Numerical Investigations of Seismic Resonance Phenomena in Fluid-Filled Layers

Hans F. Schwaiger and David F. Aldridge

Geophysics Department

Sandia National Laboratories

Albuquerque, New Mexico, USA

Seismological Society of America 2008 Annual Meeting

Sante Fe, New Mexico

April 16-18, 2008

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration

under contract DE-AC04-94AL85000.

• Numerical Algorithm– Features and limitations

• Problem formulation• Preliminary results• Conclusions and ongoing work

Outline

• 1-D Layered Model– N layers, 2 half-spaces

– Homogeneous, isotropic, anelastic

– Welded interfaces between layers

Layered Earth Model

interfacedepth

x

yz

1

2

n-1

n

n+1

N

N+1

0

layer #

2

3

n-1

n

n+1

N

N+1

1

interface #

z2

z3

zn-1

zn

zn+1

zN

zN+1

z1

. . .

. . .

. . .

. . .

• Solution Strategy

– Fourier transform x, y, t to kx, ky,

– Obtain analytic solution for layers

– Apply interface conditions

– Solve for coefficients a, b, c, d, e, f

– Inverse FFT back to x, y, z, t

Numerical Algorithm

interfacedepth

x

yz

1

2

n-1

n

n+1

N

N+1

0

layer #

2

3

n-1

n

n+1

N

N+1

1

interface #

z2

z3

zn-1

zn

zn+1

zN

zN+1

z1

. . .

. . .

. . .

. . .

)()()()(

2

2

zzdz

zd

dz

zddCu

uB

uA

,

/

1

0

/

0

1

1

/

/

/

1

0

/

0

1

1

/

/

)(Hzr

Sy

zr

Sx

zrPy

Pxzr

Sy

zr

Sx

zrSy

Px

SSPSSP e

rik

fe

rik

eerik

rik

de

rik

ce

rik

berik

rik

az

u

,)(

))(()()()(2

1)(

2NH

zh dz

zdizzzzz e

MkMfSPu

• Cartesian Coordinates– Enables use of 2D FFT (FFTW)

– Advantageous for eventual inclusion of anisotropy

• Global matrix approach– Solved by LU decomposition (Lapack)

• Uses layer-local coordinates– Allows arbitrary number of layers

• Allows both force and moment point sources• Top surface can be specified as stress-free

Algorithm Features

• Variation of material properties limited to 1-D• Thickness of individual layers is limited• Solution is periodic in x, y and t

Algorithm Limitations

• Rectangular spectrum of attenuation mechanisms– Attenuative, dispersive wave propagation

– Characterized by 8 parameters per layer:

Anelastic Geological Layers

, lo, hi, ref

Vp(ref), Qp(ref) Vs(ref), Qs(ref)

High Q medium Low Q medium

Example Synthetic Seismograms

300 m

VP = 350 m/s V

S = 1 m/s ρ = 1 kg/m3

VP = 2000 m/s V

S = 1500 m/s

ρ = 2200 kg/m3

Vp = 3500 m/s V

s = 2100 m/s ρ = 2400 kg/m3

Rayleigh Head Waves

MultiplesPPPP

3P+1S2P+2S1P+3SSSSS

PP

Amplitude scale: x4

SS

PS+SP

Amplitude scale: x1

Air overlying earth model

Fullsolution

Differencesolution

Problem Geometry

VP = 3500 m/s

VS = 2020 m/s

ρ = 2400 kg/m3

VP = 1500 m/s V

S = 1 m/s ρ = 1000 kg/m3

VP = 350 m/s V

S = 1 m/s ρ = 1 kg/m3

Fluid layer

20 m

d

Sources:Fx, Fz, E,Torque

Receivers: Vx, Vz200 m

Water:

Air:

Problem geometry

Reverberations

Direct ArrivalTop-bed reflectionIntra-bed multiple

Seismic Source Wavelet: Berlage

Berlage waveletcharacteristics:• causal• high-frequency content

Fz / Vz Responses (20 Hz Berlage Wavelet)

Half-space 5 m Water layer 5 m Air layer

Z = 2.5 m

Z = 7.5 m

Z = 12.5 m

Z = 17.5 m

Z = 22.5 m

Z = 27.5 m

Sourcedepth

Spectra: Fz / Vz (Direct wave subtracted)

Spectra: Fz / Vz (Direct wave subtracted)

Spectra: Fz / Vz (Source in layer)

Spectra: E / Vz (Direct wave subtracted)

Spectra: E / Vz (Direct wave subtracted)

Spectra: E / Vz (Source in layer)

Torque Source Spectra (z = 10 m)

5 m Air layer20 Hz Berlage

Source depthz = 10 m

Fz / Vz Traces (50 Hz Berlage Wavelet)

Half-space 5 m Water layer 5 m Air layer

Z = 2.5 m

Z = 7.5 m

Z = 12.5 m

Z = 17.5 m

Z = 22.5 m

Note: Direct wave subtracted

Spectra: E / Vz (Direct wave subtracted)

Spectra: E / Vz (Direct wave subtracted)

Spectra: E / Vz (Source in layer)

Orientation Dependence on Force Source

5 m Air layer

Source depthz = 22.5 m

• Receiver spectrum is dominated by reverberations in the layer containing the source– No significant generation of reverberations in fluid layer when

source is in overburden • Some sources are more effective than others in generating

reverberations– Explosion sources are efficient at generating reverberations

– Frequency of reverberations from force sources are angle dependent

– Torque sources are inefficient

• Reverberations are stronger in air layers than water layers

• Investigation of magma-filled layers are underway

Conclusions and Ongoing Work

Spectra: Fz / Vz (Direct wave subtracted)

Spectra: Fz / Vz (Direct wave subtracted)

Spectra: Fz / Vz (Source in layer)