2D Laplace-Domain Waveform Inversion of Field Data Using a Power Objective Function

Transcription

1 Pure Appl. Geophys. Ó 213 Springer Basel DOI 1.17/s Pure and Applied Geophysics 2D Laplace-Domain Waveform Inversion of Field Data Using a Power Objective Function EUNJIN PARK, 1 WANSOO HA, 1 WOOKEEN CHUNG, 2 CHANGSOO SHIN, 1 and DONG-JOO MIN 1 Abstract The wavefield in the Laplace domain has a very small amplitude except only near the source point. In order to deal with this characteristic, the logarithmic objective function has been used in many Laplace domain inversion studies. The Laplacedomain waveform inversion using the logarithmic objective function has fewer local minima than the time- or frequency domain inversion. Recently, the power objective function was suggested as an alternative to the logarithmic objective function in the Laplace domain. Since amplitudes of wavefields are very small generally, a power \1 amplifies the wavefields especially at large offset. Therefore, the power objective function can enhance the Laplacedomain inversion results. In previous studies about synthetic datasets, it is confirmed that the inversion using a power objective function shows a similar result when compared with the inversion using a logarithmic objective function. In this paper, we apply an inversion algorithm using a power objective function to field datasets. We perform the waveform inversion using the power objective function and compare the result obtained by the logarithmic objective function. The Gulf of Mexico dataset is used for the comparison. When we use a power objective function in the inversion algorithm, it is important to choose the appropriate exponent. By testing the various exponents, we can select the range of the exponent from to in the Gulf of Mexico dataset. The results obtained from the power objective function with appropriate exponent are very similar to the results of the logarithmic objective function. Even though we do not get better results than the conventional method, we can confirm the possibility of applying the power objective function for field data. In addition, the power objective function shows good results in spite of little difference in the amplitude of the wavefield. Based on these results, we can expect that the power objective function will produce good results from the data with a small amplitude difference. Also, it can partially be utilized at the sections where the amplitude difference is very small. Key words: Power objective function, field data, inverse theory. 1 Department of Energy Systems Engineering, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul, South Korea. wansooha@gmail.com 2 Department of Energy and Resources Engineering, Korea Maritime University, 727 Taejong-ro, Yeongdo-gu, Busan, South Korea. 1. Introduction Since LAILLY (1983) and TARANTOLA (1984) suggested the back-propagation technique for waveform inversion, many studies on waveform inversion have been performed in the time domain (MORA 1987; BUNKS et al. 1995; SHIPP and SINGH 22) and in the frequency domain (GELLER and HARA 1993; PRATT et al. 1998; OPERTO et al. 24; SHIN and MIN 26; SHIN et al. 27). However, both domains of waveform inversion have problems such as the existence of local minima and the absence of low-frequency components. In particular, it is difficult to generate or measure lowfrequency components in seismic exploration. Therefore, conventional waveform inversion fails in recovering long-wavelength velocity models (SYMES 28). SHIN and CHA (28) proposed an acoustic waveform inversion technique in the Laplace domain. The objective function in the Laplace domain has fewer local minima than in the frequency domain (SHIN and HA 28), and it can construct the longwavelength velocity model reflecting the characteristics of the subsurface medium. In particular, when we apply the method to the field data that has small low-frequency components or high noise levels, it shows better results than other waveform inversions (SHIN and CHA 28). Based on these advantages, the Laplace-domain waveform inversion has been expanded to 3D acoustic mediums (PYUN et al. 211), elastic mediums (CHUNG et al., 21) and acousticelastic coupled mediums (BAE et al. 21). To increase the resolution of inversion results, the Laplace Fourier domain waveform inversion has also been studied (SHIN and CHA 29). For a waveform inversion based on the gradient method, the difference between the observed and

2 E. Park et al. Pure Appl. Geophys. modeled data is defined by using an objective function. We obtain the velocity model by iteratively minimizing the objective function. Therefore, the choice of the objective function has a great deal of influence on the inversion results (SHIN and HA 28). Waveform inversions in the time or frequency domain generally use an l 2 -norm objective function (TARANTOLA 1984; PRATT et al. 1998; SHIPP and SINGH 22). However, in the Laplace domain, it is unsuitable to use the l 2 -norm as an objective function because the Laplace-transformed wavefield has very large amplitude only near the source. For this reason, most of the Laplace-domain waveform inversion algorithms use the logarithmic objective function suggested by SHIN and MIN (26). SHIN and HA (28) performed the Laplacedomain waveform inversion using various objective functions. The power objective function is suitable for finding the global minimum, when the exponent has a small value. SHIN and HA (28) applied the power objective function to a synthetic dataset in the Laplace domain, and they obtained very similar results to the logarithmic objective function. In this paper, we focus on the power objective function and conduct the in-depth study for the Laplace domain waveform inversion. We develop the Laplace-domain waveform inversion algorithm using the power objective function, and the algorithm is applied to the field dataset in acoustic media. From this research result, we can discuss the applicability of the power objective function in the Laplace domain waveform inversion. 2. Theory 2.1. Wave equation in the Laplace domain The 2D acoustic wave equation in the time domain for a homogeneous and isotropic medium can be expressed as r 2 uðx; tþ 1 o 2 uðx; tþ c 2 ot 2 ¼ f ðx; tþ; ð1þ where c is the velocity, uðx; tþ is the wavefield in the time domain, and f ðx; tþ is the source function in the time domain. with and The Laplace transform of Eq. (1) is given by r 2 ~u ðx; s Þ s2 ~u ðx; sþ ¼ ~f ðx; sþ ¼ ~u x; s c2 ð Þ ¼ f ~ ðx; sþ; ð2þ Z 1 Z 1 uðx; tþe st dt; ð3þ f ðx; tþe st dt; ð4þ where s is a Laplace damping constant. In Eq. (3), uðx; tþe st is the damped wavefield at a given s. This equation can be written as ~u ðx; sþ ¼ lim x! Z1 Z1 ¼ lim x! 1 uðx; tþe st dt uðx; tþe st e ixt dt; ð5þ where x is the angular frequency. We can know that the Laplace-transformed wavefield is the zero-frequency component of the damped wavefield. Because of this characteristic, a long-wavelength velocity model is recovered and smooth inversion results are provided. By using the finite element method (FEM), Eq. (2) can be written as with S~u ¼ ~ f; S ¼ K þ s 2 M; ð6þ ð7þ where M is a mass matrix, K is a stiffness matrix, u is the wavefield, and f is the source vector Objective function and gradient direction In this paper, we use the power objective function suggested by SHIN and HA (28). The objective function for a damping constant is expressed as EðmÞ ¼ 1 X N sr X N rc ~u p jk 2 ~d p jk 2 ; ð8þ j¼1 k¼1

3 Power Objective Function in the Laplace Domain where m is the model parameter vector representing characteristics of a medium, ~u is the modeled data, ~d is the field data, N sr and N rc are the number of sources and receivers. Because the characteristics of the wavefield depend on the exponent, it is important to select an appropriate exponent (SHIN and HA 28). After determining the objective function, we focus on minimizing it. We first have to calculate the steepest-descent direction of the objective function. The derivative of Eq. (8) with respect to the l-th model parameter m l can be expressed as oe om l ¼ " XNsr X N rc o~u # jk p ~u p 1 jk ~u p jk om ~d p jk l j¼1 k¼1 : ð9þ Likewise, differentiating Eq. (6) with respect to a model parameter m l, we obtain os ~u þ S o~u ¼ ; l ¼ 1; 2;...; N p ; ð1þ om l om l where N p is the number of parameters. Equation (1) is simply expressed as with o~u om l ¼ S 1 v l ; l ¼ 1; 2;...; N p ð11þ v l ¼ os ~u ; ð12þ om l where v l is the virtual source vector with respect to the l-th model parameter m l (SHIN and MIN 26). By substituting Eq. (11) by Eq. (9), we obtain and oe om l ¼ XNsr j¼1 ð Þ T S 1 r v l ð13þ 2 p ~u p 1 j1 ~u p j1 3 ~d p j1 p ~u p 1 j2 ~u p j2 ~d p j2. r ¼ p ~u p 1 jn r ~u p jn r ~d p jn r : ð14þ The gradient direction of the logarithmic objective function is same as Eq. (13), but r is given by ~u ij1 ln ~u ij1 ~d ij1 1 ~u ij2 ln ~u ij2 ~d ij2. r ¼ 1 ~u ijnr ln ~u ijnr ~d ijnr : ð15þ In Eq. (13), S 1 r is the back-propagated wavefield vector. We can obtain the steepest-descent direction by calculating the zero-lag convolution between the virtual source vector and the back-propagated wavefield. The back-propagation algorithm helps the gradient direction to be calculated efficiently because the partial derivative wavefield doesn t have to be calculated directly (TARANTOLA 1984; PRATT et al. 1998; PRATT 1999) Normalization and inversion flow In order to minimize the objective function, we iteratively update the model parameter m l. The model parameter is updated by the preconditioned gradient method (PRATT et al. 1998). It is expressed as m nþ1 l ¼ m n l þ dm ð16þ dm ¼ H 1 r m E; ð17þ where H is a Hessian matrix. Although optimizing techniques have been developed for efficiently accounting for the Hessian matrix in recent studies (ABUBAKAR et al. 211; MÉTIVIER et al. 212), the Hessian matrix commonly requires large amount of computations. Therefore, we use the diagonal elements of the pseudo-hessian matrix suggested by SHIN et al. (21) instead of the Hessian. Equation (16) can be rewritten as ml nþ1 ¼ m n l þ dm ¼ m n l þ a n g n ; ð18þ where a n is the step length at n-th iteration and g n is the steepest-descent direction normalized by the

4 E. Park et al. Pure Appl. Geophys. Figure 2 a The wavefield, b the powered wavefield (p =.5), and c the logarithmic wavefield of field data in the Laplace-domain Figure 1 a The first common shot gather and b its frequency spectrum obtained from the Gulf of Mexico dataset pseudo-hessian. The preconditioned gradient direction g n can be expressed as (HA et al. 29) " " ( P g n ¼ NRM XN s Nsr j¼1 ðv l Þ T S 1 )## r NRM P Nsr i¼1 j¼1 ðv l Þ T^v ; l þ k ð19þ where N sr is the number of the damping constant, n is the iteration number, NRM is the normalizing operation and k is a stabilizing factor in the inversion. The NRM (values) is to divide each value by the largest absolute value. The stabilizing factor k is used to avoid singular values of the pseudo-hessian (MAR- QUARDT 1963; LEVENBERG 1994). In order to obtain g n, we first divide the steepestdescent direction by the pseudo-hessian. Then we can normalize it using the maximum absolute value. Second, we repeatedly perform previous steps for each Laplace damping constant. Finally, we sum all values and normalize once more. In this manner, we can update the velocity by using the gradient direction. And by repeating the algorithm, we can identify subsurface structures.

5 Power Objective Function in the Laplace Domain 3. Numerical results 3.1. Preprocessing and wavefield analysis The field datasets generally have a lot of noise, so the preprocessing processes are needed. We use a preprocessing technique for filtering and muting. We cut the low-frequency components using a low-cut filter and mute the upper part of the direct wave, because the signal is almost unclear but the noise is evident in these parts. Figure 1 show the first shot gather of the Gulf of Mexico dataset and its frequency spectrum after preprocessing. The frequency spectrum is expressed except for a.5 % value from the maximum and a.5 % value from the minimum. We can note that the frequency components lower than 4 Hz are missing in this data. When the data doesn t have low-frequency components, it is difficult to obtain correct subsurface structure. One way to improve this problem is applying the Laplace-domain waveform inversion, which generates the long-wavelength velocity model. The Laplace-transformed wavefield obtained from the shot gather (Fig. 1a) is shown in Fig. 2. The Laplace damping constant is 7. As shown in Fig. 2a, the Laplace-transformed wavefield has values close to zero except for a very large amplitude near source. Figure 2b, c shows the wavefield to the power of.5 and the logarithmic wavefield in Laplacedomain. Likewise, by taking the logarithm or the exponent to \1 at the wavefield, we can amplify the wavefield close to zero. Through this process, the difference of the modeled data and observed data is Figure 3 Laplace-transformed wavefield a with various damping constants (p =.5) and b with various exponent values (s = 7)

6 E. Park et al. Pure Appl. Geophys. Figure 4 The comparison of the observed data and the inverted data of a first shot gather a in the powered wavefield (p =.5) and b in the logarithmic wavefield Figure 5 The starting velocity model. (1st layer: 1.5 km/s, 2nd layer: 3.3 km/s) remarkably more clear. Then, each objective function is less likely to converge to local minima. Figure 3 expresses the wavefield depending on various parameters in the Laplace domain. We first fix the exponent at.5 and vary the damping constants (Fig. 3a). We can know that the lager damping constant includes a lot of relatively low wave-number components. Next, we fix the damping constant at seven and use several exponents. When the exponent is smaller, the difference of the amplitude is smaller as shown in Fig. 3b Application on the field dataset We tested our waveform inversion algorithm on a field dataset acquired from the Gulf of Mexico. There were 399 shots with a shot interval of 5 m and 48 receivers with a receiver interval of 25 m. The maximum recording length is 12 s and the time

7 Power Objective Function in the Laplace Domain sampling interval is 4 ms. The offset ranges from 137 to 1,321 m and the water depth ranges from 42 to 91 m. In order to confirm the applicability of the algorithm, we compare the observed data and the inverted data of a first shot gather in the Laplace domain. Figure 4 is the result of each comparison in the powered wavefield and the logarithmic wavefield. The Laplace damping constant is five and the exponent of the powered wavefield is.5. Because the inverted data has a similar result with the observed data, we verify that it is reasonable to apply this algorithm. The starting model is a two-layer velocity model (Fig. 5). The first layer, the seawater layer, is fixed at 1.5 km/s and the second layer has a homogeneous velocity of 3.3 km/s. The grid interval for inversion is 25 m. Nine Laplace damping constants are used, ranging from 1 to 13 with an interval of 1.5. Our algorithm performed 1 iterations. The source wavelet and velocity are updated in each iteration. The source wavelet is estimated along the method suggested by SHIN et al. (27). We inverted the velocity model by changing the exponents in the power objective function. SHIN and HA (28) found from a test of a salt model that the power objective function was very smooth and desirable for local optimization methods when the exponent was equal to or smaller than.1. From this result, we choose the first exponent of.5 and next values are reduced to a tenth of the time of the preceding value. We tested the exponent from to The result was good and almost the same with each other when the exponent was from to and the rest of the exponents were not good. Therefore, we selected three exponents for analysis of the power objective function in this test. The velocity model using the logarithmic wavefield is shown in Fig. 6d. When the exponent p is (Fig. 6a), the result is poorer than in Fig. 6d. However, inverted velocity models are similar to Fig. 6d when the exponent p is and The reason why Fig. 6a shows a lowquality result is that it used too large an exponent value. As shown in Fig. 3b, too large an exponent can t amplify the wavefield enough in the power objective function. If the value of the wavefield is too small, the objective function is more affected by variations in the background velocity than variations in the anomalous-body velocity, and then the objective function has more of a possibility to converge to the local minima. If the exponent is equal to one in the power objective function, it is same for the l 2 -norm objective function. As mentioned earlier, the l 2 -norm objective function is not recommended for the Laplace-domain waveform inversion because it is very vulnerable to the local minima problem. In contrast, if the exponent is too small it is also difficult to obtain good inversion results, since most of the values of the wavefield converge to one. In this test, all values of the wavefield are calculated into one when the exponent is , so the algorithm failed to find the velocity model of the substructure. We calculate the velocity error of the each power objective function based on the logarithmic objective Figure 6 The inverted velocity model using the power objective function with a p = , b p = and c p = , and d using the logarithmic objective function after the 1th iteration

8 E. Park et al. Pure Appl. Geophys. Figure 7 The velocity error of several power objective functions about the logarithmic objective function at 8 km (a and b) and 1 km (c and d) offset function in order to know the similarity between two objective functions. The equation is as follows; error ð% Þ ¼ v p v l 1; ð2þ v l where v p is the velocity obtained by using the power objective function and v l is the velocity obtained by using the logarithmic objective function. We choose the velocity profile at 8 and 1 km from the left, extracted from Fig. 6a d. Figure 7a, c show that the power objective function with p = has a relatively large difference about the logarithmic objective function. In contrast, the power objective function with p = and with p = have small errors, so we do not distinguish it with the naked eye. Therefore, we draw only these two power objective functions in Fig. 7b, d. The error of the power objective function with p = is \.1 percent, so we can know that it has very similar results to the logarithmic objective function. The result obtained from the logarithmic objective function is not an exact solution, but its accuracy has been verified through many experiments about various synthetic data. Therefore, we can determine that the power objective function with suitable exponent gives good results in the Laplace-domain waveform inversion. In the previous Fig. 3b, we know that the smaller the exponent is, the smaller the difference of the amplitude is. Despite the power objective function with p = having almost no difference for the amplitude, it shows a good inversion result. From

9 Power Objective Function in the Laplace Domain Figure 7 continued this result, we can see that the power objective function has strengths for the data with small amplitude differences. The migrated images generated from the velocity models in Fig. 5 and Fig. 6 are shown in Fig. 8 and Fig. 9. We use the simple reverse-time migration (RTM) algorithm. Figure 9d is obtained from the logarithmic objective function and shows overall improved results compared to Fig. 8 generated from the starting velocity model. Figure 9b, c obtained from the power objective function show similar results to Fig. 9d because of very small error in the velocity model. Finally, in order to verify the accuracy of the velocity models and the migrated images, we make the common image gathers (CIGs). While the Figure 8 The migrated images generated from the starting velocity models in Fig. 5 migrated images are obtained by overlapping images of all shot points, CIGs are obtained by horizontally arraying images of a specific shot point. Therefore, theoretically, CIGs obtained by exact velocity should

10 E. Park et al. Pure Appl. Geophys. Figure 1 The common-image gathers (CIGs) obtained by a using the initial velocity model and using the inverted velocity model with b p = and c p = Figure 9 The migrated images generated from the corresponding velocity models in Fig. 6 have flat structures. Figure 1 shows the CIGs obtained from the 225th shot. Figure 1a shows the CIGs using the initial velocity model, whereas Fig. 1b, c show the CIGs using the inverted velocity model with p = and with p = Most reflection events in the CIGs using the inverted velocity model are flattened. Therefore, we can confirm that the proposed algorithm of the power objective function can be applied to the field dataset. 4. Conclusions We performed the Laplace-domain waveform inversion on the field dataset. The field datasets generally have a lot of noise, so we applied the preprocessing step and analyzed the Laplace wavefield before inversion. The Laplace wavefields generally have very small absolute values; therefore, they are amplified by taking the logarithm or the exponent as \1. Through the wavefield analysis, we can know that the logarithmic wavefield and the powered wavefield have less risk of falling into local minima than the Laplace wavefield. Also, we confirm the accuracy of this algorithm through comparing the inverted wavefield and the real wavefield. We tested our algorithm on field data by using several wavefield exponents and comparing the results to those of the logarithmic objective function. In the case of the Gulf of Mexico dataset, exponents from to in the power objective function yielded very similar results to that of the logarithmic objective function. The inversion result has low quality when the exponent is larger than or smaller than , because too large an exponent can t amplify the wavefield enough and too small an exponent makes all values of the wavefield converge to one.

11 Power Objective Function in the Laplace Domain Here, we can see that subsurface structures are well recovered by using the power objective function algorithm with an appropriate exponent. Although the power objective function does not show significantly better results than the logarithmic objective function, we can confirm the applicability of the Laplace-domain waveform inversion using the power objective function on field datasets. Especially, the power objective function shows good results despite the wavefield having almost no difference in the amplitude. Therefore, we can draw a conclusion that the power objective function has strengths on the data with small amplitude difference. Based on these results, we can expect that the proposed algorithm using the power objective function will produce good results from the data with small amplitude differences. Also, it can partially be utilized at the sections where the amplitude difference is very small. Acknowledgments This work was supported financially by the Energy Efficiency & Resources (No. 21T12376) and Human Resources Development program (No ) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy. We thank GX Technology for providing us with the field data. REFERENCES ABUBAKAR, A., LI, M., LIU, J., and HABASHY, T.M. (211), Application of the compressed implicit jacobian scheme for elastic full waveform inversion, In Expanded Abstracts, EAGE. BAE, H.S., SHIN, C., CHA, Y.H., CHOI, Y., and MIN, D.J. (21), 2D acoustic-elastic coupled waveform inversion in the Laplace domain, Geophysical Prospecting 58, BUNKS, C., F.M. SALECK, S. ZALESKI and G. CHAVENT. (1995), Multiscale seismic waveform inversion, Geophysics 6, CHUNG, W., SHIN, C., and PYUN, S. (21), 2D Elastic Waveform inversion in the Laplace Domain, Bulletin of the Seismological Society of America 1, No. 6, GELLER, R.J., and T., HARA. 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