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High-order schemes in numeriical problems of seismic exploration in the Arctic D. I Petrov, P.V. Stognii, N. I. Khokhlov Laboratory of Applied Computational Geophysics, Moscow Institute of Physics and Technologies 1 Dolgoprudny, 2016

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High-order schemes in numeriical problems of seismic exploration in the Arctic

  • D. I Petrov, P.V. Stognii, N. I. Khokhlov

Laboratory of Applied Computational Geophysics, Moscow Institute of Physics and Technologies

1 Dolgoprudny, 2016

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Contents

  • The aim of study
  • Mathematical model of medium
  • Numerical method
  • Obtained results 2D and 3D
  • Conclusion
  • Further research

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Aims of Study

  • Modeling of wave propagation in

elastic media by grid-charactersitic method.

  • Correct definition and calculation of

boundary and interface conditions

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Mathematical model Elastic medium

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 

т tv

  σ

   

 

т t

v v v            σ I

Components of vector of velocity and components

  • f stress tension describing the state of linear-elastic

medium are the solutions of the following system of equations:

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Mathematical model Acoustic medium

For numerical modeling of sea water we use the prefect fluid approximation, solve acoustic wave equation and find components of vector of velocity and pressure.

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v p t     

2 (

) p с v t      

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Grid-charactristic method

Method for solving hypergolic systems of equations. We use it for solving both acoustic and elastic wave equations. In 2D-case these systems could be written in the following form

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2 2 2 2 2 1 2 1 2 e e e e e

q q q t x x          A A

2e

q

  • vector of unknown fields
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Grid-characteristic method

We use splitting on spatial directions and obtain 2 systems of equations

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2 2 2

=

e e e j j

q q t x     A

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Grid-characteristic method

Both of these systems:

  • is hyperbolic
  • obtains 5 real eigenvalues
  • So we can write it in the following form

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 

2 2 1 2 2 2

=

e e e e e j j j j

q q t x

    Ω Λ Ω

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Grid-characteristic method

Change of unknown fields:

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All of obtained systems becomes the system of 5 independant transport equations:

2 2 2

= 0

e e e

p p t x      Λ

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Grid-characteristic method

Then one can find the solution of the given system

  • f equations:

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2D Model

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  • Spatial step 0.2 м
  • Time step
  • 15 000 time steps.
  • Region for integration 1200 х 600 м
  • System “ice-water-ground-carbon reservoir-

ground

  • Absorbing conditions at the sides and at the

bottom of the region

  • Free boundary condition on the top side of

the region

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  • 1. Sources in the water

and at the seabed, the case without ice

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Problem definitions

Source in the water Source in the water, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

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Wave patterns

Source in the water Source in the water, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

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Seismograms, receivers in the water, V

Source in the water Source in the water, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

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Seismograms, receivers in the water, Vy

Source in the water Source in the water, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

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Seismograms, receivers at the seabed, V

Source in the water Source in the water, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

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Seismograms, receivers at the seabed, Vx

Source in the water Source in the water, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

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Seismograms, receivers at the seabed, Vy

Source in the water Source in the water, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

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  • 2. Sources in the ice

and at the seabed, the case with ice

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Source in the ice Source in the ice, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

Problem definitions

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Source in the ice Source in the ice, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

Wave patterns

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Source in the ice Source in the ice, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

Seismograms, receivers in the ice, V

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Source in the ice Source in the ice, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

Seismograms, receivers in the ice, Vx

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Source in the ice Source in the ice, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

Seismograms, receivers in the ice, Vy

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Source in the ice Source in the ice, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

Seismograms, receivers at the seabed, V

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Source in the ice Source in the ice, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

Seismograms, receivers at the seabed, Vx

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Source in the ice Source in the ice, without carbon reservoir

Source at the seabed Source at the seabed, without carbon reservoir

Seismograms, receivers at the seabed, Vy

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  • 3. Influence of ice.

Sources in the ice and in the water.

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Source in the ice Source in the ice, without carbon reservoir Source in the water Source in the water, without carbon reservoir

Problem definitions

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Wave patterns

Source in the ice Source in the ice, without carbon reservoir Source in the water Source in the water, without carbon reservoir

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Seismograms, receivers in the water/ice, V

Source in the ice Source in the ice, without carbon reservoir Source in the water Source in the water, without carbon reservoir

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Seismograms, receivers in the water/ice, Vx

Source in the ice Source in the ice, without carbon reservoir Source in the water Source in the water, without carbon reservoir

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Seismograms, receivers in the water/ice, Vy

Source in the ice Source in the ice, without carbon reservoir Source in the water Source in the water, without carbon reservoir

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Seismograms, receivers at the seabed, V

Source in the ice Source in the ice, without carbon reservoir Source in the water Source in the water, without carbon reservoir

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Seismograms, receivers at the seabed, Vx

Source in the ice Source in the ice, without carbon reservoir Source in the water Source in the water, without carbon reservoir

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Seismograms, receivers at the seabed, Vy

Source in the ice Source in the ice, without carbon reservoir Source in the water Source in the water, without carbon reservoir

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  • 4. Influence of ice.

Sources at the seabed.

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With ice With ice, without carbon reservoir Without ice Without ice, without carbon reservoir

Problem definitions

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Wave patterns

With ice With ice, without carbon reservoir Without ice Without ice, without carbon reservoir

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Wave patterns

With ice With ice, without carbon reservoir Without ice Without ice, without carbon reservoir

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Seismograms, receivers in the water/ice, V

With ice With ice, without carbon reservoir Without ice Without ice, without carbon reservoir

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Seismograms, receivers in the water/ice, Vx

With ice With ice, without carbon reservoir Without ice Without ice, without carbon reservoir

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Seismograms, receivers in the water/ice, Vy

With ice With ice, without carbon reservoir Without ice Without ice, without carbon reservoir

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Seismograms, receivers at the seabed, V

With ice With ice, without carbon reservoir Without ice Without ice, without carbon reservoir

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Seismograms, receivers at the seabed, Vx

With ice With ice, without carbon reservoir Without ice Without ice, without carbon reservoir

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Seismograms, receivers at the seabed, Vy

With ice With ice, without carbon reservoir Without ice Without ice, without carbon reservoir

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Iceberg under explosion

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3D Model

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Wave propagation in the ice

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Wave propagation in the water

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Wave propagation in the reservoir

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Wave propagation in the ground

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Conclusions

  • Numerical modeling of wave

processes in the elastic and acoustic media was done

  • We solve problems of seismic

exploration in the Arctic shelf.

  • We made synthetic seismograms.
  • We study wave propagation in the

icebergs under explosion.

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Further research: immerse interface method

Problem: loss of precision at the boundaries Possible solution (Chaoming Zhang, Randall J. LeVeque, Charles S. Peskin, Xin Wen, Shi Jin, and

  • thers):
  • “crop” the boundary and the nearest nodes
  • apply local scheme in the “cropped” area, taking

into consideration its features

  • smooth the results obtained in the inner and
  • uter grids.

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Thank you for your attention!

Petrov Dmitry Igorevich diapetr@gmail.com Stognii Polina Vladimirovna Khokhlov Nikolay Igorevich Laboratory of Applied Computational Geophysics (MIPT)

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