Laplace-domain full waveform inversion using a low ...

Laplace-domain Full Waveform Inversion using a Low-frequency Time-domain Extrapolator Eunjin Park*, Seoul National University, Wansoo Ha, Pukyong National University, Jiwoong Kim and Changsoo Shin, Seoul National University Summary We propose a Laplace domain waveform inversion method using a time-domain modelling scheme. This method calculates the forward-modelled and the back-propagated wavefields in the time domain. We Laplace-transform the wavefields to obtain the gradient direction in the Laplace domain. By propagating the wave in the time domain, we maintained the advantages of the time domain approach. We reduced the calculation time by using a low frequency source wavelet and large grids in the time domain. A 2D and 3D numerical example of the proposed approach showed a smooth inversion result similar to that of the Laplace domain inversion using a Laplace-domain modelling algorithm. Introduction Laplace domain full waveform inversion is a method for recovering large-scale subsurface material parameters. The wavefield in the Laplace domain is equivalent to the zerofrequency content of the damped wavefield (Shin and Cha, 2008). The Laplace domain inversion has been used in several studies, including those based on synthetic data (Shin and Cha, 2008; Shin and Ha, 2008; Chung et al., 2010; Bae et al., 2010) and field data (Shin and Cha, 2008; Ha et al., 2012). Except for the damping constant or frequency, the algorithm of the Laplace domain inversion is similar to that of the frequency domain inversion (Ha et al., 2010). Therefore, time-related approaches such as time windowing are not applicable in the Laplace domain inversion. Moreover, time domain modelling using explicit schemes is easier to parallelize than Laplace or frequency domain modelling using matrices. Sirgue et al. (2008) introduced a time domain modelling approach to the frequency domain inversion to make 3D applications more affordable. We adopted their approach to the Laplace domain; however, inverse Laplace transformation is not a firmly established process, unlike the inverse Fourier transformation (Lakhtakia, 1994), which is required in the back-propagation step (Kim and Shin, 2012). We used Green's function to avoid using the inverse Laplace transformation. We reduced the computational cost by using low frequency wavelets and coarse grids. In this study, we first examine the Laplace transformation of time domain wavefield. Then we review the inversion theory and evaluate the algorithm using a numerical example.

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Time domain extrapolator for the Laplace domain wavefield We first examined the possibility of using the time domain extrapolator to generate Laplace domain wavefields. Figure 1 shows a 2D section of the SEG/EAGE salt velocity model (Aminzadeh et al., 1994). We used a 4thorder finite-difference modelling algorithm to generate the time domain wavefield. We generated the wavefield using two first-derivative Gaussian source wavelets with different maximum frequencies. Figure 2 shows the resultant seismograms. These are obtained from the source wavelets with the maximum frequency of 2.0 (Figure 2a) and 45.0 Hz (Figure 2b). Note that the shapes of the seismograms are significantly different from each other. We Laplace-transformed these seismograms and obtained Laplace domain wavefields for a damping constant of 2.0 and 15.0 s −1 . The shapes of the Laplace domain wavefields are same, and only their amplitudes are different. After we normalized the wavefields, we obtained the same wavefields as shown in Figure 3. The results showed that the frequency or shape of the time domain source wavelet did not affect the shape of the Laplacetransformed wavefield (Ha and Shin, 2012). The frequency or shape of the source wavelet affects the overall amplitude of the Laplace domain wavefield only. Accordingly, we do not need to use a high-frequency source wavelet and fine grids to obtain smooth Laplace domain wavefields using time domain modelling. Instead, we can use low frequency source wavelets and coarse grids to save computational resources. The amplitude difference was compensated by a deconvolution of the source wavelet (Ha and Shin, 2012), which is equivalent to a division in the Laplace domain:

 = u i , G i A

(1)

 is Green's function in the Laplace domain, u is where G the Laplace-transformed wavefield, A is the Laplacetransformed source wavelet, and indicates the source position. The Laplace domain Green's function obtained by deconvolution can be used to calculate the Laplace domain modelled-wavefield and residual. The gradient calculation step of the waveform inversion in the Laplace domain is as shown blow.

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Figure 1: A 2D section of the SEG/EAGE Salt model.

(a)

(b) Figure 3: Normalized Laplace-domain wavefields transformed from seismograms shown in Figure 2 for the damping constant of (a) 2.0 and (b) 15.0 s-1.

(a) The logarithmic objective function for a damping constant s in the Laplace domain (Shin and Min, 2006; Shin and Cha, 2008) can be expressed as: 2

E=

1  uij  ∑ ln  , 2 i , j  dij 

(2)

where u is the forward-modelled wavefield in the Laplace domain, d is the observed wavefield in the Laplace domain, i and j are indices of the sources and receivers.

(b) Figure 2: Seismograms obtained from the source wavelets with the maximum frequency of (a) 2.0 and (b) 45.0 Hz.

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The gradient direction in the Laplace domain inversion is a multiplication between the virtual source and the backpropagated wavefield (Shin and Cha, 2008), which can be expressed as: T ∂E = ∑ vTi S −1  ri , ∂mk i

(3)

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Time-domain Modeling for Laplace-domain Inversions where mk is the k -th model parameter, v is the virtual source vector, S is the impedance matrix, and r is the residual. Each iteration in the inversion requires two modelling steps: one for the calculation of the forwardmodelled wavefield, and the other for the back-propagation of the residual wavefield. In the present study, we moved the wave-propagation step to the time domain. We obtained the forward-modelled wavefield by applying the Laplace transformation to the time domain wavefield. Because the amplitude of the transformed wavefield depends on the time domain source wavelet, we calculated Green's function first (Equation 1) and multiplied the results by the estimated source wavefield (Shin and Cha, 2008). Although the surface seismogram of each shot was only required in this step, we saved the Green function of the entire domain with the source at the shot and receiver positions for later use in the back-propagation step. The back-propagation in the time domain requires time domain residuals; however, inverse Laplace transformations of Laplace domain residuals are not simple problems (Lakhtakia, 1994). Accordingly, we used Green's function to obtain the back-propagated wavefield: Nr

T −1  r S −1  ri S= ri = ∑ G = j ij

Nr

u j

∑ A r

=j 1 =j 1

ij

.

(4)

Note that we omitted the transpose of the impedance matrix by assuming that the wave equation satisfies the reciprocity. Equation 4 shows that we can obtain the backpropagated wavefield by wave propagation modelling with the amplitude of the residual followed by the deconvolution shown in Equation 1.

Numerical Examples We generated time domain wavefields from a 2D section of the SEG/EAGE salt model (Figure 1) using 4th-order finite difference modelling and 5 m grid spacing. The maximum frequency of the source wavelet was 45 Hz. The dataset consisted of 193 shots with an interval of 80 m. The time sampling rate was set to 0.5 ms. We inverted the data in the Laplace domain using time domain modelling. The grid size was set to 80 m. The maximum frequency used for time domain modeling was equal to 2.3 Hz. To avoid dispersion, the time sampling rate was varied from 10.8 ms to 16.0 ms, depending on the velocity of the medium. Figure 4a shows the initial velocity used to invert the synthetic salt data. The velocity of the linear velocity model was varied from 1.5 km/s to 3.0 km/s, and Figure 4b shows the inversion result. Similar

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to the results obtained in the previous studies, a smooth large-scale velocity model was obtained (Shin and Cha, 2008; Shin and Ha, 2008). We also generated Laplace-domain wavefields from the 3D SEG/EAGE salt model (Aminzadeh et al., 1994) using Laplace-domain forward modelling and 100 m grid spacing (Figure 5). The synthetic dataset consisted of 144 shots with the x and y interval of 1 km. We inverted the data in the Laplace domain using time domain modelling. The grid size was set to 100 m. The maximum frequency used for time domain modelling was equal to 0.75 Hz. To avoid dispersion, the time sampling rate was varied from 19.2 ms to 12.9 ms, depending on the velocity of the medium. Figure 6a shows the initial velocity used to invert the synthetic salt data. The velocity of the linear velocity model was varied from 1.5 km/s to 3.0 km/s, and Figure 6b shows the large-scale inversion result. The computation time is reduced to two-thirds of that of an inversion using Laplace-domain modelling algorithm.

Conclusions We introduced a low frequency time domain extrapolator for full waveform inversion in the Laplace domain. The Laplace domain inversion yields large-scale velocity models, and the grid size can be large. Because the frequency content of the time domain source wavelet does not change the shape of the Laplace-transformed wavefield, low frequency sources and large grids can be used for wave propagation in the time domain to reduce the computational cost for inverse problems. Numerical examples using the synthetic salt model showed that the inversion results of the method are consistent with the original inversion using Laplace-domain modelling algorithms. Further studies are needed to examine the application of the method for 3D field problems.

Acknowledgments This work was supported by the Energy Efficiency & Resources Core Technology Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) granted financial resource from the Ministry of Trade, Industry & Energy, Republic of Korea (20132510100060).

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(a)

(b) Figure 4: The linear velocity model used as the starting model (a) and the inversion results (b).

(a)

(b) Figure 6: A linear velocity model used as the starting model (a) and the recovered velocity model (b). Figure 5: The SEG/EAGE 3D Salt velocity model.

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http://dx.doi.org/10.1190/segam2014-0470.1 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2014 SEG Technical Program Expanded Abstracts have been copy edited 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

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