Least-squares reverse time migration with and without

Journal of Applied Geophysics 134 (2016) 1–10

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Least-squares reverse time migration with and without source wavelet estimation Qingchen Zhang a, Hui Zhou a,⁎, Hanming Chen a, Jie Wang b a b

State Key Laboratory of Petroleum Resources and Prospecting, CNPC Key Laboratory of Geophysical Exploration, China University of Petroleum, 102249 Beijing, China SINOPEC Geophysical Research Institute, 211103 Nanjing, Jiangsu, China

a r t i c l e

i n f o

Article history: Received 16 January 2016 Received in revised form 5 July 2016 Accepted 17 August 2016 Available online 20 August 2016 Keywords: LSRTM Source wavelet estimation Deconvolution Convolution Hybrid norm

a b s t r a c t Least-squares reverse time migration (LSRTM) attempts to find the best fit reflectivity model by minimizing the mismatching between the observed and simulated seismic data, where the source wavelet estimation is one of the crucial issues. We divide the frequency-domain observed seismic data by the numerical Green's function at the receiver nodes to estimate the source wavelet for the conventional LSRTM method, and propose the source-independent LSRTM based on a convolution-based objective function. The numerical Green's function can be simulated with a dirac wavelet and the migration velocity in the frequency or time domain. Compared to the conventional method with the additional source estimation procedure, the source-independent LSRTM is insensitive to the source wavelet and can still give full play to the amplitude-preserving ability even using an incorrect wavelet without the source estimation. In order to improve the anti-noise ability, we apply the robust hybrid norm objective function to both the methods and use the synthetic seismic data contaminated by the random Gaussian and spike noises with a signal-to-noise ratio of 5 dB to verify their feasibilities. The final migration images show that the source-independent algorithm is more robust and has a higher amplitudepreserving ability than the conventional source-estimated method. © 2016 Elsevier B.V. All rights reserved.

1. Introduction LSRTM, similar to the full waveform inversion (FWI), attempts to minimize the misfit between the observed and simulated data by an iterative algorithm to refine seismic images towards the true reflectivity (Dong et al., 2012; Zeng et al., 2014). Therefore, LSRTM can produce migration images with better quality than conventional migrations (Ji, 2009; Dai et al., 2012, 2013; Luo and Hale, 2014; Dutta and Schuster, 2014; Zhang and Schuster, 2014; Tan and Huang, 2014a; Aldawood et al., 2015; Wong et al., 2015; Y. Zhang et al., 2015) by removing migration artefacts, improving the illumination and revealing more subsurface details. Furthermore, LSRTM is currently the only effective way to directly migrate the blended seismic data from simultaneous-source acquisition which becomes more and more appealing for its tremendous acquisition cost reduction and the quality improvement of the seismic data (Chen et al., 2015). It can be conducted both in the time and frequency domains (Dai et al., 2013; Ren et al., 2013) for the isotropic and anisotropic (Dai et al., 2012) or the acoustic and visco-acoustic (Dutta and Schuster, 2014; Li et al., 2016) media. However, LSRTM encounters the problems similar to FWI. The primary problems are the great computation cost and memory requirement. Fortunately, many methods, e.g., source-encoding method (Dai et al., ⁎ Corresponding author. E-mail address: [email protected] (H. Zhou).

http://dx.doi.org/10.1016/j.jappgeo.2016.08.003 0926-9851/© 2016 Elsevier B.V. All rights reserved.

2012, 2013; Zhang et al., 2013; Chen et al., 2015), plane-wave method (Dai and Schuster, 2013; Li et al., 2014) and boundary-wavefield extrapolation method (Tan and Huang, 2014b; Zhang et al., 2014), have been developed to solve these two problems. In addition, the preconditioning (Dai et al., 2011; Liu et al., 2013) and regularization (Xue et al., 2016) methods are usually utilized to accelerate the convergence rate of LSRTM, by which the iteration numbers can be efficiently reduced. Besides the problems of memory requirement and computation cost, the actual source wavelet is difficult to extract from practical seismic data and the incorrect wavelet would contaminate the migration images (Kim et al., 2011, 2013; Tang and Wang, 2012) or inversion results (Song et al., 1995; Yuan and Wang, 2011; Luo et al., 2014). Conventional LSRTM supposes that the source wavelet is known beforehand. However, an incorrect source wavelet would produce some artefacts to the migration images, such as polarity reversal and layer dislocation. To solve the problem of source wavelet, several techniques are usually adopted in FWI, e.g., the iterative estimation of source signature (IES) method (Song et al., 1995), the deconvolution-based method (Lee and Kim, 2003; Zhou et al., 2014), and the convolution-based method (Choi and Alkhalifah, 2011; Zhang et al., 2015b, 2016). With regard to the migration, Kim et al. (2011) proposed the frequency-domain source estimation method using optimization technique for reverse time migration (RTM). Kim et al. (2013) developed the source estimation idea to the time-domain RTM by the deconvolution method. Tu et al. (2013) performed the least-squares migration (LSM) with source

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Q. Zhang et al. / Journal of Applied Geophysics 134 (2016) 1–10

Fig. 1. (a) The true Marmousi velocity model, (b) the smoothed (or migration) velocity model, and (c) the corresponding true reflectivity model.

Fig. 3. Trace 300 of the 20th source at X = 2 km chosen from: (a) the conventional observed (solid black line) and simulated (dashed blue line) seismograms, (b) the convolution-based observed (solid black line) and simulated (dashed blue line) seismograms. The red line denotes the residuals between the observed and simulated seismograms. The superscript of asterisk represents the convolution-based method. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2. Theory estimation by the generalized variable projection method. Due to the similarity of FWI and LSRTM, the techniques for solving the source problem used by FWI can also be used by LSRTM. In this paper, we propose the time-domain source-independent (without source estimation) LSRTM (Zhang et al., 2015a) based on the convolution-based objective function to suppress the artefacts caused by the incorrect source wavelet, and then make a comparison with the deconvolution-based source estimation method for conventional LSRTM. The hybrid-norm objective function is applied to improve the robustness of both methods. Synthetic data with and without noises are used to verify the advantages of the sourceindependent algorithm compared to the conventional sourceestimated method. The rest of this paper is organized as follows. First, we review the theory of the conventional LSRTM, and then give an introduction to the deconvolution-based source estimation method and the convolutionbased source-independent method. Second, we use the synthetic data generated with the Marmousi model to verify the feasibilities of both the algorithms and make a sensitivity analysis to the source wavelet phase for the source-independent method. Finally, we give the discussions and draw some conclusions.

2.1. Conventional LSRTM We review the theory of LSRTM (Dai et al., 2012, 2013) by the acoustic wave equation with a constant density 2

∂ p0 ðx; t; xs Þ −∇2 p0 ðx; t; xs Þ ¼ sðt; xs Þ; ∂t 2 c0 ðxÞ 1

ð1Þ

2

where xs denotes the source location, p0(x, t; xs) is the background wavefield associated with the background (migration) velocity model c0(x) and the source wavelet s(t; xs). A perturbation δc(x) in the velocity model c(x) = c0(x) + δc(x) will generate a perturbation δp(x,t;xs) in the wavefield p(x, t; xs) = p0(x, t; xs) + δp(x, t; xs). In the following text, we use q(x, t; xs) to represent δp(x, t; xs). Based on the Born approximation, we can obtain the equation for the wavefield perturbation q(x, t; xs) 2

2

∂ qðx; t; xs Þ 1 ∂ p0 ðx; t; xs Þ −∇2 qðx; t; xs Þ ¼ mðxÞ ; ∂t 2 ∂t 2 c0 ðxÞ2 c0 ðxÞ2 1

ð2Þ

Fig. 2. (a) The true source wavelet (solid line) generated by the first derivative of Gaussian function, the estimated wavelet (dotted line), and the estimated errors (dashed line). (b) The incorrect source wavelet generated by the second derivative of the Gaussian function. Both of them have a dominant frequency of 20 Hz.

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3

a × mean(|qobs(xr,t;xs)|), a is a constant ranging from 0.1 to 5. For noiseincluded data, a little threshold parameter is advised. Since the implementation of LSRTM based on Eq. (3) or (4) is a waveform matching process between the synthetic and observed seismic data, the problem of waveform mismatching is not a negligible issue. Unfortunately, the incorrect source wavelet would result in this problem even though the background velocity is accurate, which would contaminate the migration images, such as reversing the polarity, dislocating the layers, and destroying the layer continuity. In the following context, we apply the source estimation method by

(a)

(b)

Fig. 4. Conventional LSRTM images generated with (a) the true wavelet, (b) the incorrect Ricker wavelet, and (c) the estimated wavelet; (d) source-independent LSRTM image obtained by the incorrect Ricker wavelet.

(c)

where m(x) is defined as 2δcðxÞ c0 ðxÞ . Since the background wavefield p0(x,t;xs) contains almost only the direct waves and m(x) is responsible for the reflected waves, it is natural to call m(x) the reflectivity model (Symes and Kern, 1994). LSRTM attempts to minimize the differences between the observed and simulated data to refine the migration image towards the true reflectivity. The migration velocity model c0(x) is assumed to be accurate enough and does not change with iterations. Naturally, the objective function is established by
2 1


χ 2 ðmÞ ¼
qsyn ðxr ; t; xs Þ−qobs ðxr ; t; xs Þ
; 2

(d)

ð3Þ

where the subscript 2 denotes L2-norm objective function, the superscripts syn and obs represent the simulated data of an estimated model and the observed data respectively, xr is the location of receivers. To improve the robustness to noises, the hybrid norm objective function (Brossier et al., 2010) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
syn

q ðxr ; t; xs Þ−qobs ðxr ; t; xs Þ
2 −1; χ h ðmÞ ¼ 1 þ ε2

ð4Þ

is usually adopted, where the subscript h indicates the hybrid-norm objective function, ε is the threshold parameter chosen experimentally by

Fig. 5. Profiles at X = 4.42 km: conventional LSRTM results with (a) the true wavelet, (b) the incorrect wavelet, and (c) the estimated wavelet; (d) source-independent LSRTM result with the incorrect Ricker wavelet.

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deconvolution (Kim et al., 2011, 2013) to the conventional LSRTM and propose the source-independent LSRTM (Zhang et al., 2015a) by convolution (Choi and Alkhalifah, 2011) without the source estimation procedure. 2.2. Source estimation method by deconvolution Kim et al. (2011, 2013) proposed to estimate the source wavelet before RTM based on the deconvolution of observed seismic data by the Green's function. The numerical Green's function on the receiver nodes can be obtained in the time or frequent domain with a delta source wavelet, or by dividing one simulated wavefield by the used source wavelet. Assuming the background velocity is accurate, the real source wavelet in the frequency domain can be obtained by dividing the observed seismic data by the calculated Green's function d ðωÞ G ðωÞd ðωÞ ; ¼  sðωÞ ¼ GðωÞ G ðωÞGðωÞ obs

obs

ð5Þ

where s(ω) is the estimated real source wavelet, dobs(ω) is the observed seismic data, G(ω) is the Green's function, the superscript ∗ represents the conjugate. We apply Eq. (5) to estimate the source wavelet before the LSRTM. If the source wavelet of different shots differs from each other, the source estimation should be conducted for each shot. As a result, including the additional numerical modelling for calculating the Green's function, the estimation procedure would increase the computation amount of RTM or LSRTM although its elapsed time is less than the migration.

Fig. 6. The true wavelet (solid line) and that after 30° (dotted line) and 90° (dashed line) phase rotation.

where G(x, t; xs), G0(x, t; xs) are the Green's functions associated with c(x) and c0(x), and Gδ(x, t; xs) = G(x, t;xs) − G0(x, t; xs). Usually, the near-offset trace is a good choice for the reference trace. We suggest to just use the near-offset direct waves as the reference trace. Since seismic data are usually contaminated with noises, an average reference trace is better than a single one because averaging can improve the signal-to-noise ratio (SNR). The definition and physical interpretation of SNR could be referred to Chen (2015); Chen and Fomel (2015), and Chen et al. (2016). However, it should be noted that a normal-move-out (NMO) correction for the first arrivals should be conducted to make sure of an in-phase stack. In this paper, we utilize

2.3. Convolution-based source-independent method In order to avoid the procedure of source estimation, we define a convolution-based objective function (Choi and Alkhalifah, 2011; Zhang et al., 2015a) for the time-domain LSRTM as ℜ 2 ðmÞ ¼


2 1


syn obs ðxr ; t; xs Þ  rsyn ðt; xs Þ
;
q ðxr ; t; xs Þ  r obs ref ðt; xs Þ−q ref 2

ð6Þ

where robs and rsyn represent the reference trace chosen from the observed and simulated seismograms respectively, the subscript ref denotes the location of the reference trace, ∗ is the temporal convolution symbol. Through the Green's function (Tarantola, 1984; Dai et al., 2012), we have pðx; t; xs Þ ¼ Gðx; t; xs Þ  sðt; xs Þ;

ð7Þ

p0 ðx; t; xs Þ ¼ G0 ðx; t; xs Þ  sðt; xs Þ;

ð8Þ

qðx; t; xs Þ ¼ pðx; t; xs Þ−p0 ðx; t; xs Þ ¼ Gðx; t; xs Þ  sðt; xs Þ−G0 ðx; t; xs Þ  sðt; xs Þ ¼ Gδ ðx; t; xs Þ  sðt; xs Þ;

ð9Þ

Table 1 The errors between the LSRTM image and the true reflectivity model. Single means that only one reference trace is used and average means that several traces are averaged as a reference trace. Examples

Strategy

Wavelet

Error

Noise-free

Conventional Conventional Conventional Source-independent Conventional Conventional Source-independent Source-independent Conventional Source-independent Source-independent

True False Estimated False (single) True-30° True-90° True-30° True-90° Estimated False (single) False (average)

0.46 1.36 0.49 0.50 0.52 1.10 0.47 0.48 0.80 0.55 0.52

Noise-included

Fig. 7. The conventional (a) and source-independent (b) LSRTM images generated with the true wavelet after 30° phase rotation. The conventional (c) and source-independent (d) LSRTM images generated with the true wavelet after 90° phase rotation.

Q. Zhang et al. / Journal of Applied Geophysics 134 (2016) 1–10

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obs syn obs where G1 = Gδsyn ∗ Gobs ∗ ssyn. Obviously, 0 , ref, G2 = Gδ ∗ G0 , ref, f = s Eq. (10) can be regarded as the misfit between the different data excited by the same source signature f. As a result, the influence of the mismatch between the simulated and real source wavelets is eliminated. Similarly, to improve the robustness of our algorithm, we modify the hybrid norm objective function as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
u

syn u obs  r syn
q  r obs t ref −q ref
−1; ℜh ðmÞ ¼ 1 þ 2 ε

ð11Þ

where ε is the threshold chosen experimentally by a × mean(|qobs ∗ r-ref |), the value of a is chosen as described in the previous section.

syn

2.4. The gradient of the convolution-based objective function The gradient of ℜh(m) to the reflectivity model can be written tersely as " #T ∂rsyn ∂ℜh ∂qsyn obs ref obs R; ¼  rref −q  ∂m ∂m ∂m

ð12Þ

qsyn  robs −qobs  rsyn

ref ref where R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . Unfolding the convolution operator,

ε2

2

1þkqsyn  r obs −qobs  r syn k =ε2 ref ref

we obtain syn  ∂ℜ h X ∂qis;ir ðτÞ  obs ∬ ris;ref ðt−τÞ Ris;ir ðt Þdτdt ¼ ∂m ∂m is;ir 1 0  syn ∂ r is;ref ðτÞ X @ Aqobs ðt−τ ÞRis;ir ðt Þdτdt; ∬ − is;ir ∂m is;ir

ð13Þ

where is and ir denote the indexes of the sources and receivers respectively. Note that we choose the direct waves as the reference trace, then ∂r∂m

syn

¼

∂psyn 0 ∂m

¼ 0 due to the assumption of accurate background velocity. As a

Fig. 8. Profiles at X = 4.42 km: conventional (a) and source-independent (b) LSRTM results with the true wavelet after 30° phase rotation; conventional (c) and sourceindependent (d) LSRTM results with the true wavelet after 90° phase rotation. obs direct waves of p0(xr,t; xs) for reference trace selection, i.e., rref (t; xs) = p0(xref, t; xs). Then based on Eqs. (8) and (9), Eq. (6) can be rewritten briefly as


2 1


syn syn syn obs obs
Gδ  s  G0;ref  sobs −Gδ  sobs  G0;ref  ssyn
2 1 2 ¼ kG1  f−G2  f k ; 2

ℜ 2 ðmÞ ¼

ð10Þ Fig. 9. The seismogram contaminated by the random Gaussian and spike noises (SNR = 5 dB) of the 20th source at X = 2.0 km.

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result, we can rewrite Eq. (13) compactly as  syn T ∂ℜh ∂q R 0 ¼ LT R 0 ; ¼ ∂m ∂m

ð14Þ

  gn ¼ P LT R 0n ;

where, R0 ¼ robs ref ⊗R;

We calculate the gradient, Eq. (14), by the adjoint state method (Plessix, 2006; Zhou et al., 2009) using the back-propagated scheme like RTM and adopt the conjugate-gradient method to conduct the LSRTM:

ð15Þ

the symbol ⊗ stands for the temporal cross-correlation, R′ denotes the back-propagated residual seismogram.

β¼

gTn gn T gn−1 gn−1

cn ¼ mnþ1

;

n ¼ 0; −g0 ; βcn−1 −gn ; n N0; ¼ mn þ αcn ;

ð16Þ

Fig. 10. (a) The comparison between the estimated wavelet (dotted line) and the true wavelet (solid line), and the estimated errors (dashed line). The seismogram of the direct waves with the SNR of 5 dB before (b) and after (c) NMO. The single reference trace (d) chosen from (b) at the source location, and the averaged reference trace (e) obtained from (c).

Q. Zhang et al. / Journal of Applied Geophysics 134 (2016) 1–10

where P is the preprocessing operator (Zhang et al., 2012), gn represents the gradient after preprocessing, α is the step length that can be calculated by Dai et al. (2012). Moreover, the differences between the LSRTM image and the true reflectivity model are used to assess the quality of the migration images,

error ¼

kmlsrtm −mtrue k2 kmtrue k2

:

ð17Þ

3. Synthetic data examples Based on the above theory, we perform the following tests using the synthetic data with and without noises. 3.1. Noise-free synthetic data example The Marmousi model with a size of 7.67 km × 2.57 km and a spatial interval of 10 m is used to conduct the following tests, in which 80 sources and 767 receivers are evenly distributed at the depth of 20 m and 10 m respectively. The source interval is 90 m and the receiver interval is 10 m. The record length is 3.0 s with an interval of 0.5 ms. The absorbing layers are set around the modelling area so that the free-surface multiples are not considered in this paper. Fig. 1a and b shows the true and smoothed (migration) velocity models. Fig. 1c represents the corresponding true reflectivity model. Fig. 2a displays the true wavelet (solid line) which is obtained by the first derivative of the Gaussian function and used to generate the mimic observed seismic data. Fig. 2b exhibits the incorrect source wavelet which is the second derivative (Ricker wavelet) of Gaussian function. Both wavelets have a dominant frequency of 20 Hz.

7

in the frequency domain, the source wavelet can be estimated through dividing the observed seismic data by the Green's function according to Eq. (5). After inverse Fourier transform, the estimated time-domain source wavelet (dotted line in Fig. 2a) is derived, from which we can find the estimated wavelet fits well with the true one (solid line in Fig. 2a) with little errors (dashed line in Fig. 2a). Using the estimated source wavelet, the conventional LSRTM is conducted again and the corresponding migration image is shown in Fig. 4c. As to the sourceindependent LSRTM, a single reference trace of the direct waves at the source location is selected in the noise-free examples and the corresponding migration result with the incorrect Ricker wavelet (Fig. 2b) is shown in Fig. 4d. It should be pointed that all the reflectivity images shown in this paper are under the same magnitude scale. Obviously, when the source wavelet is estimated incorrectly (Fig. 2b), the conventional LSRTM algorithm collapses and the migration image (Fig. 4b) is unsatisfactory. While by conducting the source estimation with the Green's function, the true source wavelet can be reconstructed precisely (Fig. 2a) and the conventional LSRTM can recover the subsurface reflectivity model accurately (Fig. 4c). By contrast, the source-independent LSRTM image (Fig. 4d) can still keep the high quality although using the incorrect wavelet without source estimation. To show the migration results clearly, the profiles at X = 4.42 km are chosen from Fig. 4 into Fig. 5 represented by the solid lines with dots. It can be noticed that the conventional LSRTM result (Fig. 5b) with the incorrect Ricker wavelet mismatches with the true model visibly and exists some problems, such as polarity reversals (solid arrows), layer dislocations (dashed arrows). Consequently, the corresponding image errors are increased to more than 1 (error ≈ 1.36 in Table 1). Similar to the conventional LSRTM result (Fig. 5a) with the true wavelet, the LSRTM result (Fig. 5c) with the estimated source wavelet and the source-independent LSRTM result (Fig. 5d) match well with the true reflectivity model. The whole errors in Table 1 also show that the source-estimated (error ≈ 0.49) and source-independent

3.1.1. Analysis of the convolution-based objective function In this part, we use the correct (solid line in Fig. 2a) and incorrect (Fig. 2b) source wavelets to generate the mimic observed and simulated seismic data both with the true reflectivity model (Fig. 1c). Fig. 3a shows the 300th trace chosen from the 20th source at X = 2 km obtained by the conventional method, where the solid black line and the dashed blue line respectively represent the mimic observed and simulated seismic data, the solid red line denotes the corresponding residuals. Fig. 3b displays the convolution-based results at the same place, where the superscript of asterisk means the result of the convolution-based objective function and the residuals (red line) have been enhanced by 100 times. We can find the conventional residuals (red line in Fig. 3a) have the stronger amplitude than the original observed or simulated seismic data. Note that we assume the initial reflectivity model as zero in the tests, so the initial residual amplitude has the same absolute value with the observed seismic data. As a result, the conventional LSRTM built by the misfit function Eq. (3) or (4) with the incorrect wavelet would not converge to the true reflectivity model that minimizes the misfit function as zero. On the contrary, the residuals obtained by the convolution-based method (Eq. (15)) are still almost zeros although after the enhancement of 100 times, which means that our sourceindependent method can eliminate the influence of the incorrect source wavelet and can be ensured to converge to the true reflectivity. 3.1.2. Comparison between the conventional and source-independent LSRTM In the following tests, the initial reflectivity model is assumed to be zero. All the tests are conducted 40 iterations with the conjugategradient method. Fig. 4a and b shows the conventional LSRTM images generated with the true (solid line in Fig. 2a) and incorrect (Fig. 2b) wavelets. In order to estimate the true source wavelet before LSRTM, the dirac wavelet and the migration velocity are used to obtain the numerical Green's function at the receiver nodes in the time domain. Then

Fig. 11. (a) Conventional LSRTM image generated with the estimated wavelet (dotted line in Fig. 6a). Source-independent LSRTM image obtained with the incorrect Ricker wavelet using (b) the single reference trace, and (c) the averaged reference trace.

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(error ≈ 0.50) LSRTM images have a similar quality with the conventional source-known LSRTM image (error ≈ 0.46), which further demonstrates the feasibility of both the methods. However, compared to the conventional LSRTM with the estimated wavelet, the source-independent LSRTM can be implemented successfully without the procedure of source estimation. 3.1.3. The influence of the phase of the source wavelet Usually, the dominant or peak frequency of the simulated source wavelet is decided by that of the observed seismic data, as a result we do not pay attention to the influence of its peak frequency. In fact, the convolution-based objective function plays a role of low-pass filter (Choi and Alkhalifah, 2011; Zhang et al., 2016), therefore, the image resolution depends on the observed or simulated seismic data with the lower peak frequency. Next, we change the phase of the true source wavelet to analyse the sensitivity of our source-independent method. Fig. 6 shows the original true wavelet (solid line) and that after the phase rotation of 30° (the dotted line) and 90° (the dashed line). Then we use the phase rotated wavelet to conduct the tests with the single reference trace at the source location. Fig. 7a and b displays the conventional and source-independent LSRTM images at the 40th iteration

(a)

(b)

(c)

Fig. 12. Profiles at X = 4.42 km: (a) conventional LSRTM result with the estimated wavelet, source-independent LSRTM results of the incorrect wavelet using (b) the single reference trace, and (c) the average reference trace.

generated with the source wavelet after 30° phase rotation. Fig. 7c and d represents the corresponding images generated with the wavelet after 90° phase rotation. The profiles at X = 4.42 km chosen from Fig. 7 are plotted in Fig. 8 (solid lines with dots). Due to the wavelet after 30° phase rotation (dotted line in Fig. 6), with a little time shift and amplitude errors, is similar to the original true wavelet (solid line in Fig. 6), the conventional method can obtain a satisfactory image (Figs. 7a and 8a). Compared to the conventional result (Fig. 8a), the source-independent image (Fig. 8b) has a higher amplitude-preserving ability (error ≈ 0.47 in Table 1). When the wavelet after 90° phase rotation is used, the conventional LSRTM image becomes blurring (Fig. 7c) and matches poorly with the true reflectivity (Fig. 8c). Nonetheless, our source-independent LSRTM image can still keep a high quality (Figs. 7d and 8d). Therefore, our sourceindependent LSRTM algorithm is insensitive to the phase of the source wavelet, which can also be demonstrated by the data in Table 1. 3.2. Noise-included synthetic data example In the following examples, the random Gaussian and spike noises are added into the mimic observed seismic data using the Seismic Unix with the SNR of 5 dB. To improve the anti-noise ability of both the sourceestimated and source-independent algorithms, the hybrid-norm objective functions in Eqs. (4) and (11) are adopted during the 40 iterations with the conjugate-gradient method. Same as the noise-free examples, the initial reflectivity model is also set to zero. Both the single and average reference traces are used to carry out the source-independent experiments. Fig. 9 exhibits the noise-included seismogram of the 20th source at the distance of 2.0 km. Performing the source estimation procedure again, Fig. 10a shows the estimated source wavelet (dotted line) and the corresponding errors (dashed line) compared to the true one (solid line). Compared to Fig. 2a, there are some more oscillations in the estimated source wavelet (dotted line in Fig. 10a) due to the influence of the noises. Fig. 10b and c shows the selected reference traces before and after the NMO correction. Fig. 10d and e displays the single reference trace at the source location and the average reference trace obtained from Fig. 10c, which indicates that the average reference trace (Fig. 10e) has a higher SNR than the single one (Fig. 10d). Note that only the direct waves are selected in the reference trace so that the noises after the direct wave are muted. Fig. 11a displays the corresponding conventional LSRTM image generated with the estimated source wavelet (dotted line in Fig. 10a) and the hybrid-norm objective function. Fig. 11b shows the hybrid-norm source-independent LSRTM image with the incorrect Ricker wavelet and the single reference trace (Fig. 10d). It can be seen from Figs. 10a and 11a that the source estimation method is sensitive to the noises so that the conventional LSRTM image is still influenced by those noises, even though the hybrid-norm objective function in Eq. (4) is adopted. Compared to the source-estimated LSRTM image (Fig. 11a), the source-independent LSRTM image (Fig. 11b) has a higher illumination. In addition, the average reference trace (Fig. 10e) with higher SNR can further improve the resolution of the migration image (Fig. 11c) than the single one. Fig. 12 displays the profiles at X = 4.42 km chosen from Fig. 11, where the solid line and dashed line represent the true and initial reflectivity models, the solid line with dots denotes the LSRTM result. It is obvious that the conventional LSRTM result (Fig. 12a) based on the source estimation is inferior to the source-independent LSRTM result (Fig. 12b). The errors in Table 1 also show that the sourceindependent LSRTM image (error ≈ 0.56) has a higher quality than the conventional one (error ≈ 0.80). As indicated by the arrows, the source-independent LSRTM result (Fig. 12c) of the average reference trace matches to the true one better than that (Fig. 12b) of the single reference trace, and the corresponding error (≈ 0.56) is less than that (≈ 0.53) of the single one.

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4. Discussions In this study, we apply the source estimation method (Kim et al., 2011, 2013) to LSRTM to solve the waveform mismatching problems caused by the incorrect source wavelet. Besides, in order to solve the source problem, we propose the source-independent LSRTM (Zhang et al., 2015a) based on the convolution-based objective function (Choi and Alkhalifah, 2011). The noise-free examples show that the migration image would be contaminated (Fig. 4b) if an incorrect wavelet is used, which might result in the problems of polarity reversals, layer dislocations and so on, as indicated by the arrows in Fig. 5b. Through additional once modelling for calculating the numerical Green's function at the receivers node, the source wavelet can be reconstructed based on the frequency-domain deconvolution between the observed seismic data and the Green's function. As shown in Fig. 2a, the estimated source wavelet (dotted line) can matches the true wavelet (solid line) very well. Using the estimated source wavelet, the conventional LSRTM image (Fig. 4c) can be dramatically improved. In contrast, without source estimation procedure, our source-independent LSRTM algorithm can still obtain a high-quality migration image (Fig. 4d). In addition, the analysis of the wavelet phase (Figs. 6–8) further verifies the insensitivity of our source-independent method to the source wavelet. The noise-included examples (Fig. 10a) show that the source estimation method is sensitive to the noises which would decrease the resolution of the conventional LSRTM image (Fig. 11a). The average reference trace is proved to be a better choice for source-independent LSRTM (Figs. 11c and 12c) than the single one (Figs. 11b and 12b), because stacking can improve the SNR of the reference trace (Fig. 10d and e). However, there are still some problems for future practical application, such as the free-surface multiples, surface topography (Tang et al., 2013; Wang et al., 2015). Besides, the simultaneous-source acquisition, becoming more and more appealing for its tremendous acquisition cost reduction and the quality improvement of the seismic data, should be taken into account for the imaging or inversion method (Chen et al., 2015). Therefore, more works should be implemented before applying the source-independent algorithm to the real seismic data, which is out of the scope of this study. 5. Conclusions We apply the deconvolution-based source estimation method to the conventional LSRTM and propose the convolution-based sourceindependent LSRTM in the time domain. Conventional LSRTM needs additional source estimation procedure to give full play to its amplitudepreserving ability, while the source-independent algorithm can obtain the high-quality image using the incorrect wavelet without source estimation and is insensitive to the phase of the source wavelet. In order to improve the robustness of the algorithm, we apply the robust hybridnorm objective function to both the methods. The noise-included examples show that our source-independent LSRTM algorithm has higher anti-noise and amplitude-preserving abilities than the conventional LSRTM with the source estimation method which is sensitive to the noises. With regard to the reference trace in the convolution-based objective function, averaging can improve the SNR of the reference trace, which can further improve the resolution of the migration image. To apply the source-independent algorithm to the real seismic data, we need to improve its practicability, such as the irregular surface topography, the surface multiples and the simultaneous-source seismic data, which will be included in our future works. Acknowledgements This work was supported in part by the 973 Program of China (2013CB228603), National Science and Technology Program (2016ZX05010001-002), National Natural Science Foundation of China

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(41174119), and the Research of Novel Method and Technology of Geophysical Prospecting, CNPC (2014A-3609).

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