Initial model construction for elastic full waveform inversion using envelope inversion method Jingrui Luo* and Ru-Shan Wu, University of California at Santa Cruz Summary We developed an elastic envelope inversion method. The envelope of the wavefields carries ultra low frequency information which can be used to recover the large-scale component of the model, and the initial model dependence of waveform inversion can be reduced. We derived the misfit function and the corresponding gradient operator for elastic envelope inversion. Numerical tests using synthetic data for the Marmousi II model proved the validity and feasibility of the proposed approach. The inverted p-wave velocity and s-wave velocity from the combined envelope inversion plus waveform inversion (EI+WI) indicated that it can deliver much improved results compared with regular elastic full waveform inversion. Furthermore, to test the independence of the elastic envelope inversion to the source frequency band, we used a low-cut source wavelet (cut from 5Hz below) to generate the synthetic data. The envelope inversion and the combined EI+WI showed no appreciable difference from the full-band source results. Introduction Elastic full waveform inversion (FWI) has been investigated by many authors ( Mora, 1987; Crase et al., 1992; Sears et al., 2010). However, as in acoustic situation, good initial models are always needed in elastic full waveform inversion in order to get good inversion results. To generate low-frequency signal below 5Hz is very expensive, so effort has been focused to the recovery of long wavelength background without very low frequencies. Travel time inversion and migration velocity analysis are two traditional methods in this area. In recent years, Shin and Cha (2008) has developed Laplace domain full waveform inversion which can give a smooth background model from an inaccurate initial model. Zhou et al. (1997) combined traveltime and waveform inversion. Biondi and Almomin (2014) combined full waveform inversion with wave equation migration velocity analysis. Liu et at. (2011) proposed the normalized integration method. Wu et al. (2013, 2014) (Luo and Wu, 2014) developed envelope inversion (EI) method which can give a very smooth background structure. Most of the approaches above are developed in the acoustic situation. However, the real earth is elastic. In this paper, we extend the envelope inversion method to elastic situation. We first introduce the elastic envelope inversion method and derive the gradient for the elastic envelope inversion, then we use the Marmousi II model to show the validity of this method.

Trace envelope and Hilbert transform First we introduce what is the envelope of a signal. We extract envelope by taking the amplitude after the analytical signal transform using Hilbert transform. A signal having no negative-frequency components is called and analytic signal

!f (t) which can be constructed from a real signal

f (t) and its Hilbert transform H{ f (t)} ,

!f (t) = f (t)+ iH{ f (t)}, (1) The envelope of f (t) can then be derived by,

2 2( ) ( ) { ( )} ,e t f t H f t= + (2)

Envelope inversion for elastic situation The 2-D isotropic elastic wave equation can be expressed as,

ρ ∂2ui

∂t2 − ∂τ ij

∂x j = f i ,

τ ij = M ij + λδ ijuk ,k + µ(ui, j + uj ,i ).

(3)

Where iu is the thi component of the displacement vector and ijτ is the ij component of the stress tensor. ρ is the

density; λ and µ are Lamé coefficients. fi is the body

force and ijM is the traction. The p-wave velocity and s-wave velocity are given by,

vp =

λ + 2µρ

, vs =µρ

.

(4)

In the elastic envelope inversion, we define the misfit function as follow,

σ (m) = 1

4[esyn

i (t)]2 − [eobsi (t)]2

0

T∫

i∑

sr∑

2dt, (5)

where m is the model parameter, isyne and i

obse are the envelope of the thi component of the synthetic wavefield and the observed wavefield respectively. Using Hilbert transform, the above equation can be written as,

σ (m) = 14

[si(t)]2 + [sHi (t)]2{ }− [ui(t)]2 + [uH

i (t)]2{ }0

T∫

i∑

sr∑

2dt

= 14

Ei2

0

T∫

i∑

sr∑ dt

(6)

where is and iu are the thi component of the synthetic wavefield and the observed wavefield respectively, i

Hs and iHu are the corresponding Hilbert transforms.

iE is the instant envelope data residual. Here we applied squared envelope, because it has better performance in long-

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Elastic envelope inversion

wavelength background recovery (Wu et al. (2014); Luo et al. (2014)). Gradient calculation for elastic envelope inversion We first assume the density is a constant. If we considerλ and µ as the model parameters. Calculating the derivative

of the misfit function σ with λ and µ , we get,

∂σ∂λ

= Eisi (t)− H{EisH

i (t)}⎡⎣ ⎤⎦0

T

∫i∑

sr∑ ∂si (t)

∂λdt = Jλ

Trenv ,

∂σ∂µ

= Eisi (t)− H{EisH

i (t)}⎡⎣ ⎤⎦0

T

∫i∑

sr∑ ∂si (t)

∂µdt = Jµ

Trenv .

(7)

where

Jλ =∂si

vp

Jµ =∂si

vs

⎧

⎨⎪⎪

⎩⎪⎪

, renv = Eisi (t)− H{EisH

i (t)}. (8)

from equation (8) we see that the gradient for the elastic envelope inversion has exactly the same form as that in the conventional elastic full waveform inversion (Mora, 1987), except that the residual vector renv

is different. The gradient for the elastic envelope inversion can be obtained using the backpropagation method as follows,

∂σ∂λ

= − d ′t sl ,l∫!sj , j ,

∂σ∂µ

= − d ′t (sj ,k + sk , j )∫!sj ,k .

(9)

where “← ” means backpropagation wavefields which is obtained using the envelope residual vector renv

as the source wavelet. Using equation (4) we can get the gradient with respect to the velocity as follows,

∂σ∂vp

= −2ρvp sk ,k

!si,i∫ ,

∂σ∂vs

= 4ρvs sk ,k

!si,i∫ − 2ρvs (si, j + sj ,i )!si, j∫ .

(10)

because renv is related to the effective envelope residual not the seismic data residual, the low frequency information in the envelope can be lower than the lowest frequency in the source wavelet. This low frequency information can be used to construct the large-scale background structure. The low-frequency information coded in the envelope is not directly from the source wavelet. Numerical examples Now we show the validity of this method using the Marmousi II model, which is shown in Figure 1. Figure 2

shows the initial model for the inversion. We used a 1-D linear initial model.

(a)

(b)

Figure 1 The Marmousi II elastic model. (a)vp; (b) vs.

(a)

(b)

Figure 2 Linear initial model. (a) vp; (b) vs.

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Elastic envelope inversion

We first use a full-band Ricker wavelet as the source wavelet, and the dominant frequency is 10Hz. Starting from the linear initial models, we invert the p-wave velocity and s-wave velocity using envelope inversion. Figure 3 shows the envelope inversion results. From these results, we can already see the large-scale component of the model. Using the results in Figure 3 as the new initial model, we perform elastic waveform inversion. Figure 4 shows the final inversion results (EI+FWI results). We see that these results are very similar to the true models. As comparison, we also performed the conventional elastic full waveform inversion starting directly from the linear initial models. Figure 5 shows the inversion results. We see that the inversion has converged to a local minimum. From the comparison we can see that the combined EI+FWI can deliver much better results than the conventional FWI method.

Distance (km)

Dept

h (k

m)

2 4 6 8

1

22000

3000

4000

(a)

Distance (km)

Dept

h (km

)

2 4 6 8

1

2500

1000

1500

2000

2500

(b) Figure 3 Envelope inversion results using the full-band source. (a) vp; (b) vs. To further demonstrate the validity of this method, we cut the low frequencies below 5Hz off from the source wavelet, in this way there will be no low frequency information in the data. We also invert the p-wave velocity and s-wave velocity using envelope inversion starting from the linear initial models. Figure 6 shows the envelope inversion result in this case, and we can also see the large-scale component of the model.

Distance (km)

Dep

th (k

m)

2 4 6 8

1

2

2000

3000

4000

(a)

Distance (km)

Dep

th (k

m)

2 4 6 8

1

2

500

1000

1500

2000

2500

(b) Figure 4 EI+FWI results using the full-band source wavelet. (a)vp; (b) vs.

Distance (km)

Dep

th (k

m)

2 4 6 8

1

2

2000

3000

4000

(a)

Distance (km)

Dep

th (k

m)

2 4 6 8

1

2

500

1000

1500

2000

2500

(b) Figure 5 The conventional FWI result using the full-band source. (a) vp; (b) vs. Using the results in Figure 6 as the new initial model, and we perform elastic waveform inversion. Figure 7 shows the

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Elastic envelope inversion

final inversion results (EI+FWI results). We can see that although there’s no low frequency Fourier components in the data, using envelope inversion method we can still get good inversion results. As comparison, we performed the conventional elastic full waveform inversion starting directly from the linear initial models. Figure 8 shows the inversion results. We can see that these results are even worse than those in Figure 5. From the comparison we can see that the combined EI+FWI can deliver much better results than the conventional FWI method.

Conclusions

We developed the elastic envelope inversion method to reduce the initial model dependence of elastic waveform inversion. This is demonstrated by the Marmousi II model tests, where we inverted the p-wave velocity and s-wave velocity using a 1-D linear gradient starting model. The elastic envelope inversion is independent of the frequency band of the source wavelet. The model parameter can be inverted when the low frequency is removed from the source wavelet.

Acknowledgements

This work is supported by the WTOPI Research Consortium at University of California, Santa Cruz, USA.

(a)

(b)

Figure 6 Envelope inversion results using the low-cut source. (a) vp; (b) vs.

(a)

(b) Figure 7 EI+FWI results using the low-cut source wavelet. (a)vp; (b) vs.

(a)

(b)

Figure 8 The conventional FWI result using the low-cut source. (a) vp; (b) vs.

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EDITED REFERENCES Note: This reference list is a copyedited version of the reference list submitted by the author. Reference lists for the 2015 SEG Technical Program Expanded Abstracts have been copyedited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

Biondi, B., and A. Almomin, 2014, Efficient and robust waveform-inversion workflow: Tomographic FWI followed by FWI: 84th Annual International Meeting, SEG, Expanded Abstracts, 917–921.

Crase, E., C. Wideman, M. Noble, and A. Tarantola, 1992, Nonlinear elastic waveform inversion of land seismic reflection data: Journal of Geophysical Research, 97, B4, 4685–4703, http://dx.doi.org/10.1029/90JB00832.

Liu, J., H. Chauris, and H. Calandra, 2011, The normalized integration method — An alternative to full waveform inversion?: 17th European Meeting of Environmental and Engineering Geophysics, EAGE, Extended Abstracts, B07, doi:10.3997/2214-4609.20144373.

Mora, P., 1987, Nonlinear two-dimensional elastic inversion of multioffset seismic data: Geophysics, 52, 1211–1228, http://dx.doi.org/10.1190/1.1442384.

Sears, T. J., P. J. Barton, and S. C. Singh, 2010, Elastic full waveform inversion of multicomponent ocean-bottom cable seismic data: Application to Alba field, U. K. North Sea: Geophysics, 75, no. 6, R109–R119.

Shin, C., and Y. H. Cha, 2008, Waveform inversion in the Laplace domain: Geophysical Journal International, 173, no. 3, 922–931, http://dx.doi.org/10.1111/j.1365-246X.2008.03768.x.

Zhou, C., G. T. Schuster, S. Hassanzadeh, and J. M. Harris, 1997, Elastic wave equation traveltime and waveform inversion of cross well data: Geophysics, 62, 853–868, http://dx.doi.org/10.1190/1.1444194.

Wu, R.-S., J. Luo, and B. Wu, 2013, Ultra-low-frequency information in seismic data and envelope inversion: 83rd Annual International Meeting, SEG, Expanded Abstracts, 3078–3082.

Wu, R.-S., J. Luo, and B. Wu, 2014, Seismic envelope inversion and modulation signal model: Geophysics, 79, no. 3, WA13–WA24, http://dx.doi.org/10.1190/geo2013-0294.1.

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