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Compensating for attenuation by inverse Q filtering
Carlos A. MontañaDr. Gary F. Margrave
Motivation
Assess and compare the different methods of applying inverse Q filterUse Q filter as a reference to assess phase restoration in Gabor deconvolution
Outline
Anelastic attenuation: Constant Q modelInverse Q filter approaches
Inverting the Q matrixDownward continuationPseudodifferential operators
Comparison of the methodsConclusion
Attenuation mechanisms (Margrave G. F., Methods of seismic data processing, 2002)
Geometric spreadingAbsorption (Anelastic attenuation)Transmission lossesMode conversionScatteringRefraction at critical angles
Anelastic attenuation
In real materials wave energy is absorbed due to internal frictionAbsorption is frequency dependentAbsorption effects:
waveform change, amplitude decay phase delay
The macroscopic effect of the internal friction is summarized by Q
Constant Q theory (Kjartansson, 1979)
Q is independent of frequency in the seismic bandwidthAbsorption is linear (Q>10)Dispersion: each plane wave travels at a different velocityThe Fourier transform of the impulse response is
⎥⎦⎤
⎢⎣⎡−
⎥⎦
⎤⎢⎣
⎡−=
Vifx
VQfxfB ππ 2expexp)(frequency
Velocity
Distance Attenuation Phase
Impulse response (Q=10)
(Aki and Richards, 2002)
With dispersionWithout dispersion
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
2
1
2
1 log11)()(
ωω
πωω
QVV
Nonstationary convolutional model
∫∞
∞−
−= τττα τπ derffwfs ifQ
2)(),()()(
t
τ τ
τ tt
W matrix Q matrix r s
Fourier transform attenuated
signal
Fourier transform source wavelet
Pseudodifferentialoperator
( ))2/(2/exp),( QiHQQ ωτωττωα +−=
Outline
Anelastic attenuation: Constant Q modelInverse Q filter approaches
Inverting the Q matrixDownward continuationPseudodifferential operators
Comparison of the methodsConclusion
Q filter and spiking deconvolution
WQrs =
t
τ τ
τ tt
W matrix Q matrix r s
sWQr 11 −−=Exact inversion
[ ]sWQsQWr 1111 , −−−− +=
sQWr 11 −−≈
Approximate inversion
Forward modeled traces
QrsIWWQrs === )(
Error estimation: Crosscorrelation
Error estimation: L2 norm of error
Outline
Anelastic attenuation: Constant Q modelInverse Q filter approaches
Inverting the Q matrixDownward continuationPseudodifferential operators
Comparison of the methodsConclusion
Inverting the Q matrix
A conventional matrix inversion algorithm gets unstable for Q<70
sQrsQWr
1
11
−
−−
≈
≈
Hale’s inversion matrix, Hale (1981)
PsPQWr 11 )( −−≈Each column of P is the convolution inverse of the corresponding column of Q.
Q-1
Hale’s inversion matrix
Outline
Anelastic attenuation: Constant Q modelInverse Q filter approaches
Inverting the Q matrixDownward continuationPseudodifferential operators
Comparison of the methodsConclusion
Downward continuation inverse Q filter, Wang (2001)
τ
∆τ
1D-Earth modelτ0
τ1
τj
τn
u(τ1)u(τ0)
u(τj)
u(τn)
Attenuated signal
Time varying frequency limitfor the low pass f ilter
τ
∆τ
1D-Earth modelτ0
τ1
τj
τn
u(τ1)u(τ0)
u(τj)
u(τn)
Attenuated signal
Time varying frequency limitfor the low pass f ilter
0),(),( 22
2
=+∂
∂ ωω zUkzzU
⎟⎟⎠
⎞⎜⎜⎝
⎛∆⎟⎟
⎠
⎞⎜⎜⎝
⎛∆=∆+ t
QVVt
VVitUttU
)(2)(exp
)()(exp),(),( 00
ωωω
ωωωωω
∫ ∆+=∆+ ωωπ
dttUttU ),(21)(
Phase correction
Amplitude correction
Downward continuation inverse Q filter
Outline
Anelastic attenuation: Constant Q modelInverse Q filter approaches
Inverting the Q matrixDownward continuationPseudodifferential operators
Comparison of the methodsConclusion
Inverse Q filtering by pseudo-differential operators, Margrave (1998)
∫∞
∞−
−= τττ dhtatg )()()(
In a stationary linear filter the output can be obtained byconvolving an arbitrary inputwith the impulse response
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡)( )( )( tghta ττ−
Stationary linear filter theory
Inverse Q filtering bypseudodifferential operators
∫∞
∞−
−= ττττ dhtbts )(),()(
∫∞
∞−
−= τττ dhttbts )(),()(~
Nonstationary linear filter theory
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
Nonstationary convolution
Nonstationary combination
b(t-τ, τ) h(τ) s(t)
Inverse Q filtering by pseudodifferential operators
Nonstationary convolution and combination in the frequency domain
Nonstationary convolution
Nonstationary combination
∫∞
∞−
ΩΩΩΩ−= dHBS )(),()( ωω
∫∞
∞−
ΩΩ−= dHBS )(),()(~ ωωωω
From Margrave (1998)
These integrals are the nonstationary extension of the convolution theorem
Inverse Q filtering by pseudodifferential operators
Nonstationary convolution in mixed domains
Nonstationary combinationin mixed domains
∫∞
∞−
−= τττωβω ωτdehS i)(),()(
∫∞
∞−
Ω ΩΩΩ= deHtts ti)(),(21)(~ βπ
( ))2/(2/exp),(),( QiHQQ ωτωττωατωβ +−==
( ))2/(2/exp),(),( 1 QiHQQ ωτωττωατωβ −=≈ −Inverse Q filter:
Forward Q filter:
Generalized Fourier integral (direct)
Generalized Fourier integral (inverse)
Anti-standard pseudo-differential
operator
Standard Pseudo-differential operator
Inverse Q filtering by using pseudodifferential operators
Outline
Anelastic attenuation: Constant Q modelInverse Q filter approaches
Inverting the Q matrixDownward continuationPseudodifferential operators
Comparison of the methodsConclusion
Comparison for Q=50
Comparison for Q=20
Conclusions
AA+AC+
Pseudo-differential operator
A+A+A-C
Downward continuation
CA+ (Q>40)D (Q<40)
A+ (Q>40)D (Q<40)
A+ (Q>40)D (Q<40)
Hale’s matrix inversion
Computat. cost
Numerical stability
Phase recovery
Amplitude recovery
Future work
Consider as variablesUncertainty in Q estimationVariations of Q with depthNoise
Improve amplitude recovery and efficiency in pseudodifferentialoperator Q filtering
ReferencesAki, K., and Richards, P. G., 2002, Quantitative seismology, theory and methods. Hale, D., 1981, An Inverse Q filter: SEP report 26.Kjartansson, E., 1979, Constant Q wave propagation and attenuation: J. Geophysics. Res., 84, 4737-4748Margrave, G. F., 1998, Theory of nonstationary linear filtering in the Fourier domain with application to time-variant filtering: Geophysics, 63, 244-259.Montana, C. A., and Margrave, G. F., 2004, Phase correction in Gabor deconvolution: CREWES Report, 16Wang, Y.,2002, An stable and efficient approach of inverse Q filtering: Geophysics, 67, 657-663.
Acknowledgements
CREWES sponsorsPOTSI sponsorsGeology and Geophysics DepartmentMITACSNSERCCSEG
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