efficient space and time discretization of the wave equation ......e cient space and time...

Efficient Space and Time Discretization of theWave Equation. Application to the Reverse Time

Migration

J. Diaz1,2

In collaboration with C. Baldassari1,2, H. Barucq1,2,H. Calandra3, B. Denel3

(1) INRIA Bordeaux Research Center, Project Team Magique3D(2) LMA, CNRS UMR 5142, Universite de Pau(3) TOTAL

08/06/2010

Outline

1 What is the Reverse Time Migration?

2 Appropriate space discretization of the wave equation

3 Improvement of the time discretization of the wave equation

The seismic reflection

Oil exploration by seismic reflection :

Why? Have a good image of the sub-surface andconsequently know where there is oil or not.

How? Seismic acquisition campaigns.

Seismic acquisition : principle

Seismic Imaging (summary)

1 Get an initial velocity model from acquisition campain:

Tomography step

2 Using same sources and receivers as for tomography

Solve the waveequation twice foreach source

(a) Propagate the sources into the velocitymodel

(b) Retropropagate the recorded waves(reverse time)

3 Cross-correlation of (a) and (b): imaging condition(Claerbout)

Produce an image of ground

4 Compare with velocity model: if different, go to 2 aftermodification of the velocity model.

Seismic Imaging (example)

Seismogram from theacquisition campaign


Seismogram from theacquisition campaign

=⇒

Initial guess


Computational Domain


Computational Domain

=⇒

RTM Image

Numerical methods for solving the wave equation

∂2u

∂t2− c2∆u = f (t, x), (t, x) ∈ [0 ; T ]× Ω

• Finite difference methods: fast but not really effective inheterogeneous medium.

• Finite element methods with the choice criteria:

• Efficient to take the topography effects into account;

• Optimized computational burden;

• Limitation of the dispersion effects.


M∂2U

∂t2+ KU = F

• Finite difference methods: fast but not really effective inheterogeneous medium.

• Finite element methods with the choice criteria:

• Efficient to take the topography effects into account;

• Optimized computational burden;

• Limitation of the dispersion effects.


Most popular Finite Element Method (FEM) for geophysicalapplications:

Spectral Element Method (SEM) (Seriani, Priolo, Cohen, Komatitsch...)

• Explicit scheme (diagonal mass matrix)

• In general, FEM =⇒ implicit scheme;

• Combined with a lumping process: the order of convergence isdestroyed except for order one or SEM.

Numerical methods for solving the full-wave equation

Discontinuous Galerkin Method (DG)

• Quasi-explicit scheme: mass matrix is block-diagonal;

• Based on tetrahedras (easy to handle);

• Adapted to strongly varying velocities;

• Fast computational methods:

• Volume calculi are made locally;

• Communications between elements thanks to conditions on theedges only ((N − 1)D calculi);

• High-order elements to overcome dispersion effects.

SEM versus DG

1 The SEM method• Gauss Lobatto quadrature rule : diagonal mass matrix, do not

hamper the order of convergence.• Meshes made of quadrangles in 2D or hexahedra in 3D :

difficult to compute, not always suitable to complextopographies.

2 The DG method• Representation of the solution is quasi-explicit because the

mass matrix is block-diagonal without any approximation.• To compute easily its coefficients, we use an exact quadrature

formula which does not hamper the order of convergence.• Meshes made of triangles in 2D or tetrahedra in 3D. Thus the

topography of the computational domain is easily discretized.• Handle polynomial velocities inside each element.

Interior Penalty Discontinuous Galerkin Method (IPDG)

• V hl :=

v ∈ L2 (Ω) : v|K ∈ Pl (K ) ∀K ∈ Th

• M is block-diagonal

• Kij =∑K∈Th

∫K

1

ρ∇v i · ∇v j dx −

∑F∈F

∫F

[[ v j ]] · 1

ρ∇v i dF

-∑F∈F

∫F

[[ v i ]] · 1

ρ∇v j dF +

∑F∈F

∫F

γ[[ v i ]] · [[ v j ]] dF

(Ainswoth et al., Grote et al.)


• V hl :=

v ∈ L2 (Ω) : v|K ∈ Pl (K ) ∀K ∈ Th


• Kij =∑K∈Th

∫K

1


∑F∈F

∫F

[[ v j ]] · 1

ρ∇v i dF

-∑F∈F

∫F

[[ v i ]] · 1

ρ∇v j dF

+∑F∈F

∫F

γ[[ v i ]] · [[ v j ]] dF



• V hl :=

v ∈ L2 (Ω) : v|K ∈ Pl (K ) ∀K ∈ Th


• Kij =∑K∈Th

∫K

1


∑F∈F

∫F

[[ v j ]] · 1

ρ∇v i dF

-∑F∈F

∫F

[[ v i ]] · 1

ρ∇v j dF +

∑F∈F

∫F

γ[[ v i ]] · [[ v j ]] dF


“Drawbacks” of the DG methods

• The solution to the wave equation is not discontinuous.

The solution is neither piecewise polynomial. Actually, thediscontinuities of the approximate solution are controlled bythe penalization parameter and they do not hamper theaccuracy of the method.

• DG requires more degrees of freedom than SEM.

That is true for a given mesh. But there is no reason to usethe same mesh for both methods.

• The stability condition (which restricts the time space of thescheme) is smaller for DG than for SEM.

That is true for a given mesh.

What are the good criteria to compare IPDG and SEM?

• Compare the computational burden for a given accuracy.

• Compare the accuracy for a given computational burden.

The computation cost of one iteration is directly related tothe number of degrees of freedom.=⇒ compare the accuracy for a given number of degrees of

freedom



• Compare the accuracy for a given computational burden.The computation cost of one iteration is directly related tothe number of degrees of freedom.

=⇒ compare the accuracy for a given number of degrees offreedom



• Compare the accuracy for a given computational burden.The computation cost of one iteration is directly related tothe number of degrees of freedom.=⇒ compare the accuracy for a given number of degrees of

freedom

Comparison IPDG versus SEM

To obtain a given accuracy:

• The number of degrees of freedom and the number ofmultiplications required by IPDG are (slightly) smaller.

• The stability condition of IPDG is higher (and the number ofiterations is smaller).

IPDG performs as well as (and sometimes better than) SEM.

A more realistic experiment

Improvements of the method

• Use P1 polynomials in the fine mesh and P3 polynomials inthe coarse mesh (p-adaptivity)


• Use P1 polynomials in the fine mesh and P3 polynomials inthe coarse mesh

• Use a local time stepping strategy

• Use a second order time scheme in the fine mesh and a fourthorder time scheme in the coarse mesh.

Local Time Stepping : Bibliography

POems Team: Becache, Collino, Fouquet, Joly, Rodrıguez

• Conservation of energy

• Optimal stability condition

• Requires the introduction of a Lagrange Multiplier

• Implicit scheme on the interface

Piperno

• First-order Maxwell system



• Explicit or implicit scheme on the interface


ADER Schemes, Kaser, Dumbser et al.

• High-order Explicit Time Schemes ;

• First order Systems ;

• No energy conservation ;

• Difficult to implement (one time step by element).


Hairer, Lubich and Wanner (2002), Leimkuhler and Reich (2004) :Local Time Stepping for ODE’s (second order scheme)

Diaz, Grote (2009) : High-Order Local Time Stepping for theWave Equation.



Local Time Stepping

Conclusions

• The performances of IPDG and SEM are similar with highorder polynomials in 2D.

• It is much easier to use elements of various order with IPDG.

• The local time stepping strategy does not hamper theaccuracy of the method.

• Comparison of IPDG and SEM in 3D.

• Extension to elastodynamics.

Time discretization of the wave equation

Md2U

dt2+ KU = 0

We consider space discretization methods such that M and K aresymmetric positive matrices and M is (block-)diagonal (FEM withmass lumping or DG methods).

Classical Leap Frog Scheme:

Y (t + ∆t)− 2Y (t) + Y (t −∆t)

∆t2= + O(∆t2)

Energy Conservation

En+ 12 =

⟨Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩+⟨AY n+1,Y n

⟩


M12d2U

dt2+ M−

12KM−

12︸︷︷︸

A

M12U︸︷︷︸Y

= 0


Y (t + ∆t)− 2Y (t) + Y (t −∆t)

∆t2= + O(∆t2)

Energy Conservation

En+ 12 =

⟨Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩+⟨AY n+1,Y n

⟩


d2Y

dt2+ AY = 0


Y (t + ∆t)− 2Y (t) + Y (t −∆t)

∆t2= + O(∆t2)

Energy Conservation

En+ 12 =

⟨Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩+⟨AY n+1,Y n

⟩


d2Y

dt2+ AY = 0


Y (t + ∆t)− 2Y (t) + Y (t −∆t)

∆t2=

d2Y

dt2(t) + O(∆t2)

Energy Conservation

En+ 12 =

⟨Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩+⟨AY n+1,Y n

⟩


d2Y

dt2+ AY = 0


Y (t + ∆t)− 2Y (t) + Y (t −∆t)

∆t2= AY (t) + O(∆t2)

Energy Conservation

En+ 12 =

⟨Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩+⟨AY n+1,Y n

⟩


d2Y

dt2+ AY = 0


Y n+1 − 2Y n + Y n−1

∆t2= −AY n.

Energy Conservation

En+ 12 =

⟨Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩+⟨AY n+1,Y n

⟩

Time discretization of the wave equationd2Y

dt2+ AY = 0


Y n+1 − 2Y n + Y n−1

∆t2= −AY n.

Energy Conservation

En+ 12 =

⟨(I − ∆t2

4A

)Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨AY n+1 + Y n

2,Y n+1 + Y n

2

⟩


Energy Conservation

En+ 12 =

⟨(I − ∆t2

4A

)Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨AY n+1 + Y n

2,Y n+1 + Y n

2

⟩CFL Condition

The scheme is stable, if and only if :

I − ∆t2

4A and A are symmetric positive


Energy Conservation

En+ 12 =

⟨(I − ∆t2

4A

)Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨AY n+1 + Y n

2,Y n+1 + Y n

2

⟩CFL Condition

The scheme is stable, if and only if :

I − ∆t2

4A and A are symmetric positive

0 ≤ λA ≤4

∆t2


Energy Conservation

En+ 12 =

⟨(I − ∆t2

4A

)Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨AY n+1 + Y n

2,Y n+1 + Y n

2

⟩CFL Condition

The scheme is stable under the CFL condition :

∆t ≤ αLFh


Energy Conservation

En+ 12 =

⟨(I − ∆t2

4A

)Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨AY n+1 + Y n

2,Y n+1 + Y n

2

⟩CFL Condition

We want the new scheme to satisfy:

∆tcoarse ≤ αLFhcoarse and ∆tfine ≤ αLFh

fine

≈ αLFhcoarse/p


Energy Conservation

En+ 12 =

⟨(I − ∆t2

4A

)Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨AY n+1 + Y n

2,Y n+1 + Y n

2

⟩CFL Condition

We want the new scheme to satisfy:

∆tcoarse ≤ αLFhcoarse and ∆tfine ≤ αLFh

fine ≈ αLFhcoarse/p

Higher Order Schemes, Global Time Stepping

d2Y

dt2+ AY = 0

Modified Equation Scheme:.

Energy Conservation

En+ 12 =

⟨(I − ∆t2

4

(A− ∆t2

12A2

))Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨(A− ∆t2

12A2

)Y n+1 + Y n

2,Y n+1 + Y n

2

⟩


d2Y

dt2+ AY = 0

Modified Equation Scheme:

Y (t + ∆t)− 2Y (t) + Y (t −∆t)

∆t2=

d2Y

dt2(t) +

∆t2

12

d4Y

dt4(t) + O(∆t4).

Energy Conservation

En+ 12 =

⟨(I − ∆t2

4

(A− ∆t2

12A2

))Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨(A− ∆t2

12A2

)Y n+1 + Y n

2,Y n+1 + Y n

2

⟩


d2Y

dt2+ AY = 0


Y (t + ∆t)− 2Y (t) + Y (t −∆t)

∆t2= AY (t) +

∆t2

12Ad2Y

dt2(t) + O(∆t4).

Energy Conservation

En+ 12 =

⟨(I − ∆t2

4

(A− ∆t2

12A2

))Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨(A− ∆t2

12A2

)Y n+1 + Y n

2,Y n+1 + Y n

2

⟩


d2Y

dt2+ AY = 0


Y (t + ∆t)− 2Y (t) + Y (t −∆t)

∆t2= AY (t) +

∆t2

12A2Y (t) + O(∆t4).

Energy Conservation

En+ 12 =

⟨(I − ∆t2

4

(A− ∆t2

12A2

))Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨(A− ∆t2

12A2

)Y n+1 + Y n

2,Y n+1 + Y n

2

⟩


d2Y

dt2+ AY = 0


Y n+1 − 2Y n + Y n−1

∆t2= −AY n +

∆t2

12A2Y n.

Energy Conservation

En+ 12 =

⟨(I − ∆t2

4

(A− ∆t2

12A2

))Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨(A− ∆t2

12A2

)Y n+1 + Y n

2,Y n+1 + Y n

2

⟩


Energy Conservation

En+ 12 =

⟨(I − ∆t2

4

(A− ∆t2

12A2

))Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨(A− ∆t2

12A2

)Y n+1 + Y n

2,Y n+1 + Y n

2

⟩

CFL Condition

The scheme is stable, if and only if

I − ∆t2

4

(A− ∆t2

12A2

)and A− ∆t2

12A2 are symmetric positive


Energy Conservation

En+ 12 =

⟨(I − ∆t2

4

(A− ∆t2

12A2

))Y n+1 − Y n

∆t,Y n+1 − Y n

∆t

⟩

+

⟨(A− ∆t2

12A2

)Y n+1 + Y n

2,Y n+1 + Y n

2

⟩

CFL Condition

The scheme is stable under the CFL condition

∆t ≤ αMEh =√

3αLFh


CFL Condition

The scheme is stable under the CFL condition

∆t ≤ αMEh =√

3αLFh


CFL Condition

We want the new scheme to satisfy

∆tcoarse ≤ αMEhcoarse and ∆tfine ≤ αMEh

fine

≈ αMEhcoarse/p


CFL Condition

We want the new scheme to satisfy

∆tcoarse ≤ αMEhcoarse and ∆tfine ≤ αMEh

fine ≈ αMEhcoarse/p

Auxiliary Function

At each time step n we define an auxiliary function

Qn(τ) =Y (n∆t − τ) + Y (n∆t + τ)

2

for τ ∈ [−∆t ; ∆t].

This function is obviously even and satisfy:d2Qn

dτ2(τ) = −AQn(τ),

Qn(0) = Y (n∆t),dQn

dτ(0) = 0,

After having solved this equation, Y ((n + 1)∆t) can be computedusing Y ((n + 1)∆t) = −Y ((n − 1)∆t) + 2Qn(∆t)

Two different ways to solve the auxiliary equation

First Way

Solve d2Qn

dτ2(τ) = −AQn(τ),


dτ(0) = 0,

by a fourth order modified equation scheme of time step ∆t/p andcompute Y ((n + 1)∆t) = −Y ((n − 1)∆t) + 2Qn(∆t).

Remark

Is is equivalent to solve the original equation by a fourth ordermodified equation scheme of time step ∆t/p.

Two different ways to solve the auxiliary equation

Second Way

Use a fourth order approximation of AQn(τ):

AQn(τ) ≈ AQn(0) +τ2

2Ad2Qn(0)

dτ2= AY (n∆t)− τ2

2AAY (n∆t),

then solved2Qn

dτ2(τ) = −AY (n∆t) +

τ2

2AAY (n∆t),


dτ(0) = 0,

by a fourth order modified equation scheme of time step ∆t/pand compute Y ((n + 1)∆t) = −Y ((n − 1)∆t) + 2Qn(∆t).

Remark

Is is equivalent to solve the original equation by a fourth ordermodified equation scheme of time step ∆t, whatever is p.

Local Time-Stepping

d2Qn

dτ2+ AQn = 0

Local Time-Stepping

d2Qn

dτ2+ AQn = 0

Let us now split Qn in two parts :

Qn =

[Qcoarse

n

Qfinen

]

Local Time-Stepping

d2Qn

dτ2+ AQn = 0


Qn =

[Qcoarse

n

0

]+

[0

Qfinen

]

Local Time-Stepping

d2Qn

dτ2+ AQn = 0


Qn =

[Qcoarse

n

0

]+

[0

Qfinen

]= (I − P)Qn + PQn, with P2 = P

Local Time-Stepping

d2Qn

dτ2+ A(I − P)Qn + APQn = 0


Qn =

[Qcoarse

n

0

]+

[0

Qfinen

]= (I − P)Qn + PQn, with P2 = P

Local Time-Stepping

d2Qn


Idea

Approximate only A(I − P)Qn(τ) by

A(I − P)Qn(τ) ≈ A(I − P)Qn(0) +τ2

2A(I − P)

d2Qn(0)

dτ2

Local Time-Stepping

d2Qn


Idea


A(I − P)Qn(τ) ≈ A(I − P)Y (n∆t)− τ2

2A(I − P)AY (n∆t)

Local Time-Stepping

d2Qn


Idea


A(I − P)Qn(τ) ≈ A(I − P)Y (n∆t)− τ2

2A(I − P)AY (n∆t)

So that Qn is the solution to∣∣∣∣∣∣∣∣∣d2Qn(τ)

dτ2+ A(I − P)Y (t)− τ2

2A(I − P)AY (n∆t) + APQn(τ) = 0

Qn(0) = Y (t)

Q ′n(0) = 0

Algorithm of the fourth order local time-stepping scheme

Computation of Q(∆t)

We solve

∣∣∣∣∣∣∣∣∣d2

dτ2Qn(τ) + A(I − P)Y n − τ2

2A(I − P)AY n + APQn(τ) = 0

Qn(0) = Y n

Q ′n(0) = 0from τ = 0 to τ = ∆t, using a fourth order Modified Equation Scheme

with a time step∆t

p.

Algorithm of the fourth order local time-stepping scheme

Computation of Q(∆t)

Q0n = Y n

V 1 = −A(I − P)Y n − APQ0n = −AY n

V 2 = A(I − P)AY n − APV 1

Q1pn = Q0

n +∆t2

2p2V 1 +

∆t4

24p4V 2

For i = 1..p − 1

V 1 = −A(I − P)Y n + 12

(i∆tp

)2A(I − P)AY n − APQ

ipn

V 2 = A(I − P)AY n − APV 1

Qi+1p

n = 2Qipn − Q

i−1p

n +∆t2

p2V 1 +

∆t4

12p4V 2

Endfor

efficient space and time discretization of the wave equation ......e cient space and time...

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