Elastic and Acoustic Wave Phenomena

by

Phil BordingMemorial University of Newfoundland

Swastika Sharma and Steve ButtDalhousie University

SIAM GEOSCIENCES - 2007

Acoustic Wave Equations1D

2D

3D

2.5D

Source – Causality ?

Uniform Grid

Nyquist Criteria

• Two grid points – support wave without aliasing

• Dominate frequency at source

• Does the earth act as a filter?• Yes, deeper has higher velocities and lower

frequencies

Stability Limit

Courant – Nifty Mathematical Fellow– had a condition!

Grid points per Wave length

• Wave length = Velocity/Frequency• Nyquist requires 2 points to support ..

– Using at least two points we can reconstruct the waves without aliasing

– Modeling requires more than 2 to reduce the error in dispersion, numerical error

Kirchhoff PSDM 100% Velocity - inline 300

(Lazarevic, 2004)

Horizons approximated by stair steps!

Faults

0

1

2

3

4

Marfurt, 2004

So just how many Grid Points per Wavelength

do we really need???

Nyquist says………. 2

We can image to…. ¼ Lambda 4

Dispersion error requires too many…. 10+

So just how many Grid Points per Wavelength

do we really need???

The difference operators control everything!

Error = D(spatial) – D(time)

Taylor Series -- Weights(T=0)Optimal Weights – Weights(T=t)Spectral Methods -- D(spatial) = 0Frequency Domain -- D(time) = 0

Complexity – how long will it run??

n(x,y,z) are the spatial grid sizesn(t) is the number of time stepsw(s) is the work per stepn(s) is the number of shots

Realistic Models

Early Elastic Modeling

• Amoco Group -- Kelly, Alford, and Treitel

Elastic Staggered Grid

Alan Levander – Rice

P-Sv waves with higher Poisson’s ratio

Elastic P-S Modeling

Levander

Elastic - Continued

Levander

Elastic Wave Equation

Levander, 1988

Elastic today

• 4th order staggered grid - Veiiux• Topography Surface - Hestholm

• Slow shear waves dominate the grid sizing (spacing is small)

• 3D is at least 10X more computing• Visco-elastic schemes in development –

another 10X more? -- Aldridge at Sandia

Imaging Machine

Wave Machine

Acknowledgements

• ACOA – AIF• Husky• IBM

DATA TIME