CSIRO PUBLISHING
www.publish.csiro.au/journals/eg
Exploration Geophysics, 2010, 41, 189–197
A comparison between Laplace domain and frequency domain methods for inverting seismic waveforms Wansoo Ha1,3 YoungHo Cha2 Changsoo Shin1 1
Department of Energy System Engineering, Seoul National University, San 56-1, Shillim-dong, Gwanak-gu, Seoul 151-742, South Korea. 2 Present address: ExxonMobil Upstream Research Company, 3120 Buffalo Speedway, Houston, TX 77098, USA. 3 Corresponding author. Email:
[email protected]
Abstract. We compared the ability of full waveform inversion to recover background velocity models from data containing no low-frequency information using the frequency and Laplace domains. Low-frequency information is crucial for recovering background velocity when using frequency-domain waveform inversions. However, the dearth of low-frequency information in field data makes frequency-domain inversion impractical without accurate starting velocity models. Instead, by performing waveform inversion in the Laplace domain, one can recover a smooth velocity model that can be used for either migration or for subsequent frequency-domain inversion as an accurate initial velocity model. The Laplace-transformed wavefield can be thought of as the zero-frequency component of a damped wavefield over a range of damping constants. In this paper, we compare results obtained from both frequency- and Laplace-domain inversions and confirm that the Laplace-domain inversion can be used to recover background velocity from real data without low-frequency information. We also demonstrate that the Laplace-domain inversion can provide the frequency-domain inversion with smooth initial velocity models for better inversion results. Key words: acoustic, frequency domain, Laplace domain, smooth velocity model, waveform inversion.
Introduction Lailly (1983) and Tarantola (1984) introduced an efficient backpropagation approach to waveform inversion. Since then, there has been a continuous research effort to further develop and extend these techniques. For instance, Kolb et al. (1986) and Gauthier et al. (1986) studied waveform inversions in the time domain, while Geller and Hara (1993), Pratt et al. (1998) and Pratt (1999) presented efficient waveform inversions in the frequency domain. Mora (1987) and Sheen et al. (2006) also suggested an inversion method suitable for elastic media. Several studies have also sought to eliminate some of the salient problems with conventional waveform inversions by exploring the use of new objective functions (Crase et al., 1990; Amundsen, 1991; Symes and Carazzone, 1991; Causse, 2002; Guitton and Symes, 2003; Shin and Min, 2006). These techniques have generally yielded satisfactory results for synthetic datasets (e.g. Sirgue and Pratt, 2004; Shin et al., 2007), and it was once expected that waveform inversions would eventually replace conventional approaches to seismic data processing in terms of velocity model building. However, the practical application of waveform inversion to real data has not been satisfactory. Obstacles that prevent accurate waveform inversions using real data include the high computational cost of three-dimensional problems, the non-uniqueness of solutions, sensitivity to noise, a lack of low-frequency information, and the highly non-linear nature of the problem. In this study, we compared the frequency-domain waveform inversion to an alternative technique of Laplace-domain waveform inversion as suggested by Shin and Cha (2008). Specifically, we emphasised the ability of each technique to reconstruct the background velocity model. An appropriate macroscopic velocity model of the subsurface structure is crucial for migration and inversion imaging (Billette and ASEG 2010
Lambaré, 1998; Chauris and Noble, 2001). We argue that waveform inversion in the Laplace domain provides a more accurate reconstruction of background velocity models, even for data that contain no low-frequency information. One can think of a Laplace-domain wavefield as the zerofrequency component of the damped wavefield. This makes it impossible for a Laplace-domain waveform inversion to recover the true velocity model, even for synthetic problems. However, it can provide a smoothed background velocity model for either migration (Versteeg, 1993; Gray, 2000; Pacheco and Larner, 2005) or for use as an initial velocity model for subsequent frequency-domain waveform inversion. In this paper, we first review the theory of waveform inversion in the Laplace domain. We then demonstrate that Laplace-domain inversion yields results that are superior to frequency-domain techniques by using a 2D section of the SEG/EAGE synthetic salt model and a real dataset. We also show that the Laplace-domain inversion can provide the frequency domain smooth initial velocity models and make it derive better inversion results. Theory The acoustic wave equation can be made discrete and expressed as a matrix equation using the finite element method, resulting in M€ u þ Ku ¼ f;
ð1Þ
where M is a mass matrix, K is a stiffness matrix, u is a wavefield € is the second order time derivative of u, in the time domain, u and f is a source vector (Marfurt, 1984; Zienkiewicz and Taylor, 1991). Boundary conditions are imposed to suppress artificial reflections from the edges of the dataset. Taking the Laplace transform, we obtain S~ u ¼ ~f; 10.1071/EG09031
ð2Þ 0812-3985/10/030189
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where S ¼ K þ s2 M; ð¥ ~ðsÞ ¼ uðtÞest dt; u
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Here we introduced the complex damping constant s* = s + io. The right-hand side of equation 4 can also be obtained from the Fourier transform by using a complex frequency o* = ois and allowing o to approach zero, or ð¥ ð¥ ~ðoÞ lim uðtÞeio*t dt ¼ lim uðtÞest eiot dt: ð5Þ lim u o!0
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Equations 4 and 5 imply that a Laplace-transformed wavefield corresponds to the zero-frequency component of a damped wavefield. This fact greatly simplifies the implementation of waveform inversions in the Laplace domain in that we can actually use an algorithm for the frequency domain inversion without further modification. A damped wavefield generally has a zero frequency component even though there is no zero frequency in the corresponding undamped wavefield. For example, a shot gather shown in Figure 1a has no information below 3 Hz. Figure 1b and c show the zero-offset trace and its frequency spectrum. However, its damped version contains low-frequency information in the spectrum of the zero-offset trace (Figure 2). Several damping constants are used for the Laplace-domain waveform inversion. When the damping constant is small, the proposed strategy damps late arrivals more than early arrivals; therefore, the results are more weighted towards the early arrivals. This method is similar to early arrival waveform tomography (Sheng et al., 2006), which uses an early arrival time window for the same purpose. When the damping constant is large, information for the first arrivals can only be obtained from a wavefield, and our strategy thus mirrors first arrival traveltime
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Fig. 2. (a) The damped shot gather with a damping constant of 5.0 (equation 3), (b) the zero-offset trace of the damped shot gather until 2 s, and (c) the frequency spectrum of the zero-offset trace of the damped shot gather.
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Fig. 3. Comparison between inversion algorithms that use either the frequency or Laplace domains.
where the upper bar signifies a complex conjugate. Here m is the vector of model parameters, which contains the velocities of the subsurface structure in the acoustic case. Additionally, u is the modelled wavefield, d is the observed wavefield, Nf is the number of frequencies in the frequency domain (or the number of damping constants in the Laplace domain), Ns is the number of sources, and Nr is the number of receivers. We can easily modify the logarithmic inversion technique developed in Shin and Min (2006) and Shin et al. (2007) by holding the real frequency at zero and changing the damping constant appropriately. Figure 3 compares a frequency-domain waveform inversion algorithm to a Laplace-domain inversion algorithm. Both algorithms start with the same initial velocity model. In the frequency domain, we used a fixed damping constant to
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tomography (Aki and Richards, 2002). Using complex frequencies can reduce the number of local minima in the frequency-domain waveform inversion (Brenders and Pratt, 2007). Waveform inversion in the Laplace domain also uses complex frequencies, however, the real part of the frequency is zero. We can then invert smooth velocity models by using a range of damping constants for the imaginary portion of the complex frequency. To implement the waveform inversion, we adopted the logarithmic objective function employed by Shin and Min (2006), namely Nf X Ns X Nr X uijk uijk ð6Þ ; ln ln EðmÞ ¼ dijk dijk i j k
Frequency (Hz) Fig. 4. (a) A 2-D SEG/EAGE salt model, (b) the source wavelet used in the time domain modelling, and (c) the frequency spectrum of the source wavelet.
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~(o). In the Laplace domain, generate the forward modelled data u we set the angular frequency to zero and varied the damping ~(s). In the constant to generate the forward modelled data u frequency domain, field data were transformed via a Fast Fourier Transform. In the Laplace domain, data were transformed by a simple numerical integration technique, such as the trapezoidal rule. As in Shin and Min (2006), we used the steepest descent method to find the velocity model and computed the direction of
steepest descent by multiplying a virtual source vector by the back-propagated residual vector. Specifically, we calculated the residual vector between the forward modelled data and the observed data for each o* and shot. We normalised the steepest descent direction by a pseudo-Hessian function (Shin et al., 2001), which is damped by a real damping constant to attain stability. After each iteration, we updated the velocity model by the product of the normalised steepest descent direction and the current step size. If the root mean square error did not meet our
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Fig. 6. (a) The Laplace transformed shot gather shown in log scale and cross-sections of the transformed shot gather for damping constants of (b) 5, (c) 10, and (d) 15.
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Fig. 7. Sections of the residual spectrum Re[ln(uijk/dijk)] of the 48th shot after two iterations. The data were obtained in the frequency domain at frequencies of (a) 5 Hz, (b) 10 Hz, and (c) 15 Hz.
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domain data were generated via a fourth-order finite difference method with free surface and absorbing boundary conditions (Reynolds, 1978). To simulate the limitations of the real data, the source wavelet was obtained by applying a trapezoidal frequency filter (frequencies of 3, 5, 50, 60 Hz) to a first-derivative Gauss wavelet (Figure 4b) with a main frequency of 11.2 Hz (Figure 4c). The number of shots was 96, and the number of receivers was 779. The shot interval was 160 m, the receiver interval was 20 m, and the grid spacing in both the horizontal and vertical dimensions was 20 m. The total recording time was 10 s with a sampling interval of 1 ms.
convergence criterion, a new iteration was performed using the updated velocity model. Note that the only difference between the two algorithms is the choice of the frequency or Laplace domain. We also estimated the source wavelet following the technique suggested by Shin et al. (2007), Numerical examples A synthetic data example We applied the frequency-domain waveform inversion to a 2-D section of the SEG/EAGE salt model (Figure 4a). The time
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Fig. 13. (a) The frequency-domain inversion result obtained by using the recovered velocity model in the Laplace domain as the starting velocity model and (b) the migration image corresponding to the velocity.
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Fig. 14. A shot gather from a marine survey.
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the Laplace domain clearly showed long-wavelength velocity information (Figure 10). The inverted velocity models are shown in Figure 11. The resolution of the inverted velocity model obtained in the frequency domain was high, but the background velocity of the inverted velocity model was similar to that of the initial model. The inverted velocity provides only a qualitative image of the reflectivity. Waveform inversion in the frequency domain cannot recover background velocity due to the lack of lowfrequency data. Laplace-domain inversion provides a much smoother velocity model and successfully reconstructs longwavelength velocity. The resulting model is preferable for migration or subsequent frequency-domain waveform inversion. Figure 12 shows the reverse time migration results obtained using the two inverted velocity models. The migration results of both models show the top boundary of the salt body and the faults between the subsurface layers. However, only the Laplacedomain model recovers information for the bottom boundary of the salt body. Figure 13a shows the frequency-domain
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Figure 1a shows a shot gather. The frequency spectrum of the zero-offset trace (Figure 1b) contains no information for frequencies below 3.0 Hz (Figure 1c). Figure 5 shows the full frequency spectrum of the shot gather and three cross-sections. Note that the higher frequency oscillations exhibit more variability in amplitude. In contrast, Figure 6 shows that the Laplace-transformed wavefield has much smoother variability. Smooth wavefields result in smooth residuals, and smooth residuals result in smooth back-propagated residuals and smooth gradient directions. Accordingly, they yield smooth velocity models. We confirmed this in the following example. We inverted for the velocities in both the frequency and Laplace domains using the finite element method with free surface and absorbing boundary conditions (Clayton and Engquist, 1977). The initial velocity model was a linear function varying from 1.5 km/s to 3.0 km/s, increasing with depth. We used 85 frequencies ranging from 3.0 Hz to 20.0 Hz for the frequency-domain waveform inversion. We used 12 damping constants ranging from 2.0 to 15.0 in the Laplace domain. The gradients used to update the velocity model were calculated by back-propagating the residuals. It is interesting to analyse the relationship between the inverted velocity model and the shape of the residuals. Figure 7 shows the high oscillations of residuals produced by the frequency-domain inversion after two iterations. As the frequency increased, the model captured more information for the sharper variations of the wavefield. This tendency can also be observed in the frequency spectrum of the recorded data (Figure 5). This observation suggests that information for low frequencies is required to invert longwavelength information. Figure 8 shows the residuals of the Laplace-domain inversion after the second iteration. The Laplace transform captures the information of the longwavelength features for all damping constants. The directions of steepest descent used to obtain the next iteration of the model were obtained by back-propagating the residuals. Even by the second iteration, reflectors could be distinguished in the gradient field of the frequency-domain inversion, but this information was mainly composed of shortwavelength oscillations (Figure 9). However, the gradient field in
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inversion result obtained using the Laplace-domain inversion result as the starting velocity model. Although the corresponding migration image has some artefacts, it clearly identifies the salt boundaries and faults (Figure 13b). A field data example Figure 14 shows the first 10 s of a shot gather obtained from a marine seismic survey, which has 599 shots, each with 408 receivers. The shot interval was 100 m, and the receiver interval was 25 m. The maximum offset distance was 10 198 m, and the minimum offset was 173 m. The recording time was 18.4 s, and the time sampling interval was 4 ms. The data was inverted using the initial velocity model in Figure 15a. We fixed the velocity of the water layer and updated the velocity below the water bottom only during inversion. We used 22 frequencies ranging from 0.5 to 11.0 in the frequency-domain inversion and nine damping constants ranging from 1.0 to 13.0 in the Laplacedomain inversion. The inverted velocity model obtained in the frequency domain using the logarithmic objective function is presented in Figure 15b. It is not far from the starting velocity model. However, Laplace-domain inversion result reveals high-velocity structures and background velocity (Figure 16a). We used the Laplace-domain inversion result as the starting velocity model for a frequency-domain waveform inversion and obtained the velocity model shown in Figure 16b. Frequency-domain inversion added short-wavelength features to the background velocity obtained using the Laplace-domain inversion. Figure 17 shows three migration images corresponding to the three inversion results. Migration images in Figure 17b and c (a)
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clearly show the top of the high-velocity structure, which is not shown in Figure 17a. The image derived from the Laplacedomain result shows better continuity of layers than that of the image derived from the frequency-domain result obtained using the Laplace-domain result as the starting velocity model. A smooth velocity can yield a better depth migration image than that derived from a high-resolution velocity with errors for real data (Gray, 2000; Pacheco and Larner, 2005). Conclusions We have compared two seismic full waveform inversion methods that used either the frequency domain or the Laplace domain. Specifically, we have examined the ability of either inversion method to accurately reconstruct a smooth background velocity model. Although several studies have attempted to refine the velocity models generated from frequency-domain inversions, this technique remains unable to recover sufficient low-wavenumber information from observed data. Waveform inversions in the frequency domain also generate highly non-linear characteristics. Low-frequency waves are critical for subsurface imaging, but they are difficult to generate. Compared to frequency-domain inversions, Laplace-domain inversions can be used to construct velocity models that better capture long-wavelength velocity structures. The Laplacedomain wavefield inversion is successful because it represents a zero-frequency of a damped seismogram, which extracts smooth background velocity from the available data for various values of the damping constant. The smooth velocity model obtained in the Laplace domain can be used as an accurate initial velocity model for subsequent frequency-domain inversion. This technique can be further advanced by defining the optimum range of damping constants, the ideal interval between damping constants, and the relationship between the damping constant s and angular frequency. Furthermore, the applications of the model for 2-D acoustic approximations or circumstances that cause the Laplace-domain inversion to fail have not been determined. The method has also not been applied to a 3-D elastic inversion. Although there are still significant theoretical details to resolve, this study suggests that waveform inversion in the Laplace domain provides a new technique for solving seismic inversion problems.
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Acknowledgments
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This work was supported by the Brain Korea 21 project of the Ministry of Education. We are grateful to GX Technology for providing us with the field datasets.
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Manuscript received 15 July 2009; accepted 16 July 2010.
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