Coupling of the Discontinuous Galerkin and Finite Difference techniques to simulate seismic waves in the presence of sharp

interfaces

J. Diaz, D. Kolukhin, J. Diaz, D. Kolukhin, V. LisitsaV. Lisitsa, V. , V. TcheverdaTcheverda

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Motivation for Motivation for mathematiciansmathematicians

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σ = 1.38, I = 44.9 м

Free-surface perturbation

Motivation for Motivation for mathematiciansmathematicians

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σ = 1.38, I = 44.9 м

N

jjXN

XRMS1

21)(

Free-surface perturbation

30%

Motivation for Motivation for geophysicistsgeophysicists

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x (m)

z (m

)

vs (m/s)

0.7 0.8 0.9 1 1.1 1.2 1.3

x 104

-500

0

500

1000

1500

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500

1000

1500

2000

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3000

3500

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x (m)

z (m)v

s (m/s)

0.7 0.8 0.9 1 1.1 1.2 1.3

x 104

-500

0

500

1000

1500

2000

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500

1000

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3000

3500

Motivation for geophysicistsMotivation for geophysicists

Motivation for Motivation for geophysicistsgeophysicists

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x (m)

z (m

)

ux, t = 0.55s

0.7 0.8 0.9 1 1.1 1.2 1.3

x 104

-500

0

500

1000

1500

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2500

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-13

Original source

Motivation for Motivation for geophysicistsgeophysicists

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x (m)

z (m

)

ux, t = 1.75s

0.7 0.8 0.9 1 1.1 1.2 1.3

x 104

-500

0

500

1000

1500

2000

2500

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

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1x 10

-13

Diffraction of Rayleigh wave, secondary sources

MotivationMotivation

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x (m)

time

(s)

uz

0.7 0.8 0.9 1 1.1 1.2 1.3

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Standard staggered grid Standard staggered grid schemescheme

9

.,2

1,

tC

tuu

tt

u T

zz

yy

xx

66

55

44

332313

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c

c

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yz

xzxy

zz

yy

xx

yz

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Standard staggered grid Standard staggered grid schemescheme

10

.,2

1,

tC

tuu

tt

u T

• Easy to implement• Able to handle complex models• High computational efficiency• Suitable accuracy• Poor approximation of sharp

interfaces

Discontinuous Galerkin Discontinuous Galerkin methodmethod

Elastic wave equation in Cartesian coordinates:

dim

1

*

0

0

0

0

j

u

jj

j

f

fu

xB

Bu

tS

I

FVGt

VA

)(

Discontinuous Galerkin Discontinuous Galerkin methodmethod

FVGt

VA

)(

kkkk DDDD

dxWFdsWnVGdxVWGdxWt

VA

])([)]([

Discontinuous Galerkin Discontinuous Galerkin methodmethod

kkkk DDDD

dxWFdsWnVGdxVWGdxWt

VA

])([)]([

dim

1

)(j

inoutjj VVnBnVG

Discontinuous Galerkin Discontinuous Galerkin methodmethod

kkkk DDDD

dxWFdsWnVGdxVWGdxWt

VA

])([)]([

• Use of polyhedral meshes• Accurate description of sharp

interfaces• Hard to implement for complex

models• Computationally intense• Strong stability restrictions (low

Courant numbers)

Dispersion analysis Dispersion analysis (P1)(P1)

Courant ratio 0.25

Dispersion analysis Dispersion analysis (P2)(P2)

Courant ratio 0.144

Dispersion analysis Dispersion analysis (P3)(P3)

Courant ratio 0.09

DG + FDDG + FD

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Finite differences:•Easy to implement•Able to handle complex models•High computational efficiency•Suitable accuracy•Poor approximation of sharp interfaces

Discontinuous Galerkin method:•Use of polyhedral meshes•Accurate description of sharp interfaces•Hard to implement for complex models•Computationally intense•Strong stability restrictions (low Courant numbers)

A sketchA sketch

P1-P3 DG on irregular triangular grid to match free-surface topography

P0 DG on regular rectangular grid = conventional (non-staggered grid scheme) – transition zone

Standard staggered grid scheme

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ExperimentsExperiments

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PPW Reflection

15 ~3 %

30 ~0.5 %

60 ~0.1 %

120 ??? %

DG

FD+DG on rectangular FD+DG on rectangular gridgrid

P0 DG on regular rectangular grid

Standard staggered grid scheme

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Spurious ModesSpurious Modes2D example in Cartesian coordinates

Spurious ModesSpurious Modes2D example in Cartesian coordinates

InterfaceInterfaceIncident waves Reflected waves

Transmitted artificial waves

Transmitted true waves

Conjugation Conjugation conditionsconditions

Incident waves Reflected waves

Transmitted artificial waves

Transmitted true waves

ExperimentsExperiments

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PPW Reflection

15 1.6 %

30 0.5 %

60 0.1 %

120 0.03 %

Numerical Numerical experimentsexperiments

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P

S

Surface Xs=4000, Zs=110 (10

meters below free surface), volumetric source, freq=30HzZr=5 meters below free surfaceVertical component is presented

Source

Comparison with FDComparison with FD

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DG P1 h=2.5 m.FD h=2.5 m.

The same amplitude normalizationNumerical diffraction

Comparison with FDComparison with FD

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DG P1 h=2.5 m.FD h=1m.

The same amplitude normalization Numerical diffraction

Numerical Numerical experimentsexperiments

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Xs=4500, Zs= 5 meters below free surface, volumetric source, freq=20HzZr=5 meters below free surface

Numerical Numerical ExperimentsExperiments

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Numerical Experiment – Sea Numerical Experiment – Sea

BedBed

Source position x=12,500 m, z=5 m Ricker pulse with central frequency of 10 Hz Receivers were placed at the seabed.

Numerical ExperimentsNumerical Experiments

ConclusionsConclusions

• Discontinuous Galerkin method allows properly handling wave interaction with sharp interfaces, but it is computationally intense

• Finite differences are computationally efficient but cause high diffractions because of stair-step approximation of the interfaces.

• The algorithm based on the use of the DG in the upper part of the model and FD in the deeper part allows properly treating the free surface topography but preserves the efficiency of FD simulation.

Thank you Thank you for attentionfor attention

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