Full waveform inversion in the Laplace and Laplace-Fourier domains Changsoo Shin, Wansoo Ha, Wookeen Chung, and Ho Seuk Bae Summary We present a review of Laplace and Laplace-Fourier domain waveform inversion. The wave equation in the Laplace and Laplace-Fourier domains can be solved by changing the real frequencies from the Fourier transform into imaginary frequencies. The initial model of Laplace-domain inversion can be a scratch such as a homogeneous velocity model. The inversion provides a long-wavelength velocity model that can be used as a starting velocity model for conventional waveform inversion, which uses the zero-frequency components of the damped wavefield. Laplace-Fourier domain inversion can recover long-, medium- and short-wavelength velocity models by adjusting the complex frequencies. Careful muting of noise should be applied before the first arrival because the damped wavefield is sensitive to random noise. Numerical experiments and real data examples show that full waveform inversion in the Laplace and Laplace-Fourier domains can provide an alternative for seismic velocity estimation. 7 nd EAGE Conference & Exhibition incorporating SPE EUROPEC Barcelona, Spain, - 7 June
Introduction Since Albert Tarantola s pioneering work (98), waveform inversion has emerged as a tool for velocity estimation. However, full waveform inversion performed in the time or frequency domains could not recover large-scale velocity structures due to the high non-linearity of the objective functions and the lack of low-frequency information in the field data. Shin and Cha (8) introduced the Laplace-domain full waveform inversion, which can recover a long-wavelength velocity model from data containing little low-frequency information by minimizing the logarithmic difference between the damped wavefields of the modeled and observed data. Later, they extended the method to the Laplace-Fourier domain and were able to recover medium- to shortwavelength velocities in addition to long-wavelength velocity structures from real field data (Shin and Cha, 9). One drawback of the Laplace or Laplace-Fourier domain inversion is its sensitivity to noise appearing before the first arrival signal. In this paper, we review the Laplace-domain and Laplace-Fourier domain full waveform inversion methods. First, we review their theoretical backgrounds, and then we present numerical examples, including multi-parameter synthetic data and field data examples. Waveform inversion in the Laplace domain The wave equation can be expressed as a matrix equation using the finite element method, as given by M u + Ku = f, () where M is a mass matrix, K is a stiffness matrix, u is a wavefield in the time domain, and f is a source vector (Marfurt, 98). Taking the Laplace transform, we obtain Su = f, () where S = K + s M, u s, ()= u()e t st dt f s, ()= f()e t st dt and s is a positive, real damping constant. u () s in equation (3) can be expressed as u ()= s u()e t st dt lim u ()e t st e iωt dt. () ω Equation () implies that a Laplace-transformed wavefield corresponds to the zero-frequency component of a damped wavefield. Although there is no zero-frequency component in a seismic trace, its damped version generally contains a zero-frequency component. By exploiting this zero-frequency information, the Laplace-domain inversion can recover a large-scale velocity model. In the frequency domain, we use a fixed damping constant to generate the forward-modeled data. In the Laplace domain, we set the angular frequency to zero and vary the damping constant to generate the forwardmodeled data. The inversion procedure is identical using either method. For greater efficiency, we can use a coarse grid without significant numerical error for the inversion in the Laplace domain (Shin and Cha, 8). The logarithmic objective function in the Laplace domain has little local minima and a better convex form than conventional objective functions in the time or frequency domains (Shin and Ha, 8). These characteristics of the Laplace-domain waveform inversion make it less sensitive to the starting velocity model than the frequency-domain waveform inversion, allowing us to start the inversion from scratch and recover smooth velocity models. Waveform inversion in the Laplace-Fourier domain As the Laplace-domain inversion utilizes zero-frequency information, the Laplace-Fourier domain inversion exploits the low- and medium-frequency information of damped wavefields. We can calculate Laplace-Fourier transformed wavefields by introducing complex-valued frequencies in the Fourier transform as follows: (3) 7 nd EAGE Conference & Exhibition incorporating SPE EUROPEC Barcelona, Spain, - 7 June
u ( ω, s)= u()e t i( ω +is)t dt = u()e t st e iωt dt. (5) In the Laplace domain, only the amplitude of wavefields is used for the inversion. The complexvalued wavefields in the Laplace-Fourier domain, on the other hand, allow us to use both amplitude and phase information. By inverting the complex-valued wavefields, medium- and short-wavelength velocity structures can be recovered along with long-wavelength velocity structures. The results of Laplace and Laplace-Fourier domain waveform inversions can be used as an initial velocity for subsequent frequency-domain waveform inversions. This two-step strategy may produce accurate high-resolution velocity models for synthetic and field data. Examples Here, we demonstrate the Laplace-domain inversion algorithm for an acoustic-elastic coupled salt model modified from a D section of the SEG/EAGE salt model (Aminzadeh et al., 99). The thickness of the water layer is. km. The P-wave velocity model is the same as the original SEG/EAGE salt model, and the S-wave velocity model is derived from the P-wave velocity model (House et al., ). Figure shows the P-wave and the derived S-wave velocity models. We deployed 78 receivers with a grid size of m at the free surface of the acoustic medium. The 53 sources at.-km intervals are located beneath the water surface. We fixed the density at. g/cm 3, and we recovered the P- and S-wave velocities. Fifteen Laplace damping constants from to 5 are used. The initial P-wave and S-wave velocities under the water layer vary linearly with depth from.5 km/s to 3. km/s and from. km/s to.6 km/s, respectively. Figure shows the S-wave velocity models inverted by the Laplace-domain waveform inversion and the subsequent frequency-domain waveform inversion. The frequency-domain waveform inversion was carried out sequentially using frequencies from.5 Hz to 6. Hz. The result shows that Laplace-domain inversion results can be used successfully as starting velocity models for subsequent frequency domain waveform inversions (Bae et al.,, unpubl. results). 5 5.5 3.5.5.5 P-wave velociy (km/s) 5 5 Figure True P-wave and S-wave velocity models (courtesy Bae et al.,, unpubl. results). 5 5.6.. 5 5 Figure Left) Recovered S-wave velocity model of Laplace-domain acoustic-elastic coupled inversion. Right) Recovered S-wave velocity model of frequency domain waveform inversion obtained using the Laplace-domain results as the starting velocity models (courtesy Bae et al.,, unpubl. results). We also performed an elastic Laplace-domain inversion using the CCSS data (Zelt et al., 5). The CCSS model was designed to simulate a realistic continental crustal model. The horizontal distance is.6...6.. 7 nd EAGE Conference & Exhibition incorporating SPE EUROPEC Barcelona, Spain, - 7 June
5 km and the maximum depth is km. This dataset has 5 shots, each with,779 receivers. The shot and receiver are spaced at 5 km and 9 m. The maximum recording time is 6 s, and the time sampling interval is 6 ms. We used discrete Laplace damping constants ranging from. to 9.7 for the inversion with a grid size of 8 m. The initial P-wave and S-wave velocities vary linearly with depth from.3 km/s to 8. km/s and from.5 km/s to.7 km/s, respectively. Figure 3 shows the true and recovered P-wave velocity models. Although the inversion result suffered from a lack of data, it was able to recover the large-scale velocity structures correctly (Chung et al.,, unpubl. results). Figure 3 Left) The P-wave velocity model of the CCSS data. Right) Recovered P-wave velocity model of an elastic Laplace-domain waveform inversion (courtesy Chung et al.,, unpubl. results). Figure shows a muted shot gather of field data acquired by GX Technology. The maximum offset distance of the data is.355 km, and the maximum recording time is 8.3 s. We used,9 shots and 8 receivers for each shot in the inversion process. A total of 9 Laplace-Fourier frequencies were used with a 5-m grid spacing. We started from a two-layer velocity model with a homogeneous velocity of 3.3 km/s under the water layer. The Laplace-Fourier waveform inversion result presented on the left side of Figure 5 shows a low velocity region under the water layer and high velocity structures on the left and right sides. A migration result obtained using the recovered velocity (visible on the right side of Figure 5) shows stratified layers and meaningful subsurface structures at km and km from the left. Figure A muted shot gather from the field data. Figure 5 Left) The inverted velocity model of a Laplace-Fourier domain acoustic waveform inversion. Right) A migration result obtained using the inversion result. 7nd EAGE Conference & Exhibition incorporating SPE EUROPEC Barcelona, Spain, - 7 June
Conclusions We briefly reviewed the theories of the Laplace and Laplace-Fourier domain full waveform inversions. We also presented numerical examples of synthetic multi-parameter inversions in the Laplace domain and a real acoustic inversion in the Laplace-Fourier domain. A Laplace-domain waveform inversion technique using a logarithmic objective function was able to recover a large-scale velocity model by exploiting the zero-frequency components of the damped wavefields. The Laplace- Fourier domain inversion was able to recover a higher-resolution velocity model than Laplace-domain inversion by utilizing low- and medium-frequency components along with the zero-frequency components. The results obtained by these techniques can be used as initial velocity models for a subsequent frequency-domain full waveform inversion to obtain more accurate high-resolution velocity models. Because they are based on damped wavefields, Laplace- and Laplace-Fourier domain inversions are highly sensitive to noise appearing before the first signal arrival. Accordingly, careful muting of this noise is crucial for a successful inversion of real data. For stable Laplace transforms, the maximum recording time should be long enough that the damped signals after the maximum recording time can be safely ignored. Laplace-domain waveform inversion with coarse grids makes 3-D seismic inversion affordable. Multi-parameter inversion in the Laplace and Laplace-Fourier domains has the potential to recover elastic parameters from real land and marine data. The selection of damping constants and Laplace- Fourier frequencies is not firmly established, and it demands more study in the future. Acknowledgements We thank GX Technology for providing the field data. This work was supported by TOTAL and the Brain Korea Project. References Aminzadeh, F., N. Burkhard, L. Nicoletis, F. Rocca, and K. Wyatt, 99. SEG/EAGE 3-D modeling project: nd update, The Leading Edge, 3, 99-95. House, L., S. Larsen, and J.B. Bednar,. 3-D elastic numerical modeling of a complex salt structure, 7 th SEG Annual Meeting Expanded Abstracts, -. Marfurt, K.J., 98. Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equation, Geophysics, 9, 533-59. Shin, C., and Y.H. Cha, 8. Waveform inversion in the Laplace domain, Geophysical Journal International, 73, 9-93., 9. Waveform inversion in the Laplace-Fourier domain, Geophysical Journal International, 77, 67-79. Shin, C., and W. Ha, 8. A comparison between the behavior of objective functions for waveform inversion in the frequency and Laplace domains, Geophysics, 73, VE9-VE33. Tarantola, A., 98. Inversion of seismic reflection data in the acoustic approximation, Geophysics, 9, 59-66. Zelt, C.A., R.G. Pratt, A.J. Brenders, S. Hanson-Hedgecock, and J.A. Hole, 5. Advancements in long-offset seismic imaging: A blind test of traveltime and waveform tomography. AGU meeting, Abstract S5A-. 7 nd EAGE Conference & Exhibition incorporating SPE EUROPEC Barcelona, Spain, - 7 June