Seismic envelope inversion: reduction of local minima and noise resistance

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1 Geophysical Prospecting, 5, 63, doi:./ Seismic envelope inversion: reduction of local minima and noise resistance Jingrui Luo, and Ru-Shan Wu Institute of Wave and Information, Xi an Jiaotong University, Xi an, 749, China, and Modeling and Imaging Laboratory, Earth & Planetary Sciences, Univ. of California, Santa Cruz, CA 9564 Received September 3, revision accepted July 4 ABSTRACT Waveform inversion met severe challenge in retrieving long-wavelength background structure. We have proposed to use envelope inversion to recover the large-scale component of the model. Using the large-scale background recovered by envelope inversion as new starting model, we can get much better result than the conventional full waveform inversion. By comparing the configurations of the misfit functional between the envelope inversion and the conventional waveform inversion, we show that envelope inversion can greatly reduce the local minimum problem. The combination of envelope inversion and waveform inversion can deliver more faithful and accurate final result with almost no extra computation cost compared to the conventional full waveform inversion. We also tested the noise resistance ability of envelope inversion to Gaussian noise and seismic interference noise. The results showed that envelope inversion is insensitive to Gaussian noise and, to a certain extent, insensitive to seismic interference noise. This indicates the robustness of this method and its potential use for noisy data. Key words: Envelope inversion, Waveform inversion, misfit function, convergence rate, noise resistance. INTRODUCTION Full waveform inversion (FWI) was introduced in the early eighties in the time domain by Lailly (983) and Tarantola (984). The gradient of the misfit function was calculated using back propagation method, thus avoided the explicitly computing of the partial derivatives (Virieux and Operto, 9). Mora (987, 988) extended this technique to elastic problems. Crase et al. (99) applied this method to real data. Pratt (999) and Pratt and Shipp (999) applied the same idea to frequency domain full waveform inversion. Recently, full waveform inversion has been developed to 3D (Warner et al., 3), and high performance computation has been applied (Mao et al., ). However, in full waveform inversion, we always need a good initial model that is not far from the true model to get correct inversion results. bluebirdjingrui@gmail.com In order to overcome this problem, some people used global optimization methods (Boschetti, 996; Mallick, 999; Padhi et al., ). However, applications of these methods are restricted because of the tremendous computation cost. Several other approaches have been developed in order to overcome this problem. Bunks et al. (995) introduced multiscale full waveform inversion method, where the scale decomposition is performed and allows us to do the inversion from low frequency to high frequency. Frequency domain full waveform inversion is an intrinsic multiscale approach (Pratt, 999; Pratt and Shipp, 999). Brenders and Pratt (7) used complex-valued frequencies in frequency domain full waveform tomography to ease the local minima problem. However, the recovery of long wavelength component of the model depends on the availability of low-frequency signal in seismic source. To generate low-frequency signal below 5Hz is very expensive, even not realistic below - Hz. Therefore, effort has been focused to the recovery of long wavelength C 4 European Association of Geoscientists & Engineers 597

2 598 J. Luo and R.-S. Wu Receiver locations 5 5 Receiver locations 5 5 Time (s) Frequency (Hz) 3 3 Time (s) Frequency (Hz) 3 3 Figure Data traces and their spectra. A shot profile (common shot records) of the seismic data (top) and the envelope (bottom). Seismic data spectra (top) and envelope spectra (bottom). The data are from the synthetic data set of the Marmousi model. Trace Original trace Trace envelope Trace Original trace Trace envelope Time (s) 3 x 4 Trace spectrum Original trace Trace envelope Time (s) 3 x 4 Trace spectrum Original trace Trace envelope Frequency (Hz) Frequency (Hz) Figure A trace from the shot profile in Fig.. The top panel shows the time-domain trace and its envelope; the bottom panel shows the amplitude spectra of the trace and its envelope; same as except that the low-frequency component below 5 Hz was removed from the source wavelet. background without very low frequencies. Travel time inversion and migration velocity analysis are two traditional methods in this area. In travel time inversion, the travel time information of the wavefield is used to invert the model parameters. It can be classified to two categories. One is ray based travel time inversion (Dines and Lytle, 979; Paulsson et al., 985; Justice et al., 989) which needs the picking of travel time, and the other is wave equation travel time inversion (Luo and Schuter, 99; Woodward, 99) which is based on wave theory instead of ray theory. In migration velocity analysis, the residual moveout (Liu and Bleistein, 995; Xie and Yang, 8) or the image perturbation (Biondi and C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

3 Seismic envelope inversion Time(s) Time(s) Figure 3 Effective data residual for back propagation in the time domain (the amplitudes have been normalized). The first iteration. The th iteration. The top panels are the seismic data residuals for the conventional full-waveform inversion; the middle panels are the effective residuals for the EI using envelope misfit; and the bottom panels are the effective residuals for EI using squared envelope misfit. Sava, 999; Sava and Biondi, 4; Shen and Symes, 8) is used, and the velocity perturbation is then obtained. In recent years, Shin and Cha (8) has developed Laplace domain full waveform inversion which can give a smooth background model from an inaccurate initial model. Later, they extended this method to the Laplace-Fourier domain full waveform inversion (Shin and Cha, 9). There are also another kind of methods, which combines waveform inversion and some other types of inversion. For example, Zhou et al. (995) and Zhou et al. (997) combined traveltime and waveform inversion. Biondi and Almomin () and Almomin and Biondi () combined full waveform inversion with wave-equation migration velocity analysis. Wang et al. () used wave equation tomography and full waveform inversion. Liu et at. () proposed the normalized integration method, in which the envelope information of the data was used. In their method, they measure the misfit between the integral of the absolute value, or of the square, or of the envelope of the signal. The integration is an increasing function, and the objective function is convex. This method can provide an intermediate solution between a very oscillating solution and a very smooth model. We (Wu et al., 3, 4) have proposed an envelope inversion (EI) method. In our method, we also use the envelope information of the seismograms. However, instead of measuring the integral of the envelope, we directly measure the envelope or the squared envelope as a function of time. The low frequency information in the seismogram envelope can be used, and a very smooth background structure can be obtained. In this paper, we discuss some detailed implementation and application issues of envelope inversion. We first give the procedure about how to realize envelope inversion. Then we compare the behavior of the misfit functions between envelope inversion and the conventional waveform inversion. From the numerical results we can see that envelope inversion can avoid or reduce the local minima problem. The combined inversion of EI and WI can deliver more faithful and accurate final result with almost no extra cost compared with conventional FWI. Also, we test the noise-resistant C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

4 6 J. Luo and R.-S. Wu property of envelope inversion. Numerical results showed that this method is insensitive to white Gaussian noise, and to a certain extent insensitive to seismic interference noise. This indicates the robustness of this method and its potential application in dealing with noisy data. A BRIEF REVIEW OF THE CONVENTIONAL FULL WAVEFORM INVERSION IN THE TIME DOMAIN Full Waveform Inversion can retrieve information of subsurface by fitting the observed data and the synthetic data. The classical least squares misfit functional in the time domain is, σ (m) = sr T [y(t) u(t)] dt, () where u is the observed wavefield, y is the synthetic wavefield, m is the model parameter set. The summation is over all the sources and receivers. We consider the acoustic situation with constant density, and consider velocity v as the model parameter, the gradient of the misfit function σ with respect to v can be obtained by, σ v = sr T [y(t) u(t)] (t) dt. () v By introducing an operator J (Jacobian) and a vector (data residual) η, where J = y(t),η = y(t) u(t) (3) v equation () can be written as, σ v = JT η. (4) The Jacobian J is also called the linear Fréchet derivative. It is known that this gradient can be calculated by zero-lag correlation of the forward propagated source wavefields and the backward propagated residual wavefields (Lailly, 983; Tarantola, 984; Bunks et al., 995). ENVELOPE INVERSION METHOD Extraction of a trace envelope In order to realize envelope inversion, we should first get the envelope of a signal (seismic trace). We extract envelope by taking the amplitude after the analytical signal transform using Hilbert transform. A signal without negative-frequency components is called an 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)}. (5) The Hilbert transform is defined as, H{ f (t)} = + π P f (τ) t τ, (6) where P is the Cauchy principal value. The envelope of f (t) can then be derived by, e(t) = f (t) + H{ f (t)}. (7) So we can easily get the envelope of this signal by using Hilbert transform. Figure gives an example of the seismic data and its envelope (Figure ) of a shot profile from the Marmousi model and their spectra (Figure ). The data was generated using time domain finite difference method. The source is the Ricker wavelet with a dominant frequency of Hz. We see that the envelope traces are much smoother than the seismic data traces, and the corresponding envelope spectra are much richer in low frequency components. For a better look, in Figure we zoom in one of the traces from the shot profile. Figure shows the original trace and it s envelope in both time domain and frequency domain. To better demonstrate the ability to extract low frequency information from the envelope, we removed the low frequency component below 5Hz from the data trace (Figure ). We see that in this case, the envelope is still very rich in low frequency components, which can be used to recover the long wavelength background that is beyond the reach of the conventional full waveform inversion. For detailed signal model related to envelope inversion, please see Wu et al. (3b). In Wu et al. (3b), we have shown the inversion results for both the cases with and without low frequency source component below 5Hz (for the readers convenience, we will show the results again in later sections of this paper). Misfit function In envelope inversion, we define the misfit function as the envelope misfit (Wu et al. 3a, b), σ (m) = 4 sr T e syn eobs dt. (8) where m is the model parameter, e syn and e obs are the envelopes of the synthetic trace and the observed trace C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

5 Seismic envelope inversion Frequency (Hz) Figure 4 Same as Fig. 3 but in the frequency domain (the amplitudes have been normalized) Frequency (Hz) Figure 5 Test model for the misfit function configuration test. The test model is from the top left corner of the Marmousi model. respectively. Using the Hilbert transform which has been introduced previously, the above equation can then be written as, σ (m) = 4 sr T {[y (t) + y H] [u (t) + u H(t)]}, (9) = 4 sr T E dt, where y and u are the synthetic traces and the observed traces respectively, y H and u H are the corresponding Hilbert transforms. E is the instant envelope data residual. The summation is over all the sources and receivers. Here we applied a square to the envelope, because the squared envelope has better performance in long-wavelength background recovery. The difference of the envelope and its power in the misfit function has been discussed in Wu et al. (3b). Here we also show some simple numerical examples to briefly demonstrate this, but from the view of data residuals. We can see the difference between the envelope and squared envelope from Figure 3 and Figure 4, which show the effective data residual for back-propagation (the amplitudes have been normalized). We see that the residuals for envelope inversion are much richer in low frequency components compared to the conventional residual. Meanwhile the curves for the case of squared envelope are much smoother than that for the case of envelope, and the spectra for the case of squared envelope are richer in low frequencies than the case of envelope. C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

6 6 J. Luo and R.-S. Wu (c) Figure 6 Misfit function configurations of conventional waveform inversion; the enlarged part inside the red border in ; and (c) EI. C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

7 Seismic envelope inversion line A line B Figure 7 The Marmousi velocity model. Figure 8 Linear initial model The gradient method in envelope inversion Consider wave propagation velocity v as themodel parameter. Calculating the derivative of the misfit functionσ with respect to v, weget, σ v = = = sr = sr sr T T sr T T E [y (t) + yh (t)] dt, v E[y(t) y(t) v [Ey(t) y(t) v + y H(t) y H(t) ]dt, v H{Ey H(t)} y(t) v ]dt, [Ey(t) H{Ey H (t)}] y(t) dt. () v We introduce the Fréchet derivative operator J and the effective envelope residual vector η, where J = y(t) v,η = Ey(t) H{Ey H(t)} () So equation () can be expressed as, σ v = JT η. () From equation () we see that the derivative of the envelope misfit function has the same form as that of the conventional waveform inversion (equation (4)) but with a different data residual vector. Therefore, envelope inversion can also be realized by using back-propagation method. We see the forward modeling is done with the same frequency content as for the input seismic data. However, because η is related to the effective envelope residual not the seismic data residual, the low frequency information in signal 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. ANALYSIS AND TESTS OF MISFIT FUNCTION CONFIGURATION We first compare the configuration of the misfit functions from envelope inversion and the conventional waveform inversion. We see from Figure that the envelope fluctuates along time much smoothly and have much stronger lowfrequency components than the seismic data, so we can expect that the misfit function of the envelope varies also smoother than that of the seismic data. Figure 5 shows the model we used for this test. We use the top left corner (inside the red border) of the Marmousi model as the test model. As defined in equations () and (8), misfit function can be parameterized as a function of velocity model, which varies in the multi-dimensional parameter space. In order to show the behaviors of the misfit function, we simplify the multi-dimensional parameter space into a - dimensional space. We decompose the test model v true (x) into a background model and a perturbation to the background. The background is a linear gradient background varies from.5 km/s (Vmin) to 5.5 km/s (Vmax). We call this background as the true background v true (x). The true perturbation δv true (x) is obtained from the subtraction of v true (x) and v true (x). We set a strength parameter α. By varing α, wegeta series perturbation functions δv(x) = αδv true (x). For the true model, α = %. The two parameters are Vmax and the C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

8 64 J. Luo and R.-S. Wu (c) Figure 9 EI results after one iteration; five iterations; and (c) ten iterations Envelope inversion FWI using EI initial Normalized misfit Iterations Figure Reduction of seismic data residuals. The misfit has been normalized with the norm of the original seismic data. The solid line is the seismic data misfit for the EI, and the dashed line is for EI+WI. perturbation strength α. We keep Vmin as.5 km/s, and change Vmax from.5km/s to 7.5km/s. We also change the perturbation strength α from % to 3%. By summing the background and the perturbation, we get a series of trial models. Then we calculate the waveform and envelope data from the true model and from the trial models. The misfit for each trial model is obtained from the two data sets. Figure EI+WI inversion result. This is the result of full-waveform inversion using the EI result in Fig. 9(c) as the initial model. Figure 6 shows the configuration of the misfit functions (the variation of the misfit function with respect to the two parameters). The horizontal axis is Vmax, and the vertical axis is the perturbation strength. The amplitudes have been normalized. is for the conventional waveform inversion. To see the details, we enlarge the part inside the red border (Figure 6). We see that besides the global minimum, there are some local minima, which will lead the inversion to wrong C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

9 Seismic envelope inversion Figure Conventional FWI result. The starting model is the linear initial model in Fig. 8. results. (c) shows the configuration of the envelope misfit, we see that the misfit function is very smooth, and there is only a global minimum and no local minima. So using envelope inversion can avoid or reduce the local minimum problem, and we will show this in the next section. NUMERICAL EXAMPLES OF ENVELOPE INVERSION In this part, we will show some envelope inversion results. We use Marmousi velocity model and plotted it again in Figure 7, True model EI+FWI FWI only and a -D linear gradient initial model (Figure 8) for the tests. The source wavelet is the Ricker wavelet with a dominant frequency of Hz. There are 5 shots and 8 receivers equally spaced on the surface. We start envelope inversion (EI) directly from the linear initial model. Figure 9 shows the results after iteration, 5 iterations and iterations respectively. We can see that the large-scale structures have gradually shown up in the inversion results. After iterations, we can already see the large-scale structures clearly. We plot the curve of the seismic data residual (the residual between the observed seismic data and the calculated seismic data) reduction with iterations of envelope inversion in Figure as the solid line. The misfit value has been normalized with the norm of the observation data. We see that the result has actually converged after about iterations, showing that envelope inversion converges fast. To test the validity of the recovered large-scale structure by envelope inversion, we use the result in Figure 9(c) as the new initial model and do a subsequent waveform inversion (WI). Figure shows the final result of EI+WI. This result is obtained after iterations. The convergence curve is plotted in Figure as the dashed line. We implement the inversion on GPU, and the computation time of each iteration for EI and for WI are almost the same, which is about minutes. So the computation time for envelope inversion is less than True model EI+FWI FWI only Figure 3 Comparison of inversion results. The comparison is along two vertical profiles from the velocity model (marked in Fig. 7). Trace A. Trace B. The solid lines are from the true model; dashed lines are from EI+WI; and dotted lines are from FWI alone. C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

10 66 J. Luo and R.-S. Wu (c) Figure 4 Inversion tests with low-cut (cut from 5 Hz below) source wavelet: Smooth background obtained from EI; waveform inversion using as the starting model; and (c) conventional full-waveform inversion starting from the linear initial model. (c) (d) Figure 5 A shot profile data set with Gaussian noise of dbw (without noise); dbw; (c) dbw; and (d) 3 dbw. 5 percent of the whole EI+WI procedure. There is very little extra cost to include EI into FWI. To compare the inversion results between EI+WI and the conventional FWI, we give the result of the conventional FWI in Figure, which starts directly from the linear initial model. We see that FWI result converged to a local minimum, while the EI+WI result is more faithful to the true model. We picked two traces from the model (the red lines in Figure 7) to see the inversion accuracy (Figure 3). The solid lines are the true model, the dashed lines are the EI+WI results, and C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

11 Seismic envelope inversion 67 dbw Gaussian noise dbw Gaussian noise dbw Gaussian noise dBW Gaussian noise Time (s) Figure 6 A selected trace from the shot profiles in Fig. 5. C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

12 68 J. Luo and R.-S. Wu dbw Gaussian noise dbw Gaussian noise dbw Gaussian noise dBW Gaussian noise Time (s) Figure 7 Corresponding envelopes of the traces in Fig. 6. C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

13 Seismic envelope inversion (c) (d) Figure 8 EI results (after iterations) using data with Gaussian noise of dbw (without noise); dbw; (c) dbw; and (d) 3 dbw C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

14 6 J. Luo and R.-S. Wu Figure 9 spectra of the traces in Fig. 6. Figure spectra of the envelopes in Fig. 7. the dotted lines are the conventional FWI results. We can see that the results from EI+WI are close to the true model, and are much more accurate than that of the conventional FWI. To further demonstrate the ability of envelope inversion to recover the long-wavelength structure without low frequency in the source, we now show the results when the low frequency (below 5Hz) is removed from the source wavelet (this result has been shown in Wu et al. (3b), here we show it again just for the readers convenience). From Figure we have seen that when the low frequency is removed from the source wavelet, the envelope is still very rich in low frequencies. Figure 4 shows the envelope inversion result (starting from the linear initial model) after iterations in this situation. We can still see the large-scale structures from this result. Figure 4 is the final waveform inversion result using as the new initial model. We can see that this result is very similar to Figure. For comparison, Figure 4(c) shows the conventional full waveform inversion result starting directly from the linear initial model. We see that because of the lack of low frequencies in the data, this result is even worse than Figure. C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

15 Seismic envelope inversion Figure Inversion results with Gaussian noise of dbw. EI+WI result. FWI result Figure Inversion results with Gaussian noise of 3 dbw. EI+WI result. FWI result. (c) (d) Figure 3 A shot profile data set with SI noise. is the original data set without noise; the SNR in, (c), and (d) are,, respectively. NOISE RESISTANT PROPERTY OF ENVELOPE INVERSION Sensitivity to Gaussian noise We first test the sensitivity of envelope inversion to Gaussian noise. Figure 5 shows one of the shot profiles (waveform data) from the Marmousi model. is the original profile without noise. We gradually added Gaussian noise (with a white spectrum) with the power of dbw, dbw and 3dBW (the SNR (signal to noise ratio) are.43,.43 and.57; SNR=log (signal power/noise power)), and the resulting profiles are shown in, (c) and (d) respectively. In order to show the influence of the noise to the data more clearly, in Figure 6, we plot a trace from the shot profile. C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

16 6 J. Luo and R.-S. Wu (c) (d) Figure 4 EI results (after iterations) using data with SI noise. The SNR is. The SNR is. (c) The SNR is. Figure 7 shows the corresponding envelopes. We see that with the increase of the noise power, the data is buried in the noise gradually. Figure 8 shows the envelope inversion results. Again, is the result when no noise is added,, (c) and (d) are the results using the data contaminated by Gaussian noise with noise power of dbw, dbw and 3dBW respectively. Comparing with the result without noise, the envelope inversion results with different noise powers look almost the same as the one without Gaussian noise, showing that envelope inversion is insensitive to white Gaussian noise. Since EI+WI is a two-scale inversion method, and EI can recover the large-scale structures using only the low frequency information contained in the envelope without any interaction with the high frequency seismic data, therefore it is much less sensitive to high-frequency rich white Gaussian noise. Figure 9 shows the amplitude spectra of the traces in Figure 6, and Figure shows the amplitude spectra of the trace envelopes in Figure 7 (the amplitudes have been normalized for better look). We see that with the increase of the noise power, the high frequency component of the seismic data is contaminated by the noise gradually, however, the envelope is still dominant with low frequency information. We can use envelope inversion to provide a good initial model even in the noisy environment. Even though full waveform inversion is white noise sensitive and may be strongly affected by the existing of noises, however, based on the above analysis we expect that using EI+WI method will work better and more robustly for noisy data. We tried the cases when the powers of Gaussian noise are dbw and 3dBW respectively. Figure shows the inversion results with Gaussian noise of dbw. is EI+WI result using Figure 8 as the initial model; is FWI result starting from the linear initial model. In this case, the noise is not very strong, and these two results look just like the results without noise as shown in Figure and. On the other hand, Figure shows the inversion results with strong Gaussian noise (3dBW). is EI+WI result using Figure 8(d) as the initial model; is FWI result starting from the linear initial model. We see that in the case of strong noise, although these two results are both affected, the EI+WI result has less distortions. In Figure we can still see the main structures of the model, especially the overthrust structure. However, in Figure, the main structures are almost all buried into the noise. Based on this test we can say that EI+WI can somewhat reduce the influence of Gaussian noise to the inversion and therefore becomes a Gaussian noise resistant inversion method. C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

17 Seismic envelope inversion (c) Figure 5 EI+FWI results using data with SI noise.,, and (c) are the results using Fig. 4,, and (c) as initial models, respectively. Sensitivity to seismic interference noise We also tested the sensitivity of envelope inversion to seismic interference (SI) noise. SI usually comes from others sources operating in the same area. As we mentioned previously, there are 5 shots equally spaced on the surface. To simulate SI, we put another source behind the 5th shot, on the right side of the surface. Figure 3 shows a shot profile. is the original data,, (c), and (d) are the data with SI noise and the SNR are,, and respectively. We see that the noise becomes stronger from to (c), and the effective signal is affected by SI. Figure 4 shows the envelope inversion results with SI in the data. is the result when the SNR is, is the result when the SNR is, and (c) is the result when the SNR is. We see that when the SNR is, the noise is not very strong, the inversion result looks just the same as the one without any noise (Figure 8). The result is almost not affected by the noise at all. We can see the final EI+FWI result shown in Figure 5 is as good as the one in Figure. When the SNR is, the noise is a little stronger, and the envelope inversion result (Figure 4) is affected a little in this case. However we can still see the large-scale component in the result. Figure 5 shows the final EI+FWI result in this case, we see that the final result is still as good as with the large-scale component in the envelope inversion result. When the SNR is, the noise is very strong (same level as the data) in this case. We see that the envelope inversion is affected very much (Figure 4(c)), and we can only see some structures in the very shallow part. Figure 5(c) shows the final EI+FWI results. We can see that in this case the result has converged to a local minimum. From the above test we can see that when the SI noise is too strong, the envelope inversion will lose its effectiveness, however when the SI noise is not very strong, the envelope inversion can be noise-resistant. So we can conclude that the envelope inversion has potential to apply to noisy data. CONCLUSION Numerical examples have shown that envelope inversion is more reliable and effective to retrieve the large-scale component in the model than the conventional full waveform inversion and can avoid or reduce the local minimum problem. The combined inversion EI+WI can deliver more faithful and accurate final result with very little extra computation cost. Also, numerical examples showed that envelope inversion is insensitive to white Gaussian noise and to some extent, to seismic interference noise, indicating the robustness of this method and its potential use for noisy data. C 4 European Association of Geoscientists & Engineers, Geophysical Prospecting, 63,

18 64 J. Luo and R.-S. Wu ACKNOWLEDGEMENTS We thank Xiao-bi Xie, Rui Yan, Jinghuai Gao, Jicheng Liu, Bo Chen and Benfeng Wang for helpful discussions. The work is supported by WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Project at University of California, Santa Cruz. REFERENCES Almomin A. and Biondi B.. Tomographic Full Waveform Inversion: Practical and Computationally Feasible Approach. 8nd Annual International Meeting, SEG, Expanded Abstracts, 5. Biondi B. and Almomin A.. Tomographic full waveform inversion (TFWI) by combining full waveform inversion with waveequation migration velocity analysis. 8nd Annual International Meeting, SEG, Expanded Abstracts, 5. Biondi B. and Sava P Wave-equation migration velocity analysis. 69th Annual International Meeting, SEG, Expanded Abstracts, Boschetti F., Dentith M.C. and List D Inversion of seismic refraction data using genetic algorithms. Geophysics 6, Brenders A.J. and Pratt R.G. 7. 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