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