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GEOPHYSICS, VOL. 82, NO. 2 (MARCH-APRIL 2017); P. S159–S172, 18 FIGS. 10.1190/GEO2016-0416.1
Prestack correlative least-squares reverse time migration
Xuejian Liu1, Yike Liu2, Huiyi Lu2, Hao Hu3, and Majid Khan1
higher resolution and more balanced amplitudes (Nemeth et al., 1999; Aoki and Schuster, 2009; Dong et al., 2012; Huang et al., 2014; Zeng et al., 2014). The LSM methods combined with the Kirchhoff-type operators (Schuster, 1993; Nemeth et al., 1999; Duquet et al., 2000; Fomel et al., 2002; Liu et al., 2005; Trad, 2015; Hu et al., 2016) and one-way wave equations (Kühl and Sacchi, 2003; Kaplan et al., 2010; Ren et al., 2011; Wang et al., 2013) have been developed. Reverse time migration (RTM) based on two-way wave equations (Zhang and Sun, 2009) is considered more powerful to effectively handle various seismic waves and image steep reflectors as compared with the Kirchhoff or one-way wave-equation migration methods. At present, least-squares RTM (LSRTM) combining LSM with two-way wave equations is also widely researched (Wong et al., 2011, 2015; Dai et al., 2012; Dong et al., 2012; Dai and Schuster, 2013; Zhang et al., 2013, 2015; Dutta and Schuster, 2014; Luo and Hale, 2014; Tan and Huang, 2014; Zeng et al., 2014, 2015; Zhang and Schuster, 2014; Liu et al., 2016a, 2016b; Sun et al., 2016; Xue et al., 2016). Most of the LSRTM methods are based on the l2 -norm of amplitude differences between the modeled and observed data. We call methods that iteratively update the migration image to achieve accurate amplitude matching between modeled and observed data conventional LSRTM. However, many factors can cause the accurate amplitude-matching criteria to be impractical. First, the physics of wave propagation (such as density variation or absorption attenuation) cannot be fully considered in the modeling procedure (Dutta et al., 2014; Dutta and Schuster, 2014; Sun et al., 2016). The acquisition environment is also usually nonideal; e.g., field data typically contain unbalanced amplitudes for different shots and receivers (Lovelady et al., 1984; Yu, 1985; Yilmaz, 2001): In the acquisition of field data, amplitude anomalies may occur owing to variations in source strength and receiver sensitivity; moreover, the quality of the coupling between the source and the surface medium (land or water) can have a significant effect on signal amplitudes, as can the coupling between the receiver and the surface.
ABSTRACT In the correlative least-squares reverse time migration (CLSRTM) scheme, a stacked image is updated using a gradient-based inversion algorithm. However, CLSRTM experiences the incoherent stacking of different shots during each iteration due to the use of an imperfect velocity, which leads to image smearing. To reduce the sensitivity to velocity errors, we have developed prestack correlative least-squares reverse time migration (PCLSRTM), in which a gradient descent algorithm using a newly defined initial image and an efficiently defined analytical step length is developed to separately seek the optimal image for each shot gather before the final stacking. Furthermore, a weighted objective function is also designed for PCLSRTM, so that the data-domain gradient can avoid a strong truncation effect. Numerical experiments on a three-layer model as well as a marine synthetic and a field data set reveal the merits of PCLSRTM. In the presence of velocity errors, PCLSRTM shows better convergence and provides higher quality images as compared with CLSRTM. With the newly defined initial image, PCLSRTM can effectively handle observed data with unbalanced amplitudes. By using a weighted objective function, PCLSRTM can provide an image with enhanced resolution and balanced amplitude while avoiding many imaging artifacts.
INTRODUCTION Compared with conventional migration methods, least-squares migration (LSM), a linear inversion method, can iteratively suppress migration artifacts from acquisition footprints with a suitable regularization term (Nemeth et al., 1999; Duquet et al., 2000; Kühl and Sacchi, 2003; Xue et al., 2016) and can provide images with
Manuscript received by the Editor 3 August 2016; revised manuscript received 27 November 2016; published online 13 February 2017. 1 Institute of Geology and Geophysics, Key Laboratory of Shale Gas and Geoengineering, Chinese Academy of Sciences, Beijing, China and University of Chinese Academy of Sciences, Beijing, China. E-mail:
[email protected];
[email protected]. 2 Institute of Geology and Geophysics, Key Laboratory of Shale Gas and Geoengineering, Chinese Academy of Sciences, Beijing, China. E-mail: ykliu@mail. iggas.ac.cn;
[email protected]. 3 University of Houston, Department of Earth and Atmospheric Sciences, Houston, Texas, USA. E-mail:
[email protected]. © 2017 Society of Exploration Geophysicists. All rights reserved. S159
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Zhang et al. (2013, 2015) propose correlative least-squares reverse time migration (CLSRTM) that evaluates the closeness between the modeled and observed data with a normalized zero-lag crosscorrelation objective function. CLSRTM relaxes the amplitude constraint and emphasizes the phase matching, which is more stable and practical than conventional LSRTM to invert the acquired field data (Zhang et al., 2015). Dutta et al. (2014) discover that, in comparison with conventional least-squares reverse time migration (LSRTM), CLSRTM can invert a better image when the observed data experience strong attenuation in a viscoacoustic medium. However, CLSRTM is sensitive to velocity errors and provides the image with significant noise owing to the usage of an imperfect migration velocity model (Zhang et al., 2013, 2015). Therefore, we extend CLSRTM into prestack CLSRTM (PCLSRTM). PCLSRTM can provide better convergence and invert a cleaner image than CLSRTM when an inaccurate migration velocity is used. The initial image and the initial modeled data should be provided for CLSRTM. A conventional migration of the observed data can be used as the initial image (Zhang et al., 2015; Liu et al., 2016b); however, better results would be expected if the initial image is one that is derived to satisfy the assumptions of CLSRTM. We derive the initial images for PCLSRTM and CLSRTM from their respective objective functions. To handle the observed data with unbalanced amplitude, CLSRTM or PCLSRTM using our derived initial image provides a better final image than that with a conventional migration result as the initial image. An analytical step-length formula for CLRSTM was derived by Liu et al. (2016b), which can be computed more efficiently than a linear search method to determine the step length. In this paper, the analytical iterative step length for PCLSRTM is derived concisely in the data domain. Different numerical experiments on synthetic and field data sets demonstrate that PCLSRTM using analytical step lengths can converge stably. We discover that the data-domain gradient of CLSRTM or PCLSRTM tends to experience strong truncation effect at far offsets. Thus, a new weighted objective function is constructed for PCLSRTM, with which many artifacts can be avoided from the inverted image and good convergence is achieved in the marine field data example.
stacked image rðxÞ at each subsurface point x. The wavefield pr ðx; t; xs Þ would be recorded at the receiver position xr to form the reflection data dðxr ; t; xs Þ. The modeling procedure with a stacked image for all shot gathers can be compactly represented by the linear operator M½rðxÞ. Based on the dot-product test (Claerbout, 1992; Zhang et al., 2015) or the adjoint-state method (Plessix, 2006), conventional RTM can be derived from the modeling procedure in equation 1. Conventional RTM of observed data dobs ðxr ; t; xs Þ is then represented as
1 ∂2 2 − ∇ ps ðx; t; xs Þ ¼ fðt; xs Þ; v20 ðxÞ ∂t2
1 ∂2 2 p ðx; t; x Þ ¼ − ∂dobs ðxr ; t; xs Þ ; − ∇ r s ∂t v20 ðxÞ ∂t2
rmig ðxÞ ¼
XX xs
ps ðx; t; xs Þ · pr ðx; t; xs Þ;
(2b)
(2c)
t
where the source-side wavefield ps ðx; t; xs Þ is propagated in forward time, but the receiver-side wavefield pr ðx; t; xs Þ is propagated in backward time. In equation 2c, the zero-lag crosscorrelation imaging condition is implemented to generate prestack images, and a stacking procedure over shot gathers outputs the stacked image rmig ðxÞ. The migration procedure for a stacked image is compactly represented by the linear operator MT ðdobs ðxr ; t; xs ÞÞ, which can be considered the adjoint operator of M in the time domain. CLSRTM (Zhang et al., 2013, 2015) is proposed by minimizing the following normalized zero-lag crosscorrelation objective function:
ZZ EðrðxÞÞ ¼ −
R M½rðxÞ · dobs ðxr ;t;xs Þdt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dxr dxs ; R 2 R 2 dobs ðxr ;t;xs Þdt M ½rðxÞdt (3)
CORRELATIVE LSRTM With a stacked image rðxÞ as the input, the data dðxr ; t; xs Þ that are excited from the source position xs ¼ ðxs ; zs Þ, reflected at subsurface points x ¼ ðx; zÞ, and recorded at the receiver xr ¼ ðxr ; zr Þ can be simulated by the following modeling procedure (Zhang et al., 2015):
8 2 > > > v21ðxÞ ∂t∂ 2 − ∇2 ps ðx; t; xs Þ ¼ fðt; xs Þ; < 0 1 ∂2 2 sÞ ; pr ðx; t; xs Þ ¼ rðxÞ ∂ps ðx;t;x − ∇ 2 2 > ∂t > v0 ðxÞ ∂t > : dðxr ; t; xs Þ ¼ pr ðx; t; xs Þj x¼xr ;
(2a)
where rðxÞ represents a full-stacked image. Compared with LSRTM, CLSRTM uses a relaxed amplitude-matching criterion and highlights the phase closeness between the modeled and observed data (Zhang et al., 2013, 2015). The gradient of equation 3 with respect to the image is defined as
0 (1)
where v0 ðxÞ represents the background velocity model, ps ðx; t; xs Þ represents the source-side wavefield excited by the wavelet fðt; xs Þ and propagated in forward time, and pr ðx; t; xs Þ represents the receiver-side wavefield excited by fictitious subsurface sources and propagated in forward time. The fictitious sources on the right side of the second formula in equation 1 are generated through multiplying the time derivative of the source wavefield ps ðx; t; xs Þ by the
∂EðrðxÞÞ 1 B qRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ MT @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ∂rðxÞ M2 ½rðxÞdt d2 dt obs
R ×
1 M½rðxÞ · dobs dt C R 2 M½rðxÞ − dobs ðxr ; t; xs Þ A; M ½rðxÞdt
(4)
where the gradient of equation 3 with respect to the modeled data of each shot gather — i.e., the data-domain gradient — is migrated and stacked together. Note that the coordinate ðxr ; t; xs Þ is neglected for integral formulas in equation 4 and the subsequent equations.
Prestack correlative least-squares RTM
∂Eðfrs ðxÞgÞ 1 ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R R 2 ∂Ms ½rs ðxÞ M2 ½r ðxÞdt d dt
Given an initial reflectivity image, the stacked image can be updated with the gradient descent algorithm
×
where the superscript k represents the iteration number. Note that the operator ∂EðrðkÞ ðxÞÞ∕∂rðxÞ represents the value of the derivative function ∂EðrðxÞÞ∕∂rðxÞ at rðkÞ ðxÞ. The stacked RTM image of observed data M T ðdobs ðxr ; t; xs ÞÞ can be conveniently used as the initial image. After the gradient at the kth iteration is obtained, the stacked image rðkÞ ðxÞ can be updated with a step length λðkÞ . Compared with a linear-search method to determine an optimal step length, the analytical step length can be computed more efficiently (Liu et al., 2016b).
Ms ½rs ðxÞ · dobs dt R 2 Ms ½rs ðxÞ−dobs ðxr ;t;xs Þ : Ms ½rs ðxÞdt
More specifically, ∂Eðfrs ðxÞgÞ∕∂Ms ½rs ðxÞ in equation 7b represents the derivative of equation 6 with respect to each sampling point of the modeled data. Here, MTs compactly represents the imaging procedure for a single shot, which can be considered the adjoint operator of Ms in the time domain.
Inversion algorithm Given the initial prestack image, the gradient descent algorithm ðkþ1Þ
rs
PRESTACK CORRELATIVE LSRTM The CLSRTM is sensitive to migration velocity errors and would generate artifacts owing to the use of the imperfect migration velocity (Zhang et al., 2013, 2015). As illustrated in Figure 1, which is made with an inaccurate migration velocity, images of a CRP gather would be stacked incoherently. In CLSRTM, there is often incoherent stacking during each iteration, which leads to nonideal gradients and image smearing. PCLSRTM, which separately updates the migration image of each shot gather and only experiences incoherent stacking once at the end, is less susceptible to velocity errors. By replacing the stacked image rðxÞ in modeling equation 1 with the prestack image rs ðxÞ of the shot “s” and using the linear operator Ms to represent the modeling procedure with a prestack image for a single shot gather, the objective function in equation 3 for CLSRTM can be easily extended to an objective function for PCLSRTM:
a) x s
1
ðkÞ
ðkÞ
ðxÞ ¼ rs ðxÞ − λðkÞ
∂Eðfrs ðxÞgÞ ; ∂rs ðxÞ
(8a)
can be used to separately update the image of each shot gather with a unified step length; all updated prestack images should be stored in the memory at any one iteration and are stacked together in the last iteration:
RðxÞ ¼
X
ðKÞ
rs ðxÞ;
(8b)
s
where the capital letter K represents the largest iteration number and RðxÞ represents the finally stacked image. The prestack RTM image of the observed data MTs ðdobs ðxr ; t; xs ÞÞ can be used as the initial image for PCLSRTM. However, as derived in Appendix A, the initial image approximating the gradient descent xs
2
xs
3
d obs ( x r ; x s
1
dobs ( x r ; x s
1
1
dobs ( x r ; x s
2
(
Eðfrs ðxÞgÞ ¼ R ZZ M s ½rs ðxÞ · dobs ðxr ;t;xs Þdt q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxr dxs : − R 2 R 2 dobs ðxr ;t;xs Þdt Ms ½rs ðxÞdt
(7b)
d obs ( x r ; x s
2
2
d obs ( x r ; x s
3
d obs ( x r ; x s
3
3
(6) In equation 6, we define the operator frs ðxÞg as the ensemble of all prestack images of different shot records. Compared with equation 3, equation 6 can be considered an objective function with multiple parameters. Similar to CLSRTM, PCLSRTM also relaxes the amplitude constraint between the modeled and observed data. With the help of the adjoint-state method (Plessix, 2006; Routh et al., 2011; Zhang et al., 2015), the gradient of equation 6 with respect to the migration image of each shot gather is
∂Eðfrs ðxÞgÞ ∂Ms ½rs ðxÞ T ∂Eðfrs ðxÞgÞ ¼ ∂rs ðxÞ ∂rs ðxÞ ∂M s ½rs ðxÞ ∂Eðfrs ðxÞgÞ ¼ MTs ; (7a) ∂Ms ½rs ðxÞ where the data-domain gradient is
vmig =v0
b) x s
1
xs
2
xs
3
1
2
3
(
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(5)
obs
(
∂Eðr ðxÞÞ ; ∂rðxÞ
s
(
−
λðkÞ
s
(
¼
rðkÞ ðxÞ
R
(
rðkþ1Þ ðxÞ
ðkÞ
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vmig > < cos −θ0 þðoffsetðxr Þ−ofs0 Þ ofs1 −ofs0 ; ofs0 < offsetðxr Þ < ofs1 ; ofs1 ≤ offsetðxr Þ ≤ ofs2 ; Pd ðxr Þ ¼ 1.0; > > : cos2 ðoffsetðxr Þ−ofs2 Þ θ1 ; ofs2 < offsetðxr Þ < ofs3 ; ofs3 −ofs2 (14)
where a square cosine taper function is applied at the presupposed offset intervals ðofs0 ; ofs1 Þ and ðofs2 ; ofs3 Þ, and offsetðxr Þ denotes
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the offset of a trace at xr . In Figure 2, the weighting operators Pd ðxr Þ for a split-spread and an off-end shooting geometry are illustrated as examples. As illustrated in Figure 2a, for a split-spread shooting geometry with the same positive and negative offsets, the parameters of weighting operator Pd ðxr Þ are set as ofs1 − ofs0 ¼ ofs3 − ofs2 and P0 ¼ P1 ; a small value is assigned to P0 . As illustrated in Figure 2b, for an off-end shooting geometry, the taper is applied within a larger offset interval and has a much smaller boundary value at far offsets than at near offsets. Using the weighted objective function given in equation 13, the image of each shot gather is separately updated as ðkþ1Þ rs ðxÞ
¼
ðkÞ rs ðxÞ
−λ
ðkÞ
MTs
ðkÞ ∂Eðfrs ðxÞgÞ ; ∂Ms ½rs ðxÞ
(15a)
where the data-domain gradient at the kth iteration is ðkÞ
∂Eðfrs ðxÞgÞ 1 ¼ Pd ðxr Þ · qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 ðkÞ R 2 ∂Ms ½rs ðxÞ Ms ½rs ðxÞdt dobs dt R ðkÞ Ms ½rs ðxÞ · dobs dt ðkÞ × Ms ½rs ðxÞ−dobs ðxr ;t;xs Þ : (15b) R 2 ðkÞ Ms ½rs ðxÞdt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkÞ In equation 15b, the product Pd ðxr Þ · 1∕ ∫ M2s ½rs ðxÞdt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × ∫ d2obs dt should roughly decrease toward far offsets to avoid the strong truncation effect. For the objective function given in
Figure 5. Imaging of the synthetic data shown in Figure 4a using the accurate migration velocity: (a and b) the stacked RTM images of the observed data and of the amplitude-normalized observed data, respectively; (c and d) the images obtained by CLSRTM (10 iterations) with the stacked images in panels (a and b) as the initial images, respectively; (e and f) the images obtained by PCLSRTM (10 iterations) with the prestack RTM images of the observed data and of the amplitude-normalized observed data as the initial images, respectively. Note that the imaging results in panels (c and e) are similar to those in panels (d and f), respectively.
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the amplitude-normalized observed data should be weighted by the operator Pd ðxr Þ prior to migration. The analytical step length can also be easily derived as shown in Appendix B.
Figure 6. Imaging of the data with unbalanced amplitudes, shown in Figure 4b, using the accurate migration velocity: (a and b) the stacked RTM images of the observed data and of the amplitude-normalized observed data, respectively; (c and d) the images obtained by CLSRTM (10 iterations) with the stacked images in panels (a and b) as the initial images, respectively; and (e and f) the images obtained by PCLSRTM (10 iterations) with the prestack RTM images of the observed data and of the amplitude-normalized observed data as the initial images, respectively. As indicated by arrows, panels (d and f) show cleaner imaging results as compared with panels (c and e), respectively.
NUMERICAL EXPERIMENTS We demonstrate our method with a three-layer model, a marine synthetic and a marine field data set. For CLSRTM and PCLSRTM, we use their respective analytical step lengths. The convergence of CLSRTM or PCLSRTM is evaluated by
a)
b)
c)
d)
e)
f)
a)
b)
1
Normalized amplitude
Figure 7. Wavenumber spectra of the depth direction for the images shown in Figures (a) 5a, 5c, 5e, (b) 5b, 5d, 5f, (c) 6a, 6c, 6e, and (d) 6b, 6d, and 6f. Spectra of images obtained by RTM, CLSRTM, and PCLSRTM are illustrated using the black dotted, black, and gray lines, respectively. In panels (a and c), CLSRTM provides an image with a significantly broader spectrum than PCLSRTM; in panels (b and d), the spectrum of the image obtained by CLSRTM is only a little broader than that of the image obtained by PCLSRTM.
RTM CLSRTM PCLSRTM
RTM CLSRTM PCLSRTM
0.5
0
c)
d)
1
Normalized amplitude
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equation 13, the initial prestack image can be derived following qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Appendix A as M Ts ðPd ðxr Þ · ½1∕ ∫ d2obs dtdobs ðxr ; t; xs ÞÞ; i.e.,
RTM CLSRTM PCLSRTM
RTM CLSRTM PCLSRTM
0.5
0 0
0.02
Wavenumber (m–1)
0.04 0
0.02
Wavenumber (m–1)
0.04
Prestack correlative least-squares RTM
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WAEðkÞ ¼ RR
EðkÞ ; qðxr Þdxr dxs
(16)
where EðkÞ represents the objective function value at the kth iteration, the factor qðxr Þ ¼ 1.0 or qðxr Þ ¼ Pd ðxr Þ is assigned for one trace according to whether the weighted objective function is applied, and WAEðkÞ represents the weighted average of the objective function. Equation 16 normalizes the objective function value into the interval of [−1, 0]. When the modeled data are zero and have not been initially calculated, the initial value of the objective function is zero. In the best-case scenario, in which the modeled and observed data are identical or with a constant scaling difference (Zhang et al., 2015), the objective function value reaches its minimum and the weighted average of the objective function is −1.
A three-layer model The three-layer model, in which velocity changes only in the depth direction, is discretized into 1001 (in the horizontal direction)
a)
b)
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by 501 (in the depth direction) grid points with a 5 m grid interval. The 1D velocity models are shown in Figure 3. The layered velocity model is used for synthesizing data; the accurate migration velocity model is obtained by smoothing the layered velocity with a Gaussian filter; the migration velocity with 5% bulk error is obtained by scaling the accurate migration velocity with a factor of 95%. There are 17 shot gathers synthesized in which the sources with a 15 Hz Ricker wavelet are excited between 1.54 and 3.46 km with a 120 m interval. Each shot has 201 receivers with a split-spread geometry and a 20 m receiver interval. The recording length and time sampling interval are 3 s and 2 ms, respectively. Two types of initial images Figure 4a shows one shot of synthetic data, and Figure 4b shows one shot of the scaled data. The scaled data are obtained by randomly scaling synthetic data with different factors for different traces and shots to simulate data with unbalanced amplitudes. RTM images of the observed data and those of the amplitude-normalized observed data are used as two types of initial images in CLSRTM or PCLSRTM. To observe the influence of the initial image on the inversion results by CLSRTM or PCLSRTM, two data sets as shown in Figure 4 are inverted using the correct migration velocity model. Synthetic data (such as those shown in Figure 4a) are used as observed data, and the images obtained by CLSRTM and PCLSRTM are shown in Figure 5. The stacked RTM images of the observed data and those of the amplitude-normalized observed data are shown in Figure 5a and 5b, respectively. With the stacked images in Figure 5a and 5b as the initial images, the images obtained by CLSRTM after 10 iterations are shown in Figure 5c and 5d, respectively. Given the prestack RTM images of the observed data and those of the amplitude-normalized observed data as the initial images, the images obtained by PCLSRTM after 10 iterations are shown in Figure 5e and 5f, respectively. The imaging results in Figure 5c and 5e are similar to those in Figure 5d and 5f, respectively. The randomly scaled data (such as Figure 4b) are used as observed data, and the inverted images by CLSRTM and PCLSRTM are shown in Figure 6. The stacked RTM images of observed data and those of amplitude-normalized observed data are shown in Figure 6a and 6b, respectively. With the stacked images in Figure 6a and 6b as the initial images, the images obtained by CLSRTM after 10 iterations are shown in Figure 6c and 6d, respectively. The
c)
Figure 8. Imaging of synthetic data as shown in Figure 4a using the migration velocity with 5% bulk error: (a) the stacked RTM image of amplitude-normalized observed data; (b) the image obtained by CLSRTM (10 iterations); and (c) the image obtained by PCLSRTM (10 iterations). As indicated by arrows, there are more artifacts in panel (b) than panel (c).
Figure 9. Imaging of one shot data shown in Figure 4a using the migration velocity with 5% bulk error: (a) the prestack RTM image of amplitude-normalized observed data and (b) the image obtained by PCLSRTM (10 iterations). Although the migration velocity used is incorrect, PCLSRTM still can effectively update the prestack image by balancing the amplitude and improving the resolution.
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prestack RTM images of the observed data and those of the amplitude-normalized observed data are provided as the initial images, and the images obtained by PCLSRTM after 10 iterations are shown in Figure 6e and 6f, respectively. At the positions indicated by arrows in Figure 6, Figure 6d and 6f show cleaner imaging results as compared with Figure 6c and 6e, respectively. When the observed data have balanced amplitudes (Lovelady et al., 1984; Yu, 1985; Yilmaz, 2001), the RTM images of the observed data and those of the amplitude-normalized observed data both can be used as the initial images in CLSRTM or PCLSRTM. To handle the observed data with unbalanced amplitudes, the comparisons between Figure 6c and 6d and between Figure 6e and 6f show that the RTM image of the amplitude-normalized observed data is better than that of the observed data to be used as the initial image in CLSRTM or PCLSRTM. Comparing Figure 5c and 5e with Figure 5a, comparing Figure 5d and 5f with Figure 5b, and comparing Figure 6d and 6f with Figure 6b, the images obtained by CLSRTM and PCLSRTM show higher resolution and more balanced amplitudes with the two reflectors than the conventional RTM images. In addition, the provided wavenumber spectra in Figure 7 show a comparison of the results: With the RTM result from the observed data as the initial image, CLSRTM can provide an image with a significantly broader spectrum than PCLSRTM; however, using the amplitude-normalized observed data to compute the initial image, the spectrum of the image obtained by CLSRTM is only a little broader than that of the image obtained by PCLSRTM.
higher resolution than RTM, although an erroneous migration velocity is used. Convergence curves for CLSRTM using an accurate migration velocity and using a migration velocity with 5% error
Migration velocity with error Synthetic data (Figure 4a) are inverted using the migration velocity with a bulk error of 5%, in which the RTM image of amplitudenormalized observed data is used as the initial image. The stacked RTM image of the amplitude-normalized observed data, the image obtained by CLSRTM after 10 iterations, and the image obtained by PCLSRTM after 10 iterations are shown in Figure 8a–8c, respectively. As indicated by arrows, there are more artifacts in Figure 8b as compared with Figure 8c. In detail, the images of one shot data obtained by RTM and PCLSRTM using 10 iterations are shown in Figure 9a and 9b, respectively. In Figure 9, PCLSRTM still can effectively provide an image with more balanced amplitude and Figure 10. (a) Convergence curves for CLSRTM using the accurate migration velocity and using the migration velocity with 5% error are illustrated by the black and gray lines, respectively, which corresponds to the inverted images in Figures 5d and 8b. (b) Convergence curves for PCLSRTM using the accurate migration velocity and using the migration velocity with 5% error are illustrated by the black and gray lines, respectively, which correspond to the inverted images in Figures 5f and 8c. When the migration velocity contains 5% bulk error, CLSRTM converges more quickly than PCLSRTM.
Figure 11. (a) The interval velocity is transformed from the stacking velocity of the marine field data, and (b) the migration velocity is estimated by smoothing the interval velocity.
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Prestack correlative least-squares RTM
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are represented by the black and gray lines in Figure 10a, respectively; convergence curves for PCLSRTM using an accurate migration velocity and using a migration velocity with 5% error are represented by the black and gray lines in Figure 10b, respectively. Four convergence curves correspond to the inverted images in Figures 5d, 5f, 8b, and 8c. When the migration velocity contains bulk error, CLSRTM experiences a much greater convergence decrease than PCLSRTM. PCLSRTM seems less susceptible to velocity errors than CLSRTM.
Marine synthetic and field data sets The marine field data set has a total of 301 shot gathers with a shot interval of 75 m. Each shot has 192 receivers with a 25 m receiver interval, and the nearest and farthest offsets are −0.1 and −4.875 km, respectively. The recording length and the time-sampling interval are 6 s and 2 ms, respectively. The estimated interval and migration velocity models using the field data set are shown in Figure 11a and 11b, respectively, which consist of 1801 × 1401 grid points with a 12.5 m horizontal grid interval and a 5 m vertical grid interval. A synthetic data set is first generated with an acoustic-modeling code using the interval velocity in Figure 11a to test CLSRTM and PCLSRTM. The acquisition parameters of the synthetic data are the same as those of the field data, except that the zero offset is included and, to avoid recording refracted waves in the synthetic data, the farthest offset is reduced to −4.4 km. The synthetic data are used as the observed data and then inverted using the migration velocity in Figure 11b, in which the RTM image of amplitude-normalized observed data is used as the initial image. The stacked RTM image of amplitude-normalized observed data, the inverted image by CLSRTM after 10 iterations, and the inverted image by PCLSRTM after 10 iterations are obtained as shown in Figure 12a–12c, respectively. When compared with the RTM image in Figure 12a, the
Figure 12. A synthetic data set is first generated with an acoustic modeling code using the interval velocity in Figure 11a to test CLSRTM and PCLSRTM. Using the migration velocity in Figure 11b, (a) the stacked RTM image of the amplitude-normalized observed data, (b) the image by CLSRTM (10 iterations) with the stacked image in panel (a) as the initial image, and (c) the image by PCLSRTM (10 iterations) with the prestack RTM image of the amplitude-normalized observed data as the initial image. As indicated by arrows, CLSRTM or PCLSRTM enhances the image of the steep reflectors when compared with RTM. Note that the three panels are clipped with the same percentage.
Figure 13. To invert the synthetic data modeled with the interval velocity in Figure 11a, convergence curves for CLSRTM and PCLSRTM, which are represented by the black and gray lines, respectively, decrease to very small values after 10 iterations.
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a)
inverted image by CLSRTM or PCLSRTM shows much higher resolution, more balanced amplitudes, and enhanced imaging of steep reflectors as indicated by arrows. For the inversion of the synthetic data, an accurate migration velocity is used, and CLSRTM and PCLSRTM show ideal convergence in Figure 13. However, for imaging the field data, the migration velocity in Figure 11b is imperfect and contains many random errors. Furthermore, as explained in the theory section, the seismic-event muting at far offsets during the regular processing of the field data can cause the data-domain gradient to experience a strong truncation effect, and the image inverted by CLSRTM or PCLSRTM can be spoiled
a) b)
b)
c)
c)
Figure 14. Imaging of the field data using the migration velocity in Figure 11b. (a) The stacked RTM image with source illumination compensation: the amplitude-normalized observed data are weighted by the operator in equation 14 prior to migration. (b) The CLSRTM image (10 iterations) and (c) the PCLSRTM image (10 iterations) are obtained using their respective weighted objective functions. As indicated by white and black arrows, panel (c) shows better subsurface illumination and fewer artifacts in some areas than panel (b). Additional red arrows and ellipses put on panel (c) indicate the positions, which would be compared with those in the following Figure 16. Note that the compared panels in this figure and in the following Figures 15–17 are clipped with the same percentage.
Figure 15. (a-c) Local windows of the images shown in Figure 14a– 14c. As indicated by arrows and ellipses, CLSRTM and PCLSRTM provide images with higher resolution than RTM.
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Prestack correlative least-squares RTM by many artifacts. In the following three sections, first, we present the advantages of PCLSRTM over CLSRTM when the field data are inverted with an imperfect migration velocity; second, we show that the application of the weighted objective function can effectively reduce the imaging artifacts; third, we discuss the parameter selection for the weighting operator in equation 14 with the field data. Comparison of CLSRTM and PCLSRTM using the field data The images of the field data obtained by CLSRTM and PCLSRTM using their respective weighted objective functions are compared. The stacked RTM image is shown in Figure 14a, in which the amplitude-normalized observed data are weighted by the operator given in equation 14 prior to migration. With the image in Figure 14a used as the initial image, the image obtained by CLSRTM after 10 iterations is shown in Figure 14b. With the prestack RTM image used as the initial image, the image obtained by PCLSRTM after 10 iterations is shown in Figure 14c. Altogether, Figure 14b and 14c shows more balanced amplitudes and better resolution as compared with Figure 14a. For detailed comparisons, Figure 14a–14c is locally magnified in Figure 15a–15c, respectively. Figure 15b and 15c contains more focused events and reveals more structural details than Figure 15a; e.g., the fault structure circled by ellipses is more interpretable in Figure 15b and 15c as compared with Figure 15a. With errors existing in the migration velocity, PCLSRTM seems to better invert images in some areas than CLSRTM: Figure 14b is comparatively more noisy than Figure 14c in the area indicated by black arrows; Figure 14c seems to provide better subsurface illumination than Figure 14b in the areas indicated by white arrows. Effect of the weighted objective function In this section, we demonstrate that application of the weighted objective function can improve the imaging result of the field data. The image by PCLSRTM using the unweighted objective function in equation 6 is shown in Figure 16, which is compared with the image
Figure 16. The image by PCLSRTM using the unweighted objective function in equation 6. The image in this figure is compared with the image in Figure 14c, which is provided by PCLSRTM using the weighted objective function in equation 13. As indicated by red arrows and ellipses in this figure and in Figure 14c, many artifacts in this figure are avoided in Figure 14c.
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in Figure 14c by PCLSRTM using the weighted objective function from equation 13. Artifacts would be generated when data-domain gradients with a strong truncation effect are inverted into the image domains. Many such artifacts indicated by red arrows and ellipses in Figure 16a are avoided in Figure 14c. Data from one shot record are shown in Figure 17a, and the modeled data at the 10th iteration by PCLSRTM using the weighted objective function are shown in Figure 17b, in which the modeled data closely match the recorded data. Convergence curves for PCLSRTM and CLSRTM are shown in Figure 18a and 18b, respectively. PCLSRTM shows much better convergence than CLSRTM owing to the use of an imperfect migration velocity; PCLSRTM and CLSRTM using the weighted objective function show better convergence than those using the unweighted objective function, respectively. Note that CLSRTM using the unweighted objective function can decrease in only several iterations in Figure 18b, due to the following reason: In an imaging procedure, seismic waves at far offsets usually undergo more velocity errors than those at near offsets; with the data-domain gradient having much stronger amplitudes at far offsets than at near offsets, CLSRTM using the unweighted objective function becomes especially susceptible to velocity errors. Parameter selection for the weighting operator The weighting operator is designed as shown in Figure 2b. A square cosine taper function is applied at the offset ranges [−4.875, −2.875] and ½−0.5; −0.1 km. The left and right boundary values are 0.001 and 0.64, respectively. The right taper range is small, and the right boundary value is large, so the two randomly selected parameters cause the data-domain gradient at near offsets to be tapered slightly. However, the energy of the trace at the farthest offset accounts for only 0.035 of that at the nearest offset of the seismic-event muting
Figure 17. (a) One shot recorded data with the source approximately located at 5.8 km on the surface; at the 10th iteration, (b) the modeled data by PCLSRTM using the weighted objective function closely matches the recorded data.
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CONCLUSION
Figure 18. (a) Convergence curves for PCLSRTM using the weighted objective function and using the unweighted objective function are illustrated by the black and gray lines, respectively. (b) Convergence curves for CLSRTM using the weighted objective function and using the unweighted objective function are illustrated by the black and gray lines, respectively. Using an imperfect migration velocity, PCLSRTM shows much better convergence than CLSRTM. In panel (b), CLSRTM using the unweighted objective function converges in only several iterations. zone, and the energy of a trace in the seismic-event muting zone decreases rapidly with increasing offset. The left taper range is therefore selected to roughly cover the seismic-event muting zone at far offsets for all shot gathers (such as Figure 17a). In our tests, using the square cosine taper function with a left boundary value of 0.1, the image by CLSRTM or PCLSRTM already avoids most artifacts of the truncation effect; the inverted image becomes slightly better when the left boundary value decreases, and the left boundary value is finally determined to be 0.001.
DISCUSSION Exact amplitude matching between field and modeled data is very difficult to achieve (Zhang et al., 2015), and CLSRTM or PCLSRTM using a crosscorrelation-based objective function is more practical than conventional LSRTM methods. CLSRTM iteratively seeks one imaging model that can be used to explain all shots of recorded data, which provides a further constraint for providing an ideal imaging result than PCLSRTM. However, empirical results show that CLSRTM is more sensitive to velocity errors than PCLSRTM, which is similar to the comparison result between plane-wave prestack LSRTM and plane-wave LSRTM with a stacked image (Dai and Schuster, 2013; Huang et al., 2014). The analytical step length can cause PCLSRTM to stably converge in different experiments (as shown in Figures 10b, 13, and 18a), so it is realistic that an objective function for PCLSRTM with respect to the step length is approximately parabolic at each iteration. Apart from the square cosine function, other commonly used taper functions with selected parameters can also be used in the weighted objective function for CLSRTM or PCLSRTM. Regularization terms (Nemeth et al., 1999; Duquet et al., 2000; Kühl and Sacchi, 2003; Aoki and Schuster, 2009; Xue et al., 2016) can be used in CLSRTM and PCLSRTM to improve the inverted images, but the selection criterion for the regularization coefficient must be determined empirically.
PCLSRTM is proposed, in which we separately update the migration image of each shot gather before the final stacking. We define the initial image according to the objective function and use an analytical step length to develop a complete gradient descent algorithm for PCLSRTM, and a weighted objective function is designed for PCLSRTM to avoid the strong truncation effect of the data-domain gradient. In numerical experiments with synthetic and field data sets, PCLSRTM can provide a higher quality image than CLSRTM in the presence of velocity errors and can effectively handle observed data with unbalanced amplitudes. Using the weighted objective function, PCLSRTM can enhance the resolution and balance the illumination of the image while avoiding many artifacts. PCLSRTM, which relaxes the amplitude-matching criterion similar to and shows less susceptibility to velocity errors than CLSTM, can be expected to be a robust imaging tool for field data.
ACKNOWLEDGMENTS The research is funded by the National Natural Science Foundation of China (grant nos. 41430321 and 41374138) and is supported by Strategic Priority Research Program of the Chinese Academy of Science (grant no. XDB10050300). We would like to thank M. Sacchi and A. Cheng for their helpful comments. We are grateful to A. Malcolm for her insightful suggestions and intensive English corrections. We appreciate L. Duan and two anonymous reviewers for their comprehensive suggestions. We also appreciate the very insightful discussions with Y. Liu and Q. Liu.
APPENDIX A DERIVATION OF THE INITIAL PRESTACK IMAGE FOR PCLSRTM ð0Þ
ð0Þ
Starting from rs ðxÞ ¼ 0 and Ms ½rs ðxÞ ¼ 0, we derive the inið1Þ tial image rs ðxÞ for PCLSRTM under the property of normalized zero-lag crosscorrelation. First, the objective function given in equation 6 is extended as
Eðfrs ðxÞgÞ ¼ R ZZ Ms ½rs ðxÞ · dobs ðxr ;t;xs Þdt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − dxr dxs ; R 2 R dobs ðxr ;t;xs Þdt ðMs ½rs ðxÞ þ εÞ2 dt (A-1)
where ε is a negligibly small constant for stabilization purposes. The corresponding gradient is derived (Plessix, 2006; Routh et al., 2011; Zhang et al., 2015) as
Prestack correlative least-squares RTM
0
which also approximates the initial prestack image for PCLSRTM using the objective function in equation 6.
∂Eðfrs ðxÞgÞ 1 B ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ MTs @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 ffi R ∂rs ðxÞ d dt ðM ½r ðxÞ þ εÞ2 dt
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s
obs
s
1 R M ½r ðxÞ · dobs dt C × R s s ðMs ½rs ðxÞ þ εÞ − dobs ðxr ;t;xs Þ A: ðMs ½rs ðxÞ þ εÞ2 dt (A-2) ð0Þ rs ðxÞ
With state is
¼ 0 and
ð0Þ Ms ½rs ðxÞ
¼ 0, the gradient at the starting
0
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1
∂Eð0Þ 1 B C ¼ −MTs @qffiffiffiffiffiffiffiffiffiffiffi dobs ðxr ;t;xs ÞA: (A-3) R 2 R 2 ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂rs ðxÞ d dt ε dt
APPENDIX B ANALYTICAL STEP LENGTH FOR PCLSRTM ðkÞ
After defining dðxr ; t; xs Þ ¼ Ms ½rs ðxÞ and δdðxr ; t; xs Þ ¼ ðkÞ Ms ½−½∂Eðfrs ðxÞgÞ∕∂rs ðxÞ, the function φðλÞ is transformed as ðkÞ ∂Eðfrs ðxÞgÞ ðkÞ ; φðλÞ ¼ E rs ðxÞ−λ ∂rs ðxÞ R ZZ ðdðxr ;t;xs Þþλδdðxr ;t;xs ÞÞ · dobs ðxr ;t;xs Þdt ffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxr dxs ; ¼− R R 2 ðdðxr ;t;xs Þþλδdðxr ;t;xs ÞÞ2 dt dobs ðxr ;t;xs Þdt
(B-1)
¼ fðdþλδdÞ:
obs
qffiffiffiffiffiffiffiffiffiffiffiffi Because 1∕ ∫ ε2 dt is a constant value independent of time and trace position, equation A-3 can be rewritten as
0
1
Applying the Taylor expansion of λδdðxr ; t; xs Þ to the second order, the data-domain function fðd þ λδdÞ can be expanded as
fðd þ λδdÞ ¼ fðdÞ þ ð∇fðdÞÞT ðλδdÞ 1 þ ðλδdÞT ∇2 fðdÞðλδdÞ þ OððλδdÞ3 Þ; 2 1 ≈ fðdÞ þ λð∇fðdÞÞT δd þ λ2 δdT ∇2 fðdÞδd ; 2
∂Eð0Þ 1 1 C TB ¼ − qffiffiffiffiffiffiffiffiffiffiffi dobs ðxr ;t;xs ÞA: (A-4) R 2 ffi Ms @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 ∂rs ðxÞ ε dt d dt obs
(B-2)
ð1Þ
The initial image rs ðxÞ is then searched in the gradient descent direction with the step length λð0Þ : ð1Þ rs ðxÞ
∂Eð0Þ 1 T ¼ λð0Þ qffiffiffiffiffiffiffiffiffiffiffiffi ¼ −λð0Þ R 2 Ms ∂rs ðxÞ ε dt
0
1
ðkÞ
1 B C × @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 ffi dobs ðxr ; t; xs ÞA; dobs dt
(A-5) ð1Þ Ms ½rs ðxÞ
so that the modeled data using the initial image can cause the value of equation A-1 to achieve its minimum at the current state. The modeled data using the initial image are expanded as
1 1 ð1Þ T dobs ðxr ;t;xs Þ ; Ms ½rs ðxÞ ¼ M s λð0Þ qffiffiffiffiffiffiffiffiffiffiffi R 2 ffi M s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 ε dt dobs dt
1 1 ð0Þ T ¼ λ qRffiffiffiffiffiffiffiffiffiffiffiffi Ms M s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dobs ðxr ;t;xs Þ : R 2 ε2 dt dobs dt (A-6) Note that the value of equation A-1 can be approximated as unchanged data with the factor λð0Þ qffiffiffiffiffiffiffiffiffiffiffiffi by scaling theqmodeled ffiffiffiffiffiffiffiffiffiffiffiffi ½1∕ ∫ ε2 dt. Thus, λð0Þ ½1∕ ∫ ε2 dt can be assigned as 1.0, and the initial prestack image for equation A-1 is defined as
0
where fðd þ λδdÞ is assumed to be parabolic in the step length λ. In equation B-2, the value of λ at the vertex of the parabola can be written as
1
1 B C ð1Þ rs ðxÞ ¼ MTs @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 ffi dobs ðxr ; t; xs ÞA; dobs dt
(A-7)
λ
∂φð0Þ ð∇fðdÞÞT δd ∂λ ðkÞ ¼− T 2 ; i:e:; λ ¼ − ∂2 φð0Þ ; δd ∇ fðdÞδd 2
(B-3)
∂λ
which defines the analytical step length at the kth iteration for PCLSRTM. Applying the chain rule can conveniently demonstrate ∂φð0Þ∕∂λ ¼ ð∇fðdÞÞT δd and ∂2 φð0Þ∕∂λ2 ¼ δdT ∇2 fðdÞδd.
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