A novel full waveform inversion method based on a timeshift nonlinear operator

Transcription

1 A novel full waveform inversion method based on a timeshift nonlinear operator Journal: Manuscript ID GJI-S--0.R Manuscript Type: Research Paper Date Submitted by the Author: -Dec-0 Complete List of Authors: Gao, Zhaoqi; Xi\'an Jiaotong University, National Engineering Laboratory for Offshore Oil Exploration; Xi\'an Jiaotong University, School of Electronic and Information Engineering; University of California Santa Cruz, Modeling and Imaging Laboratory, Earth and Planetary Sciences Pan, Zhibin; Xi\'an Jiaotong University, School of Electronic and Information Engineering Gao, Jinghuai; Xi\'an Jiaotong University, National Engineering Laboratory for Offshore Oil Exploration; Xi\'an Jiaotong University, School of Electronic and Information Engineering Wu, Ru-Shan; University of California, Santa Cruz, Modeling and Imaging Laboratory, Institute of Geophysics and Planetary Physics Keywords: Waveform inversion < GEOPHYSICAL METHODS, Inverse theory < GEOPHYSICAL METHODS, Wave propagation < SEISMOLOGY

2 Page of A novel full waveform inversion method based on a time-shift nonlinear operator Zhaoqi Gao,,, Zhibin Pan*, Jinghuai Gao, and Ru-shan Wu. National Engineering Laboratory for Offshore Oil Exploration, Xi an Jiaotong University, China. School of Electronic and Information Engineering, Xi an Jiaotong University, China. Modeling and Imaging Laboratory, Earth and Planetary Sciences, University of California at Santa Cruz, USA of the authors: Zhaoqi Gao adom.la@stu.xjtu.edu.cn Zhibin Pan zbpan@mail.xjtu.edu.cn Jinghuai Gao jhgao@mail.xjtu.edu.cn Ru-shan Wu rwu@ucsc.edu Address of the corresponding author: Zhibin Pan: No. west xianning road, Xi an, Shaanxi, China. ZIP: 00.

3 Page of SUMMARY Conventional full waveform inversion (FWI), which uses the L norm to measure the misfit between the observed and the synthetic data, is known to be a nonlinear and illposed optimization problem. Usually, this problem is solved by using iterative gradientbased methods, which are difficult to converge to the global minimum of the misfit function because of cycle-skipping. We propose a new misfit function, which uses the nonlinearly transformed data instead of the original data. Because the nonlinearly transformed data can be very rich in low frequencies, the new misfit function is less prone to cycle-skipping. In this paper, we discuss the criteria to choose a suitable nonlinear operator and propose a new nonlinear operator by introducing a frequency selection mechanism into the envelope operator. We show that the proposed nonlinear operator is equivalent to the envelope operator in the first-order, but it outperforms the envelope operator in the case of higher orders. We develop a FWI method based on the new misfit function. Using the Marmousi model, we demonstrate that, compared with conventional FWI, the proposed FWI method is less prone to cycle-skipping. By using a model produced with the proposed FWI method as the initial model, it is shown that conventional FWI can achieve much better results. In addition, we show that the proposed FWI method is better than envelope inversion at mitigating cycle-skipping. Key words: Waveform inversion; Inverse theory; Wave propagation.

4 Page of INTRODUCTION Building an accurate velocity model from seismic data is an essential step for seismic imaging in complex areas. Traveltime inversion (Bishop et al. ; Justice et al. ) can be used for that purpose. These kinds of methods are computationally efficient, but they only use the traveltime information of particular events in the wavefield and therefore fail to recover the velocity of the subsurface with a high resolution. An alternative way to build a velocity model is the full waveform inversion (FWI) method, which takes into account the full information (amplitude and phase of all events) of the pre-stack seismic data to derive the velocity. As a consequence, higher resolution models are produced, compared with those from traveltime inversion. FWI tries to find the optimal model that minimizes the difference between the observed and the synthetic data, so it can be solved as an optimization problem. In practice, FWI is an ill-posed and highly nonlinear problem. In general, two kinds of methods can be used to solve FWI: global optimization methods and local optimization methods. Global optimization methods (Wang et al. 0; Tran et al. 0; Gao et al. 0; Sajeva et al. 0), such as simulated annealing and differential evolution, do not depend on the gradient of the misfit function, they can jump out of the local minimum, and therefore have the ability to search for the global minimum. However, applications of these methods are limited because the computational cost is usually tremendous. In most cases, gradient-based methods are used to solve FWI (Tarantola ; Pratt ). These methods require either sufficiently low frequency information in the data, or an

5 Page of initial model that is sufficiently close to the true model to prevent cycle-skipping and to avoid local minima (Virieux & Operto 00). Therefore, both building an accurate initial model for FWI and improving FWI to overcome the cycle-skipping problem are hot topics that are attracting the interest of many geophysicists. During the past decades, geophysicists have made a lot of progress on FWI. Multiscale ways have been proposed for FWI, which can be realized through frequency continuation in either the time domain (Bunks et al. ) or the frequency domain (Fichtner et al. 0) or, by definition, wavelet multi-resolution analysis (Yuan & Simons 0). Wave-equation traveltime inversion (Luo & Schuster ; Zhang & Wang 00; Ma & Hale 0) has also been proposed. Because the cross-correlation kind of misfit function is not sensitive to cycle-skipping, these methods are more likely to converge toward the global minimum and can recover the long-wavelength velocity model. Recently, many researchers have combined the migration velocity analysis (MVA) and FWI to propose a multi-scale inversion strategy in an extended model domain (Symes 00; Biondi & Almomin 0, 0). This has been achieved in two steps: first, by extending the wave equation and adding a subsurface offset or time-shift axis to the velocity model, and, second, by adding a regularization term that drives the solution towards the zero subsurface offset or the zero time lag. These strategies can update the long-wavelength components of the velocity model. Shin & Cha (00) have proposed the Laplace domain full waveform inversion method. In their work, they use the zero frequency component of the damped wavefield to do the inversion, and

6 Page of demonstrate that their inversion method can provide a velocity model that is equivalent to a long-wavelength velocity model. Later, they have extended their idea and proposed Laplace-Fourier domain full waveform inversion (Shin & Cha 00). This method uses the information less than Hz of the damped wavefield and can produce a more refined result compared with Laplace domain FWI. Bozdag et al. (0) have introduced instantaneous phase and envelope concepts in seismic adjoint tomography. The separation of amplitude and phase information reduces the nonlinear behavior of waveforms. Yuan et al. (0) have applied envelope difference misfit functions together with wavelet scaling in surface-wave FWI. Wu et al. (0, 0), Luo & Wu (0) and Chi et al. (0) have proposed envelope inversion (EI) for reflection data to construct an initial model for conventional FWI. They use the envelope information of the wavefield to construct the misfit function. Wu et al. (0) have proved that the envelope fluctuation and decay of seismic records carry ultralow frequency signals, which can be used to estimate long-wavelength velocity structures. In this study, we propose to use the nonlinearly transformed data to set up the misfit function for FWI. We demonstrate that the new misfit function is not only less prone to cycle-skipping, but can also recover the long-wavelength components of the velocity model. However, the effectiveness of the method requires that the nonlinear operator simultaneously satisfies the following two conditions: first, the energy of the nonlinearly transformed data is mainly located in low frequencies, and, in addition, the traveltime information of both the original and the nonlinearly transformed data are

7 Page of consistent. We construct a new nonlinear operator by introducing a frequency selection mechanism into the envelope operator, and we design an optimization problem to determine its parameters. A numerical example is conducted to compare the proposed nonlinear operator with the envelope operator. A new FWI method based on the nonlinear operator is proposed and its performance is compared with conventional FWI and EI by using three numerical examples based on the Marmousi model.

8 Page of TIME DOMAIN FULL WAVEFORM INVERSION AND THE CYCLE-SKIPPING PROBLEM Full waveform inversion (FWI) exploits the full information contained in the waveforms (amplitude and phase of all recorded events) to obtain high resolution subsurface parameters. The misfit function of conventional time domain FWI can be written as T J m u,,,,, syn t s r uobs t s r dt 0 m, () s, r where u t, s, r, m and u,, syn obs t s r are the synthetic and the observed data at the source s and the receiver r, respectively. m is the model parameter and T is the duration of the simulation. For convenience, we abbreviate u t, s, r, m and u,, syn obs t s r to u syn and u obs, respectively. The most affordable, and therefore popular process to solve FWI is the gradient-based technique, where the model parameter is updated iteratively as follows: where n and n n m m d, () n n d n are the step length and the descent direction, respectively, of the nth iteration. For the steepest descent method, the descent direction d n is opposite to the gradient direction of the misfit function J m over the model parameter m, which can be expressed as where J m usyn m is the sensitivity and usyn uobs u T syn usyn u obs dt m 0 s, r m, () is the effective residual or the so-

9 Page of called adjoint source. It is well known that the gradient can be calculated by backward propagating the effective residual and then zero-lag correlate the forward propagated wavefield with the back-propagated one (Tarantola ; Pratt ). Conventional FWI is based on weak scattering theory that requires the initial model to be close enough to the true model to make sure that the traveltime errors between the synthetic and the observed data are less than half a period of the observed data (Virieux & Operto 00). This concept is shown in Fig., in which there are three monochromatic waveforms with a period T. The red line in the middle represents the observed waveform, while the blue and green dashed lines represent two different synthetic waveforms. The time-shift of the synthetic (in blue) and the observed waveform (in red) is within half a period, while the time-shift of the synthetic (in green) and the observed waveform (in red) exceeds half a period, which is supposed to be cycle-skipped in conventional FWI. Pratt et al. (00) have used an alternative way to demonstrate the cycle-skipping problem. They have proposed that cycle-skipping can be avoided only if the following condition is satisfied: where t, () T L N T L denotes the duration of the simulation, t TL is the relative time error and N is the number of propagated wavelengths during the simulation. The condition shown in eq. () can hardly be satisfied for conventional FWI because usually there is no sufficiently low frequency information in the observed data.

10 Page of To see the important role of low frequency data in overcoming the cycle-skipping problem, we give an example. The true velocity model shown in Fig. consists of a background velocity of. km/s with a Gaussian anomaly in the center, while the initial model only consists of the background velocity. The model shown is 0 km in the horizontal direction and. km in the vertical direction. We use two tapered sine waves with different frequencies, i.e., Hz and Hz, shown in Figs (a) and (c) as the sources for the example. Figs (b) and (d) show the observed data (red lines) modeled with the true velocity model and the synthetic data (blue lines) modeled with the initial model for a single shot located at a distance of km and a depth of 0.0 km, and a single receiver at a distance of km and a depth of. km. Table shows t, T L and N of this example for the two different sources. For data based on the Hz tapered sine wave, eq. () is satisfied and, thus, cycle-skipping can be avoided. On the other hand, for data based on the Hz tapered sine wave, eq. () is not satisfied and cycle-skipping occurs.

11 Page 0 of METHOD Let us reinvestigate the condition shown in eq. (). Considering a given initial model, the relative time error t TL is fixed. Hence, if we want the FWI to be less prone to cycle-skipping, the only way is to reduce the value of N. This conclusion has prompted the following idea to mitigate the cycle-skipping problem. Instead of directly matching the observed data u with the synthetic data u syn, we propose to develop obs a FWI method to match two transformed data, i.e., F u obs and F u syn operator F should simultaneously satisfy the following two conditions:. The (). The energy of the transformed data should be mainly located in the low frequencies, and, thus, F must be a nonlinear operator; (). F does not change the traveltime information contained in the original wavefield u. In the following, these two conditions will be referred to as the first and the second condition, respectively. The first condition can lead to a smaller N for FWI based on the operator obs F, while the second condition guarantees that, if syn F u matches F u well, the estimated long-wavelength velocity model, which controls the traveltime information of the data, is correct. However, because the high frequency components of based on the operator model. F is nonlinear, Fu are quite different from that of u, thus FWI F is unable to recover a reliable short-wavelength velocity With these two conditions being satisfied, FWI based on the operator F can

12 Page of outperform conventional FWI in mitigating the cycle-skipping problem. In addition, the long-wavelength velocity model estimated by the proposed method can be considered as an initial model for conventional FWI. In the proposed method, the proper construction of F is the key component. As a nonlinear operator, the envelope operator (EO), which can extract the instantaneous amplitude information of a wavefield, is widely used in seismic data processing (Barnes ). In fact, the EO satisfies the above conditions, and therefore envelope inversion (EI), whose misfit function is based on the EO, is better than conventional FWI at mitigating the cycle-skipping problem. Motivated by the properties of the EO, we propose to construct a suitable F based on the EO. We embed a frequency selection mechanism into the EO and construct a new nonlinear operator. By adjusting its parameters, the proposed nonlinear operator can be better at producing low frequency data than the EO, without influencing the traveltime information.. Time-shift nonlinear operator Definition of the time-shift operator We define the time-shift operator D as follows: where * denotes convolution in the time domain, and D u u t d t, () d t is defined as follows:

13 Page of where d t n is the nth time-shift parameter, t t N n n an, () n a n is its corresponding coefficient and N is the number of time-shift parameters. The frequency domain expression of N sin n d t is dˆ j a, () where f is the angular frequency and j is the imaginary unit. The eq. () is derived in detail in Appendix A. Although its zero frequency component must be zero, the shape of the amplitude spectrum of n and a n. Supposing the amplitude spectrum of 0, d is given, the parameters n and n a n of n dt can be adjusted by the parameters dt within a frequency range dt can be determined as follows: n nt, n,,,..., N s ˆ an d sin n d, n,,,..., N, 0 s s where ˆd is the amplitude spectrum of sampling interval that guarantees s d () d t, t and t is a suitable s. The proof of eq. () is given in Appendix B. In practice, for numerical implementation, we discretize the integral in eq. () as a finite summation. Assuming dt is applied to a datum whose sampling interval is dt with the number of time steps being Nt. In this situation, we can choose t dt and ˆd can be discretized within 0, s Nt as follows: with the sampling interval of s ˆ ˆ Nt d m d m, m 0,,,...,, () then the integral in eq. () can be expressed as the summation:

14 Page of where N Nt. Nt ˆ m an d m sin n, n,,,..., N s m0 s, (0) The above procedure is verified with the following numerical example. First, a target amplitude spectrum, denoted by dˆt, is given for dt. Second, the parameters a n and n are determined following the above procedure. Third, the approximated amplitude spectrum denoted by dˆa is calculated using eq. (). The parameters are dt = 0.00 s and Nt = 000. Both dˆt and ˆa Fig., from which we can see they are quite consistent. Comparison of the time-shift operator and the Hilbert operator In this section, to further investigate the features of the time-shift operator compare it with the Hilbert operator. The Hilbert operator H u u t h t u t *, t where * denotes convolution in the time domain, and u and d are shown in H. is defined as D, we () Hu are a time domain signal and its Hilbert transform, respectively. The frequency domain expression of ht is j, 0 ˆ h 0, 0 j, 0. () Both the amplitude and the phase spectra of ĥ within the range -0 Hz to 0 Hz are shown in Figs (a) and (b), respectively. These results demonstrate the following

15 Page of two facts. First, ht applied to a signal the negative frequency components of frequency components of ut by ut has the effect of shifting the phase of ut by 0 and the phase of the positive 0. Second, from the point of view of amplitude, ht performs like an all-pass filter except for the zero frequency. For comparison, the amplitude and the phase spectra of d t within the range -0 Hz to 0 Hz for two different cases are shown in Figs (c) and (d), respectively. For the first case, we set ˆd to be the same as ĥ, while, for the second case, we choose be a band-pass filter. The results show that, for the first case, d t and ˆd to ht have an identical effect on the signal, while their effect on the amplitude is significantly different for the second case, although phase of the signal. d t and These results indicate that the amplitude spectrum of amplitude spectrum of the difference between ht have an identical effect on the ht is fixed, while the d t can be changed by adjusting its parameters. Considering dt and ht, we can treat the application of D to a signal as a two steps procedure. First, a filter is applied to the signal to obtain a filtered signal. Second, H. is applied to the filtered signal. In other words, we can interpret D as a Hilbert operator with a frequency selection mechanism. This key difference between D and H. makes D more flexible in the applications. Definition of the time-shift nonlinear operator Noting the difference between D and H., we propose to replace. H by

16 Page of D in the EO to construct a new EO with a frequency selection mechanism. The original EO is defined as where u and E u u H u, () Eu are a time domain signal and its envelope, respectively. The EO with a frequency selection mechanism is called the time-shift nonlinear operator and its first-order expression is defined as where u and P P u u D u, () u are a time domain signal and its first-order time-shift nonlinearly transformed signal, respectively. In the following, we also use P u. We define the higher order time-shift nonlinear operator as N N where N is the order of the time-shift nonlinear operator. Finding suitable parameters, Pu to represent P u P P u, N, () n and a n, which guarantee that. N P satisfies the two conditions is the key problem, which is solved in two steps: first, we search for the optimal ˆd, which can lead to the N P u whose energy is most concentrated in the low frequency range without changing the traveltime information contained in u, and second, we determine the parameters n and a n based on the optimal ˆd. The ˆd with only two independent parameters is used in this paper, and is defined as

17 Page of dˆ sin, 0, cos, 0,, where and are the two parameters. Fig. shows the shape of ˆd. The center frequency fc the degree of its energy concentration: max where A uˆ d, and 0 () u of the data, as defined in eq. () below, is used to measure max f ˆ c u u d A, () 0 û is the amplitude spectrum of u. For the Nthorder time-shift nonlinear operator, there are N groups of and that require to be determined. We use N ω,,,..., and N ω,,,..., to denote the N groups of and, respectively. Finding the optimal ω and ω is treated as an optimization problem by solving the following objective function: where u T and N P u min ω, ω c N N T f P u s. t. T P u u, T are the locations of peaks of u and P N () u on the time axis, respectively. The differential evolution method (Gao et al. 0) is used to solve this optimization problem.

18 Page of Comparison of the time-shift nonlinear operator and the envelope operator To demonstrate the advantages of the time-shift nonlinear operator over the envelope operator, we compare their performances with numerical examples. To compare their higher order behavior, we define the higher order envelope operator as follows: Both E N N N,, N N E ( N ) E u E E u, N. () and P N N,, have been applied to synthetic data, which is generated using the one-dimensional acoustic wave equation. The source signature is a Ricker wavelet with a dominant frequency of Hz. The velocity model consists of layers, where the first, second and third layer has a constant velocity of km/s, km/s and km/s, respectively. The parameters ω and ω for P N N,, are calculated and presented in Table. These experimental results are presented in Figs -0. Fig. shows the original data u, its first-order envelope time-shift nonlinearly transformed data Pu exactly coincide, which indicates that features. Fig. shows u, P E u and u are consistent, while that of comparison of u, E u and P E P Eu and its first-order Pu. It is clearly shown that Eu and Eu and Pu show the same u. The traveltime information of u and u is quite different from them. A detailed u in the time domain is presented in Fig.. It is obvious that there is a time-shift between the maximum and minimum of E u and

19 Page of P u. More specifically, the time-shift denoted by t is about 0.0 s and the timeshift denoted by t is about 0.0 s. The results indicate that the second condition. Fig. 0 shows u, information of indicates that E. E u and P. E does not satisfy u is significantly different from that of u and E does not satisfy the second condition. To compare more clearly the energy distribution of u, N,, u. The traveltime P u, which N,, E u N and P u N, their respective center frequencies f c are calculated and shown in Table. Along with increasing N, the energy of E N u and P N u become more and more concentrated in low frequencies. However, the difference between P. u and P and P u is negligible. Based on the two conditions, we can conclude that. P are better choices to construct a misfit function for FWI than E. and P, and, considering FWI based on. P can be more efficient (lower computation costs for calculating the gradient operator) compared with FWI based on based on P., then.. P is the better choice to construct a misfit function. FWI P may be less prone to cycle-skipping compared with both conventional FWI and envelope inversion, which is based on E.. Time-shift nonlinear full waveform inversion In this section, a full waveform inversion (FWI) method is developed. The proposed FWI method matches the forward modeled synthetic data to the observed data in the inversion process and the second-order time-shift nonlinear operator P is applied

20 Page of to both data sets. The proposed method is called time-shift nonlinear full waveform inversion (TSNFWI), whose misfit function is defined as m m, (0) T,,,,, syn obs J 0 P u t s r P u t s r dt s, r where u t, s, r, m and u,, syn obs t s r are the synthetic and the observed data at the source position s and the receiver position r, respectively, and T is the duration of the simulation. For convenience, we abbreviate to obs P u and syn P u, respectively. The gradient of TSNFWI can be formulated as J m s, r s, r where T 0 T 0,, P u t s r and obs P u syn P usyn P uobs dt m P u t, s, r, m P u D P u syn syn A P usyn D P usyn dt, m m syn obs syn A P u P u P u syn (). By further deduction, the final expression of the gradient can be shown as follows (see Appendix C for more details): Bu BD u u D 0 s, r P u syn P u syn m J T syn syn m where B APusyn DAD Pusyn, and usyn syn dt, () m is known as the sensitivity. It is clear in eq. () that the gradient of TSNFWI can also be calculated using the backpropagation method. The term in the braces of eq. () serves as the effective residual, or the so-called adjoint source. It is well known that a suitable step length will lead to convergence with a low number of iterations. In conventional FWI, we can determine the step length by solving

21 Page 0 of a quadratic equation (Pica et al. 0). However, it may be very complicated to solve such an equation in TSNFWI. In our work, we use the hybrid misfit method (Luo et al. 0). The conventional waveform misfit is used to obtain the step length for TSNFWI. It is clearly shown in eq. () and eq. () that the only difference between conventional FWI and TSNFWI is the adjoint source for back-propagation. Next, we explain why TSNFWI is less prone to the cycle-skipping problem from the point of view of the adjoint source. The velocity model shown in Fig. has been used to conduct this experiment. A single source is used, which is located at a horizontal distance of km and a depth of 0.0 km. A fixed-spread surface acquisition is used with 00 receivers equally spaced at 0 m intervals located at a depth of 0.0 km, covering a horizontal distance from 0.0 km to 0 km. A Ricker wavelet with a dominant frequency of Hz is used to generate a single shot-gather. The modeling is performed in the acoustic approximation and the density model is kept constant. The adjoint sources for both FWI and TSNFWI are calculated and shown in Fig.. The 0 th trace of these two shot-gathers are selected to analyze their features in both the time (Fig. a) and the frequency (Fig. b) domain. The adjoint source of conventional FWI is directly obtained by subtracting the observed and the synthetic data, so its spectrum is similar to that of the seismic data, which does not have very low frequency information. However, because of the nonlinear operator P, the adjoint source of TSNFWI is very rich in low frequencies, which can be very important for constructing the long-wavelength velocity model. In addition, Fig. illustrates that

22 Page of the long-periodic part of single reflection events is pronounced with P. This interesting feature makes TSNFWI less prone to the cycle-skipping problem than conventional FWI. Based on the above experiment, we also compare the computing time of conventional FWI and TSNFWI for one iteration and the results, presented in Table, indicate that TSNFWI increases the computing time only by less than two percent compared with conventional FWI.

23 Page of NUMERICAL EXAMPLES In this part, we evaluate the performance of time-shift nonlinear full waveform inversion (TSNFWI) on the Marmousi model. We conduct three different experiments. First, based on a D initial model, the convergence behavior of TSNFWI and conventional FWI are investigated. Second, the combined inversion method TSNFWI+FWI, which means conventional FWI using an initial model estimated by TSNFWI, is tested. Third, a comparison between TSNFWI and envelope inversion (EI) is conducted. All these experiments are based on a D acoustic time domain FWI code. The acoustic wave equation with constant density is v u t f t p x t x s, x, x x s, () where is the Laplacian operator. The forward modeling is performed using an eighth-order finite-difference stencil for the spatial discretization and a second-order leap-frog scheme for the time discretization. The minimization of the misfit function is performed using the steepest descent method, which requires the computation of the misfit function and its gradient. The gradient is computed as the cross-correlation in time following the adjoint-state method. In addition, a hybrid MPI/CUDA scheme is used to implement the code in parallel, where the MPI communicator is used to perform the computations associated with shotgathers in parallel. For each shot, the computation of the wavefield is further accelerated using the GPU of the spatial finite-difference loops.

24 Page of In the following three experiments, the iteration number is empirically set to 0. The true P-wave velocity for the Marmousi model is shown in Fig. (a), which extends.0 km along the x-direction and. km along the z-direction. A fixed-spread surface acquisition is used, involving shots uniformly located every 00 m and 0 receivers located every 0 m at a 0 m depth. Two different D initial P-wave velocity models have been used in the experiments. For the first D initial model, the velocity ranges from. km/s to. km/s, while for the second D initial model, the velocity ranges from. km/s to. km/s. The second D initial model deviates more from the true model especially between 0.0 km and. km horizontal distance. In all these experiments, a Ricker wavelet with a dominant frequency of 0 Hz has been adopted to generate the observed data. The spatial discretization step is set to 0 m and the time discretization step is set to 0.00 s according to the Courant-Friedrichs-Lewy (CFL) condition. The simulation is performed over 000 time steps, i.e., a total recording time of.0 s.. Comparison of TSNFWI and FWI In this test, the convergence behavior of TSNFWI and conventional FWI is compared. Starting from the first D initial model, which is presented in Fig. (b), both conventional FWI and TSNFWI are applied to recover the velocity model. The results after 0 iterations are presented in Figs (c) and (d), respectively. Their convergence curves are presented in Fig., and the residuals of the shot-gathers corresponding to

25 Page of the source position xs =. km for the initial and the two estimated models are presented in Fig.. Starting from the first D initial model, conventional FWI is trapped into a local minimum after about 0 iterations and its estimation of the velocity model has a lowvelocity artifact, typical for cycle-skipping, present in the zone near x =. km and z = 0. km. The residual shown in Fig. (b) also emphasizes that the conventional FWI fails to provide a satisfactory estimation of the velocity model. In comparison, the misfit function value of TSNFWI is decreased by three orders of magnitude after 0 iterations. The residual presented in Fig. (d) further demonstrates that the nonlinearly transformed shot-gathers (shot-gather operated by the second-order time-shift nonlinear operator) corresponding to the true velocity model and the estimated velocity model by TSNFWI are consistent with each other. These experiments demonstrate that TSNFWI can converge well and no cycleskipping artifacts are noticeable. This indicates that TSNFWI may be an effective way to mitigate cycle-skipping.. Combined inversion of TSNFWI and FWI In this section, the performance of TSNFWI+FWI is tested. In this experiment, an isotropic Gaussian filter with the parameter of 0 m is applied to remove the shortwavelength components of the velocity model estimated by TSNFWI. The filtered model shown in Fig. (a) is used as an initial model for conventional FWI. In the

26 Page of following, the new initial model will be also called the initial model from TSNFWI. The result of TSNFWI+FWI after 0 iterations is presented in Fig. (b). The convergence curve of TSNFWI+FWI is presented in Fig. and the residuals of the shot-gathers corresponding to the source position xs =. km in the initial model from TSNFWI and the estimated model by TSNFWI+FWI are presented in Fig.. Starting from the initial model from TSNFWI, the value of the misfit function of TSNFWI+FWI after 0 iterations is decreased by three orders of magnitude compared with that corresponding to the first D initial model. Compared with the velocity model estimated directly by conventional FWI, the velocity model estimated by TSNFWI+FWI is significantly closer to the true velocity model (Fig. b). Some differences can still be seen near the boundaries because of the low sensitivity and an insufficient coverage of these areas. However, in the central part, the velocity model is correctly recovered. The synthetic data seems to correctly explain the data, as can be seen in Fig. (b). In this experiment, the long-wavelength velocity model recovered by TSNFWI can be used as an initial model for conventional FWI to significantly improve the final results. This shows that TSNFWI may be an interesting method to build an initial model for conventional FWI.. Comparison of TSNFWI and envelope inversion In this section, a comparison between TSNFWI and envelope inversion (EI) is

27 Page of presented. Similar to TSNFWI+FWI, an inversion strategy named EI+FWI is proposed. Starting from the first D initial model, EI is used to recover the long-wavelength part of the velocity model, which is then used as an initial model for conventional FWI to derive an estimation of the true velocity model. The results are presented in Fig.. Starting from the second D initial model, which is presented in Fig. 0(a), both EI and TSNFWI are used to recover the velocity model. Both EI+FWI and TSNFWI+FWI yield estimations of the true velocity model. The results of EI and TSNFWI after 0 iterations are presented in Figs 0(b) and (c), respectively. The convergence curves of both EI and TSNFWI are presented in Fig., and the residuals of the shot-gathers at the source position xs =. km for both EI and TSNFWI in the second D initial and estimated models are presented in Fig.. The results of EI+FWI and TSNFWI+FWI after 0 iterations are presented in Figs (c) and (d), respectively. The convergence curves of both EI+FWI and TSNFWI+FWI are presented in Figs (a) and (b), respectively. The residuals of the shot-gathers for the source position xs =. km for both EI+FWI and TSNFWI+FWI in the estimated models are presented in Fig.. Starting from the first D initial model, the estimated velocity model by EI is very close to the estimated model by TSNFWI as presented in Fig. (d). Although a difference can be noted near x =. km and z = 0. km, its influence on the final estimated model by EI+FWI is negligible. The EI+FWI and TSNFWI+FWI yield estimations that are very close to each other.

28 Page of Starting from the second D initial model, EI is trapped in a local minimum after about 0 iterations. The residual shown in Fig. (b) also emphasizes that EI fails to provide a satisfactory estimation of the velocity model. In comparison, the estimated velocity model by TSNFWI is quite different from the estimated velocity model by EI, the value of the misfit function of TSNFWI is decreased by three orders of magnitude after 0 iterations. The residual presented in Fig. (d) further demonstrates that the nonlinearly transformed shot-gathers corresponding to the true velocity model and the estimated velocity model by TSNFWI are consistent with each other. The EI+FWI is unable to provide a satisfactory velocity estimation (Fig. c). The difference between the true velocity model and the velocity model estimated by EI+FWI can be seen in Fig. (e), especially at the left part of the model (x <. km). The residual presented in Fig. (a) also emphasizes that the estimated model by EI+FWI is incorrect. In comparison, the velocity model obtained using TSNFWI+FWI, presented in Fig. (d), is significantly closer to the true velocity model. The shotgathers corresponding to the estimated model by TSNFWI+FWI and the true velocity model seem to match quite well, as can be seen in Fig. (b). In these experiments, TSNFWI is more robust than EI. TSNFWI+FWI can derive a correct estimation of the true model, even though it starts from a very coarse approximation of the true model. This is an indication that TSNFWI may be better at mitigating cycle-skipping than EI.

29 Page of CONCLUSIONS In this paper, a new misfit function, based on a nonlinear operator, is proposed for FWI. A nonlinear operator, called time-shift nonlinear operator, is constructed by embedding a frequency selection mechanism into the envelope operator. The parameters of the proposed nonlinear operator can be determined by solving an optimization problem. We have demonstrated that the proposed nonlinear operator is better than the envelope operator at constructing a misfit function for FWI. Based on the nonlinear operator, we have proposed the time-shift nonlinear full waveform inversion (TSNFWI) method. We have demonstrated that TSNFWI increases the computing time only by less than two percent compared with conventional FWI. Numerical examples using the Marmousi model show that, TSNFWI is less prone to the cycle-skipping problem than conventional FWI and the long-wavelength velocity model recovered by TSNFWI can be used as an initial model for conventional FWI, which can significantly improve the final inverted model. We also have compared TSNFWI with envelope inversion (EI). The numerical test clearly show that TSNFWI is better at mitigating cycle-skipping than EI. Thus, the proposed method seems to be promising and should be further investigated with both more realistic synthetic data sets and field data. The introduction of viscous, elastic and anisotropic effects should also be investigated.

30 Page of ACKNOWLEDGEMENTS The authors gratefully thank the editor Wolfgang Friederich, Olaf Hellwig and an anonymous reviewer for their comments, which have helped greatly to improve the quality of this paper. We also thank Faqi Liu and Sheng Xu for their helpful comments, Jingrui Luo, Guoxin Chen and Xiaobi Xie for useful discussions, and David Larner for his help with language editing. This research was carried out partly during the visit of the first-mentioned author (Zhaoqi Gao) to the University of California, Santa Cruz, as a visiting student. We are grateful for the support of the WTOPI Research Consortium at the University of California, Santa Cruz, USA. Further, we greatly appreciate the Major Programs of the National Natural Science Foundation of China under grant Nos. 00 and 0, and the Major Research Plan of the National Natural Science Foundation of China under grant No. 00 for their financial support.

31 Page 0 of APPENDIX A: FOURIER TRANSFORM OF d(t) In this appendix, eq. () is derived. The Fourier transform of a time domain signal is defined as ˆ jt s s t e dt st, (A) where f is the angular frequency and j is the imaginary unit. The inverse Fourier transform is defined as Based on eq. (A) and the definition of written as jt s t sˆ e d. (A) dˆ n d t, the Fourier transform of N j j jx Considering the Euler formula e cos x jsin x expressed as d t can be n n e e an. (A), eq. (A) may be further N dˆ j a. (A) sin n n n

32 Page of APPENDIX B: RELATIONSHIP BETWEEN THE PARAMETERS OF d(t) AND ITS AMPLITUDE SPECTRUM In this appendix, the proof of eq. () is given. Let us rewrite the frequency domain expression of dt as follows: N where A a sin where obtain where n n n N sin dˆ j a n n ja,. We choose the time-shift parameters as follows: n (B) n nt, n,,,..., N, (B) t is a suitable sampling interval. By substituting eq. (B) into A, we s A suitable N sin A a n t n t. We define x s n N an sin n, n s and then eq. (B) can be written as N xs A a nx n (B) n sin. (B) t can guarantee x 0,. When N, the right-hand side of eq. (B) is the Fourier series expression of an odd function whose period is (for more information about the Fourier series see Oppenheim et al. ). Based on the theory of signal processing, if x s A is given, a n can be calculated as follows:

33 Page of x s an A sin nxdx, n,,,..., N 0. (B) Considering the relationship between x and, xs an A sin nxdx 0 s Asin n d. 0 s s a n can be further expressed as s s A sin n d 0 s s s (B) Further, we use dˆ and A to represent the amplitude spectrum of ˆd and A, respectively. Considering that inside the range of 0, satisfy Based on eq. (B), ˆ d A, (B), we can set s A as a nonnegative real function to ˆ s A d, 0. (B) a n can also be calculated as follows: s ˆ an d sin n d, n,,,..., N 0 s. (B) s The eq. (B) expresses the relationship between dˆ and a n.

34 Page of APPENDIX C: GRADIENT OPERATOR OF TSNFWI In this appendix, we show how to obtain the gradient operator of the time-shift nonlinear full waveform inversion (TSNFWI) method, which is given as eq. (). By calculating the gradient of the misfit function defined in eq. (0) with respect to the parameter m, we obtain J m s, r s, r where T 0 T 0 P u syn P usyn P uobs dt m P u D P u syn syn APusyn DPusyn dt, m m syn obs syn A P u P u P u D f t d t f t N n. As defined in eq. (), t t n n an f t where * denotes convolution in the time domain. Considering that we obtain n N n N n a a N n n n a d t n, t t N n n d t an t n t n t t n t t n n n, (C) (C) (C)

35 Page of Using eq. (C), eq. (C) becomes s t D f t dt s t d t t f t dtdt d t t s t dt f t dt d t t s t dt f t dt d t t s t dt f t dt D s t f t dt. P u D P u J syn A P u DP u dt m m m T syn 0 syn syn s, r s, r s, r s, r T 0 T 0 T 0 P usyn AP usyn D AD P usyn dt m P u B m B P u where B AP usyn DAD Pusyn syn syn dt u D u syn syn usyn Dusyn dt, m m (C) (C). We use eq. (C) again and eq. (C) can be further expressed as J B u D u syn u Du dt m m m T syn 0 syn syn s, r P u syn Bu BD u T syn syn D 0 s, r P usyn P usyn usyn m dt. (C)

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41 Page 0 of Tables TABLES AND FIGURES Table. Parameters of the cycle-skipping example. Frequency of the Source t (s) T L (s) Hz 0.0 Hz 0.0 Table. Parameters ω and ω for Order of N P u N,, P u N. ω (rad/s) ω (rad/s) N 0. N 0.,. N 0.,.,0 Table. Comparison of the central frequency u Eu, N,,0. Pu E f c of different wavefields. u P u E u P u f c (Hz)

42 Page of Table. Comparison of the computing time of conventional FWI and TSNFWI for one iteration. FWI TSNFWI Time (s).0.

43 Page of Figures Figure. Illustration of the cycle-skipping problem.

44 Page of Figure. True velocity model with an anomaly in its center. The initial velocity model for the inversion has a constant velocity of. km/s.

45 Page of Figure. The source wavelet and its corresponding observed (red line) and synthetic wavefields (blue line): (a) the tapered Hz sine wave, (b) the wavefields corresponding to (a), (c) the tapered Hz sine wave, and (d) the wavefields corresponding to (c). The cycle-skipping occurs in (d), where the first maximum of the blue line is closer to the second maximum of the red line in contrast to (b), where the shift is less than half a period.

46 Page of Figure. Comparison of the target amplitude spectrum (red line with a plus at each data point), and the approximated amplitude spectrum (blue line) of d t.

47 Page of Figure. Features of of ht and ht, (b) the phase spectrum of spectra of the same as is a band-pass filter. d t in the frequency domain: (a) the amplitude spectrum ht, (c) the amplitude spectra of d t. The red lines in (c) and (d) represent ht and the blue lines in (c) and (d) represent d t, and (d) the phase d t whose amplitude spectrum is d t whose amplitude spectrum

48 Page of Figure. The shape of the amplitude spectrum of dt.

49 Page of Figure. Comparison of the data u (red line), its first-order envelope first-order time-shift nonlinearly transformed data frequency domain. Pu and Eu exactly coincide. Eu (blue line) and its Pu (green line) in (a) time, and (b)

50 Page of Figure. Comparison of the data u (red line), its second-order envelope and its second-order time-shift nonlinearly transformed data and (b) frequency domain. The maxima of u and and E u do not coincide. P P E u (blue line) u (green line) in (a) time, u coincide whereas the maxima of u

51 Page 0 of Figure. Detailed comparison of the traveltime of the data u (red line), its second-order envelope P and E u (blue line) and its second-order time-shift nonlinearly transformed data u (green line) in the time domain. t is the time-shift between the minima of P u, and t is the time-shift between the maxima of E u and E P u. u

52 Page of Figure 0. Comparison of the data u (red line), its third-order envelope and its third-order time-shift nonlinearly transformed data and (b) frequency domain. The maxima of u and and E u do not coincide. P P E u (blue line) u (green line) in (a) time, u coincide whereas the maxima of u

53 Page of Figure. Comparison of the adjoint source of (a) FWI, and (b) TSNFWI.

54 Page of Figure. Comparison of one trace located at x =. km of the adjoint source of FWI (red line) and TSNFWI (blue line) in (a) time, and (b) frequency domain. The spectrum of the adjoint source of TSNFWI contains more power at low frequencies.

55 Page of Figure. (a) Marmousi true P-wave velocity model, (b) first D initial P-wave velocity model, which ranges from. km/s to. km/s, (c) P-wave velocity estimated by conventional FWI, and (d) P-wave velocity estimated by TSNFWI.

56 Page of Figure. The evolution of the misfit function value with increasing iteration number corresponding to (a) conventional FWI, and (b) TSNFWI.

57 Page of Figure. Residuals of the shot-gathers corresponding to the source position x s =. km: (a) the residuals between shot-gathers in the true P-wave velocity model and the D initial P-wave velocity model shown in Fig. (b), (b) the residuals between shot-gathers in the true P-wave velocity model and the P-wave velocity model estimated by conventional FWI, (c) the residuals between nonlinearly transformed shot-gathers in the true P-wave velocity model and the D initial P-wave velocity model shown in Fig. (b), and (d) the residuals between nonlinearly transformed shot-gathers in the true P-wave velocity model and the P-wave velocity model estimated by TSNFWI. The residual for TSNFWI is lower than that for conventional FWI.

58 Page of Figure. (a) Initial P-wave velocity model, computed from Fig. (d) using a Gaussian filter, (b) corresponding P-wave velocity estimated by TSNFWI+FWI, (c) difference between the true P-wave velocity model and the estimated P-wave velocity model by conventional FWI, and (d) difference between the true P-wave velocity model and the estimated P-wave velocity model by TSNFWI+FWI.

59 Page of Figure. The evolution of the misfit function value over the iteration number corresponding to conventional FWI (red line) and TSNFWI+FWI (blue line).

60 Page of Figure. Residuals of the shot-gathers corresponding to the source position x s =. km: (a) the residual between shot-gathers in the true P-wave velocity model and the P-wave velocity model shown in Fig. (a), (b) the residual between shot-gathers in the true P-wave velocity model and the P-wave velocity model estimated by TSNFWI+FWI.

61 Page 0 of Figure. (a) Estimated P-wave velocity model with EI, (b) long-wavelength P-wave velocity model, obtained by applying a Gaussian filter to (a), that serves as an initial model for a subsequent FWI, and (c) P-wave velocity estimated by EI+FWI.

62 Page of Figure 0. (a) Second D initial P-wave velocity model, which ranges from. km/s to. km/s, (b) P-wave velocity estimated by EI, and (c) P-wave velocity estimated by TSNFWI.

63 Page of Figure. The evolution of the misfit function value over the iteration number corresponding to (a) EI, and (b) TSNFWI.

64 Page of Figure. Residuals of the shot-gathers corresponding to the source position x s =. km: (a) the envelope residual between shot-gathers in the true P-wave velocity model and the D initial P-wave velocity model shown in Fig. 0(a), (b) the envelope residual between shot-gathers in the true P-wave velocity model and the P-wave velocity model estimated by EI, (c) the residual between nonlinearly transformed shot-gathers in the true P-wave velocity model and the D initial P-wave velocity model shown in Fig. 0(a), and (d) the residual between nonlinearly transformed shot-gathers in the true P-wave velocity model and the P-wave velocity model estimated by TSNFWI. The combination of TSNFWI and FWI yields a better inversion result than EI and FWI.

65 Page of Figure. (a) Initial P-wave velocity model, computed from Fig. 0(b) using a Gaussian filter, (b) initial P-wave velocity model, computed from Fig. 0(c) using a Gaussian filter, (c) P-wave velocity estimated by EI+FWI, (d) P-wave velocity model estimated by TSNFWI+FWI, (e) difference between the true P-wave velocity model and the estimated P-wave velocity model by EI+FWI, and (f) difference between the true P-wave velocity model and the estimated P- wave velocity model by TSNFWI+FWI.

66 Page of Figure. The evolution of the misfit function value over the iteration number corresponding to (a) EI+FWI, and (b) TSNFWI+FWI.

67 Page of Figure. Residuals of the shot-gathers corresponding to the source position x s =. km: (a) the residual between shot-gathers in the true P-wave velocity model and the P-wave velocity model estimated by EI+FWI, (b) the residual between shot-gathers in the true P-wave velocity model and the P-wave velocity model estimated by TSNFWI+FWI.