=1

Z T

0

|dcal [s, c](t)� d sobs(t)|2dt (1)

where Ns is the number of seismic sources and dcal [s, c] is computed as

dcal [s, c] = Rp[s, c] (2)

where R is a restriction operator at the receivers location and p[s, c] is thetime-domain acoustic wavefield satisfying

1c2

@ttp ��p = s (3)


Time-domain full waveform inversion : mathematical settings

Quasi-Newton technique

The minimization is performed following quasi-Newton methods: from c0

build ck such thatck+1 = ck � ↵kQ(ck)rf (ck), (4)

Gradient: adjoint state method

The gradient is

rf (ck) = � 2c3k (x)

NsX

s=1

Z T

0

@ttp[s, ck ](x , t)�p[s, ck ](x , t)dt (5)

where �p[s, c](x , t) is the adjoint wavefield solution of

1c2

@tt�p ��p = R† (dcal [s, c]� dobs) (6)

Hessian: l-BFGS formula

Q(ck) is an approximation of the inverse Hessian operator, for instanceusing the l-BFGS formula (Nocedal, 1980)


Frequency-domain full waveform inversion : mathematical settings

Exploiting time redudancy

We can work with few discrete temporal frequencies !i of the signal

minc

f (c) =12

NsX

s=1

NfreqX

!i =1

|dcal [s, c](!i )� d sobs(!i )|2 (7)

Now d s,!ical (c) is computed as

dcal [s, c](!i ) = Rp[s, c](x , !i ) (8)

where p[s, c](x , !i ) is the frequency-domain acoustic wavefield satisfyingthe Helmholtz equation

� !2i

c2(x)p ��p = s (9)


Frequency-domain full waveform inversion : mathematical settings

Frequency-domain gradient

The gradient of the misfit function rf (ck) becomes

rf (ck) = � 2!2i

c3(x)

NfreqX

!i =1

p[s, c](x , !i )�p[s, c](x , !i ) (10)

where �p[s, c](x , !i ) is the adjoint wavefield solution of

� !2i

c2(x)�p ��p = R† (dcal [s, ck ]� d s

obs) (11)


Outline





Case study description: The Valhall oil field (Operto et al., 2015)

0

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X(km)

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Y (km)

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X(km

)

Shallow-water environment (70m of water).

Gas in the overburden, forming locally a gas cloud between 1km and 1.5km depth.

Over-pressured, under-saturated Upper Cretaceous chalk reservoir at 2.5km depth (Barkvedet al., 2010).

Anisotropy as high as 16%.

2302 hydrophones, 49,954 shots (50m spacing) processed in a reciprocal way during seismicmodeling.

Covered area: 16km ⇥ 9km = 145km2; Maximum depth of investigation 4.5 km.


FWI experimental setup and starting model

Ten successive mono-frequency inversions with grid refinement.

h(m) 70 50 35f (Hz) 3.5 4 4.5 5 5.2 5.8 6.4 7 8 10Nitmax 20 20 10

Table: h(m): grid interval. f (Hz): inverted frequencies. Nitmax : maximum number of iterations.

Seismic modeling with a visco-acoustic VTI FDFD method (Operto et al., 2014)

Preconditioned steepest-descent with SEISCOPE optimization toolbox (Metivier andBrossier, 2015).

Free-surface boundary condition on top of grid. Multiples & ghosts generated bymodeling.

Mono-parameter inversion for V0. �, ✏, QP = 200, ⇢ (Gardner law) kept fixed.

Source signature estimation alternated with velocity-model update at each FWIiteration.

FWI performed with MUMPS full rank and block low rank (" = 10�5, 10�4, 10�3).

Initial model built by VTI reflection traveltime tomography (courtesy of BP).


FWI results (Operto et al., 2015)

3

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1700180019002000m/s

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1500 2000 2500 3000m/s

z=175m

z=500m

z=1000m

x=5575m

x=6250m

x=6475m


HPC strategy

Each inversion iteration requires to solve more than 2⇥ 2000 wave equationproblems with the same operator and di↵erent right hand sides (sources)

The total number of inversion iterations reaches 200 which makes 8⇥ 105

wave propagation problems solution for one inversion

We would like to rely on e�cient strategies for multiple right hand sidesproblems

We should use a direct solver


HPC strategy

Each inversion iteration requires to solve more than 2⇥ 2000 wave equationproblems with the same operator and di↵erent right hand sides (sources)

The total number of inversion iterations reaches 200 which makes 8⇥ 105

wave propagation problems solution for one inversion

We would like to rely on e�cient strategies for multiple right hand sidesproblems

We should use a direct solver

However, we need to control the memory requirement of such a strategy. This isperformed through an appropriate choice of

1 the PDE to be solved

2 the discretization scheme which is used

3 the type of direct solver which is used


VTI acoustic wave equation for FDFD modeling (Operto et al., 2014)

Step 1: eliminate particle velocities from the 1st-order system (e.g. Duveneck and Bakker,2011) 8

<

:

!2

0ph + (1 + 2✏) (X + Y) ph +

p1 + 2�Z pv = !2 sh

0s,

!2

0pv +

p1 + 2� (X + Y) ph + Z pv = !2 sv

0s,

with X = @x b@x , Y = @y b@y and Z = @z b@z .

Step 2: Explicit expression of pv as a function of ph (first equation)

pv =1

p1 + 2�

ph +2(✏� �)0

!2p

1 + 2�(X + Y) ph +

„sv �

1p

1 + 2�sh

«s.

Step 3: Eliminate pv from the system for a 4th-order equation for ph

!2

2

6664

Nearly ellipticz }| {!2

0+ (1 + 2✏) (X + Y) +

p1 + 2�Z

1p

1 + 2�

3

7775ph +

Anellipticz }| {

2p

1 + 2�Z0(✏� �)p

1 + 2�(X + Y) ph = b

with b = !4 sh0

s � !2p

1 + 2�Z“sv � 1p

1+2�sh

”s.

Step 4: p = 1/3(2ph + pv ).


A compact mixed-grid stencil (Operto et al., 2007; Brossier et al., 2010;Operto et al., 2014)


MUMPS direct solver with block low rank compression (Weisbecker, 2013;Amestoy et al., 2015b,a)

Computer nodes: two 2.5GHz Intel Xeon IvyBridge E5-2670v2 CPUs. 10 cores/CPU. 64GB memory/node.

Frequency band (Hz) Grid dimensions npml #u #n #MPI #th #rhs3.5-5 66 x 130 x 230 8 2.9 12 24 10 46045.2-7 92 x 181 x 321 8 7.2 16 32 10 46048-10 131 x 258 x 458 4 17.4 34 68 10 4604

Table: #u(106): number of unknowns. #n: number of nodes. #MPI : number of MPI process. #th: number of threads per MPI process.

F(Hz)/h(m) " MLU(Gb) TLU(s) Ts(s) Tms(s) Tg (mn) #g

5Hz/70mFR 61(1.0) 96.2(1.0) 19 22

10�5 35(1.7) 51.2(1.9) 0.064 295 17 2210�4 30(2.0) 47.6(2.0) 18 2210�3 24(2.5) 42.8(2.2) 17 22

7Hz/50mFR 206(1.0) 381(1.0) 45 22

10�5 106(1.9) 168(2.26) 0.23 1058 42 2210�4 89(2.3) 145(2.6) 38 2210�3 70(2.9) 129(3.0) 40 22

10Hz/35mFR 712(1.0) 1298(1.0) 98 12

10�5 324(2.2) 541(2.4) 0.51 2348 87 1510�4 264(2.7) 488(2.7) 85 1510�3 209(3.4) 420(3.1) 84 11

Table: MLU (Gb): Memory for LU factors (Gbytes). TLU (s): Elapsed time for LU. Ts (s): Elapsed time for 1 solve. Tms (s): Elapsed time for 4604solves. Tg (mn): time for 1 gradient. #g : Number of gradients.


Outline





Sensitivity to the initial model

Numerical optimization point of view

Local optimization techniques for nonlinear minimization problem onlyconverge to the nearest local minimum (Nocedal and Wright, 2006)

Geophysics point of view

Sensitivity to the large scale variation of the velocity only in the Fresnel zone(Woodward, 1992; Operto et al., 2006)

Phase matching with a L2 norm only if the initial model predicts the datawithin half a phase: this is a cycle skipping issue (Jannane et al., 1989)


Di↵erent approaches to reduce the sensitivity to the initial model accuracy

Hierarchical (or multi-scale) approaches

Frequency and illumination through time windowing and o↵set selection(Pratt, 1999; Bunks et al., 1995; Sirgue and Pratt, 2004; Shipp and Singh, 2002;Wang and Rao, 2009; Brossier et al., 2009)

Hybridization with tomography methods

Increase the sensitivity to first-arrival time detrimental to the resolution (Luoand Schuster, 1991; Dahlen et al., 2000; Montelli et al., 2004; Tromp et al., 2005;Nolet, 2008)

Extended domain strategies

Kinematic control through the maximization of the energy refocusing in afictitious dimension introduced in the imaging condition (o↵set, angle,time-lag) (Symes and Kern, 1994; Sava and Biondi, 2004a,b; Sava and Fomel,2006; Symes, 2008; Biondi and Almomin, 2013)


Di↵erent approaches to reduce the sensitivity to the initial model accuracy

Hierarchical (or multi-scale) approaches

Frequency and illumination through time windowing and o↵set selection(Pratt, 1999; Bunks et al., 1995; Sirgue and Pratt, 2004; Shipp and Singh, 2002;Wang and Rao, 2009; Brossier et al., 2009)

Hybridization with tomography methods

Increase the sensitivity to first-arrival time detrimental to the resolution (Luoand Schuster, 1991; Dahlen et al., 2000; Montelli et al., 2004; Tromp et al., 2005;Nolet, 2008)

Extended domain strategies

Kinematic control through the maximization of the energy refocusing in afictitious dimension introduced in the imaging condition (o↵set, angle,time-lag) (Symes and Kern, 1994; Sava and Biondi, 2004a,b; Sava and Fomel,2006; Symes, 2008; Biondi and Almomin, 2013)

What we propose: using an optimal transport measure of the distance (akaEarth’s mover distance or Wasserstein distance) between observed and

predicted data


Wasserstein 1 distance between f and g

Primal formulation8

>

>

>

<

>

>

>

:

W 1(dcal , dobs) = minM

Z

x2X

kx �M(x)kdcal(x)dx ,

where 8A ⇢ X ,

Z

x2A

dobs(x)dx =

Z

M(x)2A

dcal(x)dx .

(12a)

(12b)



Primal formulation8

>

>

>

<

>

>

>

:

W 1(dcal , dobs) = minM

Z

x2X

kx �M(x)kdcal(x)dx ,

where 8A ⇢ X ,

Z

x2A

dobs(x)dx =

Z

M(x)2A

dcal(x)dx .

(12a)

(12b)

Dual equivalent formulation

W 1 (dcal , dobs) = max' 2Lip1

Z

x2X

'(x) (dcal(x)� dobs(x)) dx , (13)

where Lip1 is the space of 1-Lipschitz functions, such that

8(x , y) 2 X , |'(x)� '(y)| kx � yk1. (14)





Z

x2X

'(x) (dcal(x)� dobs(x)) dx , (12)


8(x , y) 2 X , |'(x)� '(y)| kx � yk1. (13)





Z

x2X

'(x) (dcal(x)� dobs(x)) dx , (12)


8(x , y) 2 X , |'(x)� '(y)| kx � yk1. (13)

Extension to non conservation of the mass and non positivity of the signals

fW 1 (dcal , dobs) = max' 2BLip1

Z

x2X

'(x) (dcal(x)� dobs(x)) dx . (14)

where BLip1 is the space of bounded 1-Lipschitz functions, such that

8(x , y) 2 X , |'(x)� '(y)| kx � yk1, 8x 2 X , |'(x)| < c (15)


Numerical computation of fW 1

Naive discretization: linear programming problem with O(N2) global constraints8

>

>

>

>

>

<

>

>

>

>

>

:

fW 1 (dcal [m], dobs) = max'

NrX

i=1

NtX

j=1

'ij ((dcal [m])ij � (dobs)ij)�t�xr

8(i , j), |'ij | < c,

8(i , j), (k, l) |'ij � 'kl | < |(xr )i � (xr )k | + |tj � tl | .

(16a)

(16b)

(16c)



Nonsmooth concave maximization problem

W 1 (dcal [m], dobs) = max'

NrX

i=1

NtX

j=1

'ij ((dcal [m])ij � (dobs)ij)�t�xr�iK (A'), (16)



Nonsmooth concave maximization problem

W 1 (dcal [m], dobs) = max'

NrX

i=1

NtX

j=1

'ij ((dcal [m])ij � (dobs)ij)�t�xr�iK (A'), (16)

where K is the unit hypercube of R3N

K =n

x 2 R3N , |xi | 1, i = 1, . . . 3No

. (17)

the indicator function of K is denoted by iK ,

iK (x) =

˛

˛

˛

˛

0 if x 2 K+1 if x /2 K .

(18)

and the matrix A represents the constraints

(A')k =

˛

˛

˛

˛

˛

˛

˛

˛

˛

'k

cfor k = 1, . . . , N

'k+1 � 'k

�xrfor k = N + 1, . . . , 2⇥ N

'k+Nr � 'k

�xtfor k = 2⇥ N + 1, . . . , 3⇥ N.

(19)



Proximal splitting algorithm

We use the SDMM method (Simultaneous Descent Method Multipliers,Combettes and Pesquet (2011)) to solve the maximization problem

This iterative method requires the solution at each iteration of a linearsystem involving the matrix

Q = I + ATA, (16)

which can be proved to be equivalent to

Q = 2I �� (17)

where � is a second-order finite di↵erence discretized Laplace operator withNeumann boundary conditions



Proximal splitting algorithm

We use the SDMM method (Simultaneous Descent Method Multipliers,Combettes and Pesquet (2011)) to solve the maximization problem

This iterative method requires the solution at each iteration of a linearsystem involving the matrix

Q = I + ATA, (16)

which can be proved to be equivalent to

Q = 2I �� (17)

where � is a second-order finite di↵erence discretized Laplace operator withNeumann boundary conditions

A Poisson equation must be solved at each iteration of the SDMM algorithm.FFT based solvers (O(N logN), Swarztrauber (1974)) or multigrid based solvers(O(N), Adams (1993)) can be used.


Example of application: the 2D Marmousi model


Marmousi 2 iterative reconstruction using a L2 distance


Marmousi 2 iterative reconstruction using the optimal transport distance


Conclusion and perspectives

A direct solver strategy amenable for moderate size 3D acoustic problems

Frequency-domain strategies with sparse direct solvers are amenable for moderatesize 3D acoustic problems using appropriate choice of PDE, discretized scheme,and solver

Low rank compression will help us push up the limit of this strategy (Amestoyet al., 2015a), however 3D elastodynamics still seems out of reach: time-domainmethods are still preferred for this problem

Beyond the L2 distance: mitigating the sensitivity to the initial model accuracy

Standard FWI still relies on the assumption that the initial model is accurateenough: the sensitivity to the accuracy of the initial model could be mitigated bythe choice of an optimal transport distance instead of the standard L2 norm(Metivier et al., 2016)


Acknowledgments

Thank you for your attention

National HPC facilities of GENCI-IDRIS under grant Grant 046091

Local HPC facilities of CIMENT-SCCI (Univ. Grenoble)

SEICOPE sponsors : http://seiscope2.osug.fr


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