75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013 London, UK, 10-13 June 2013

Th-10-01Q Estimation Through Waveform InversionJ. Bai* (ION) & D. Yingst (ION)

SUMMARYThis paper presents an approach to estimate the quality factor Q through waveform inversion. In aviscoacoustic medium consisting of one standard linear solid, stress and strain relaxation times govern thedissipation mechanism. Their difference, normalized to be a unitless variable τ, determines the magnitudeof Q. In this paper we iteratively optimize a τ model by minimizing an objective function that measuresthe residuals between recorded and synthetic seismic data. The τ model is then converted to itscorresponding Q model. A viscoacoustic Marmousi model demonstrates the accuracy of the approach. Fora field data from the Gulf of Mexico we present a workflow to estimate its Q model and then optimize itsvelocity model through waveform inversions with the attenuation compensation. The workflow showssome promise to get the final seismic products with the attenuation compensation for physical materials.

75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013 London, UK, 10-13 June 2013

Introduction

Because of the conversion of elastic energy into heat seismic waves are attenuated and dispersed as they propagate. This anelastic behavior can result in amplitude decrease, wavelet distortion and the loss of high-frequency components. It therefore can cause strong footprints on seismic inversion and imaging, AVO/AVA analysis, and make interpretation more difficult. It is therefore important to compensate for the attenuation effects in the final seismic product. An attenuation model, described by the quality factor Q, is widely used in various compensation methods. Because the attenuation effects cause the loss of high-frequency components, seismic amplitude spectra naturally serve as a carrier for Q estimation. Usually a Q model can be estimated either through the amplitude-spectra-ratio method (Dasgupta and Clark, 1998) or through the measurement of the relative shift of dominant frequency (Quan and Harris, 1997). These methods assume that scattering, geometrical spreading, and other non Q-related factors have been removed from seismic data. Given a Q model, a viscoelastic mechanical model consisting of standard linear solids (SLSs) provides a powerful tool to model real earth materials (Robertsson et al., 1994). One SLS consists of a spring in parallel with a spring and a dashpot in series. It can approximate a constant Q within a defined frequency band. A series of SLSs connected in parallel can yield a quite general mechanical viscoelasticity (Day and Minster, 1984). For each SLS its relaxation mechanism describes the physical dissipation on seismic waves. Its stress relaxation time τσ and its strain relaxation time τε govern this relaxation mechanism. Their difference, expressed as a unitless variable τ (= τε/τσ -1), determines Q. Waveform inversion estimates subsurface parameters in a way that minimizes the residuals between the recorded and synthetic seismic data. It is attractive in its ability to produce parameter models with high resolution for complex geological structures. Since the gradient-based optimization methods were introduced (Tarantola, 1984), waveform inversion has achieved substantial success at the estimation of velocity models (Vigh et al., 2011; Wang et al., 2012) and anisotropic parameters (Prieux et al., 2012) in practice. This paper introduces an approach for Q estimation by waveform inversion. For a viscoacoustic model consisting of one SLS this approach estimates a τ model. The τ model is then converted to its corresponding Q model. We set up an objective function and then derivate a gradient from it. Viscoacoustic wave equations for forward modeling and its adjoint are used for the calculation of the objective function and the gradient (Bai et al., 2012). A nonlinear conjugate gradient method is employed to update the τ model. Test with a viscoacoustic Marmousi model demonstrates that the approach can produce a complex Q model with high resolution. We also present a workflow comprising a Q inversion, followed by a velocity update through viscoacoustic waveform inversion for a 3D deep-water OBC (ocean bottom cable) field data from the Gulf of Mexico (GOM). This test shows some promise in both Q estimation and velocity estimation with the attenuation compensation for physical materials.

Theory

In a viscoacoustic model consisting of one SLS, viscoacoustic wave equation can be used to simulate the attenuation effects on seismic waves during their propagation (Bai et al., 2012).

1v2

∂2P∂t 2

= (1+τ )ρ∇ ⋅ ( 1ρ∇P)− r + f (1)

with

1[ ( )]*[ ( )]

t

r e H t Pστ

σ

τρ

τ ρ

−

= ∇ ⋅ ∇ , (2)

75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013 London, UK, 10-13 June 2013

where P = P(x,t;xs) is the wavefield at time t and at a position x for a source located at xs, ρ = ρ(x) is density, v = v(x) is velocity, f is source, H(t) is the Heaviside function, and r is a memory variable. The memory variable is a causal time convolution and describes the dissipation mechanism. Its kernel is of exponential character. Since it decays, energy is dissipated. For a fixed frequency, τσ is nearly constant (Robertsson et al., 1994). Consequently the decay speed mainly depends on τ, which determines the magnitude of the quality factor Q.

22( 1) 1Qτ

= + − , (3)

This relationship clearly shows that we can invert a τ model and then convert it to a Q model. We therefore iteratively optimize a τ model by minimizing a total value (TV) regularized objective function, which measures the residuals between the recorded and synthetic seismic data

2

0( ) ( )J d dτ α λ= Γ − + T, (4) with the TV regularization constraint defined as

2 20( ) ( ) dτ τ τ β= ∇ − +∫T x , (5)

where d0 = d0(xr,t;xs) is the recorded data and d = d(xr,t;xs) is the synthetic data at the receiver position xr. α (= <d, d0>/||d||2) is a normalization scale. The operator Γ is a preconditioning operator on the residual d0-αd. λ is a weighting scale. The small value β is squared and added to the square norm of gradient between the current τ and the original τ0 to void singularity of the gradient of T. The gradient for τ update is given by

g(x) = −2α [ρ∇ ⋅ 1ρ∇(P)− 1

τr]R+ λ

t∑

xs

∑ ∇⋅∇(τ −τ 0 )

∇(τ −τ 0 )2+ β 2

, (6)

where R = R(x,t;xs) is the wavefield obtained by applying the adjoint of forward modeling on the residual Γ(d0-αd) (Bai et al., 2012). Viscoacoustic wave equations for forward modeling and its adjoint are solved by stable high-order finite-difference schemes in centered grids. A normalized gradient by the amplitude of forward wavefield accelerates convergence.

2 2

( )( )

( , ; )s

n

st

gg

P t κ=

+∑∑x

xx

x x, (7)

where κ is a whitening factor to avoid singularity. We update the τ model by using the Polak-Ribière implementation of nonlinear conjugate gradient method. A line search uses the BB formula (Barzilai and Borwein, 1988) for an initial estimate of step length. The BB formula is an efficient way to estimate a step length for the TV regularized problem. It does not require extra forward modeling for the evaluation of objective function.

Examples

We first use a viscoacoustic Marmousi model to demonstrate our method for Q estimation. The model includes a water layer from surface down to 500 m. A Q model shown in Figure 1(a) is directly mapped from its velocity model. The attenuation in water is weak (Q = 5000) while it is strong below water since Q ranges from 20 to 80. Using the velocity and Q models, we generate a viscoacoustic synthetic dataset for a constant density. The dataset has 125 shots. Shot interval is 100 m. Each shot has 161 receivers. Receiver interval is 20 m. We start waveform inversion from a constant Q model (Q = 5000) and only use frequencies below 9 Hz. A multi-scale approach is carried out from low to high frequencies to bypass both local-minima and cycle-skipping problems. In the inversion the true velocity model is used. The waveform inversion eventually generates a high-resolution Q model shown in Figure 1(b). The inverted model reveals complex Q anomalies in details.

75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013 London, UK, 10-13 June 2013

We next consider a 3D OBC field data from the deep-water GOM. The dataset has totally 19901 shots. Each shot has 239 receivers. We test a workflow to optimize a velocity model through waveform inversions with the attenuation compensation. In the inversions the data with offset range from 3500 m to 6500 m are used and frequencies range from 2 to 9 Hz. First our strategy relies on the construction of a Q model. We invert the Q model (Figure 2) from a constant Q model (Q = 5000). The high-value Q means no attenuation at the beginning. The inverted Q model indicates strong attenuation in some areas. A velocity model (Figure 3(a)) is kept constant in the first waveform inversion. Next we keep the inverted Q model constant and optimize the velocity model by a viscoacoustic waveform inversion. We only update sediment velocity. The inverted velocity model is shown in Figure 3(b). The seismic attenuation is strong in the area of interest (Figure 4(a)) where the geological structures are poorly imaged (Figure 4(b)). The image is improved by using the inverted velocity model (Figure 4(c)). Energy is better focused in the area of interest. This example shows that incorporating attenuation in model building is helpful to improve migration images in practice.

Figure 1 Viscoacoustic Marmousi model. (a) The true Q model. (b) The inverted Q model from waveform inversion.

Conclusions

In this paper we expand the application of waveform inversion on Q estimation. The approach uses raw seismic data without removing scattering, geometrical spreading, and other non Q-related factors. This makes the approach robust and reliable for real data.

The Marmousi model demonstrates that a Q model can be accurately estimated when its true velocity model is used. The inverted Q model has high resolution to reveal complex Q anomalies in details.

Using the GOM field data, we present a workflow to estimate a Q model and then optimize a velocity model through waveform inversions with the attenuation compensation. The inverted velocity model reduces the attenuation footprints in the final image.

Acknowledgements

We thank ION Geophysical for the permission to publish this work. We also thank our colleagues in ION Geophysical for their discussions. Thanks go to IFP for the Marmousi model.

References

Bai, J., D. Yingst, R. Bloor, and J. Leveille, 2012, Waveform inversion with attenuation: SEG Technical Program Expanded Abstract. Barzilai, J., and J. Borwein, 1988, Two-point step size gradient methods: IMA Journal of Numerical Analysis, 8, 141–148. Dasgupta, R., and R. A. Clark, 1998, Estimation of Q from surface seismic reflection data: Geophysics, 63, 2120-2128. Day, S. M. and Minster, J. B., 1984, Numerical simulation of wave fields using a Padé approximant method: Geophys. J. R. Astr. Soc., 78, 105–118.

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75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013 London, UK, 10-13 June 2013

Prieux, V., R. Brossier, S. Operto, J. Virieux, O.I. Barkved and J.H. Kommedal, 2012, Two-dimensional anisotropic visco-elastic full waveform inversion of wide-aperture 4C OBC data from the Valhall Field: EAGE expended abstract. Robertsson, J. O. A, Blanch J. O., and Symes W. W., 1994, Viscoelastic finite-difference modeling: Geophysics, 59, 1444–1456. Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, 1259–1266. Quan, Y., and J. M. Harris, 1997, Seismic attenuation tomography using the frequency shift method: Geophysics, 62, 895-905. Vigh, D., J. Kapoor, and H. Li, 2011, Full-waveform inversion application in different geological settings: SEG Technical Program Expanded Abstract. Wang, C., Yingst D., Bloor R., and Leveille J., 2012, Application of VTI waveform inversion with regularization and preconditioning to real 3D data: EAGE Expanded Abstract.

Figure 2 The inverted Q model for the 3D GOM field data.

Figure 3 The 3D GOM field data example. (a) The initial velocity model. (b) The inverted velocity model from waveform inversion.

Figure 4 (a) The area of interest shown in the inverted Q model. Migration images obtained from (b) the initial velocity model and (c) the inverted velocity model from viscoacoustic waveform inversion.

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