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1 Geophysical Journal International Geophys. J. Int. (2015) 202, GJI Seismology doi: /gji/ggv156 Limited-memory BFGS based least-squares pre-stack Kirchhoff depth migration Shaojiang Wu, 1,2 Yibo Wang, 1 Yikang Zheng 1,2 andxuchang 1 1 Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing , China. wangyibo@mail.igcas.ac.cn 2 University of Chinese Academy of Sciences, Beijing , China Accepted 2015 April 8. Received 2015 April 8; in original form 2014 November 17 1 INTRODUCTION Migration is an effective tool in seismic data processing by moving dipping reflection events to their true geological locations or diffracted energy back to the location of the scatter points. A standard seismic migration operator can be regarded as the adjoint of the seismic forward modelling operator (Claerbout 1971). Migration can also be thought as the first iteration of iterative inversion (Lailly 1983), in which the Hessian of the misfit functional is approximated as a diagonal matrix (Beydoun & Mendes 1989). Due to this simplest approximation, the migration image is usually obscured by migration artefacts when the data are incomplete, especially (Nemeth et al. 1999). Least-squares migration (LSM, Tarantola 1984; Nemeth et al. 1999) is proposed as an iterative inversion based on the least-squares algorithm. Compared to the conventional migration, LSM can improve the resolution of the migration image and suppress migration artefacts after a few iterative calculations. In the LSM algorithm, it first calculates data residuals between observed data and demigrated data using the inverted reflectivity model and forms reflectivity gradient by migrating data residuals; then, updates reflectivity model SUMMARY Least-squares migration (LSM) is a linearized inversion technique for subsurface reflectivity estimation. Compared to conventional migration algorithms, it can improve spatial resolution significantly with a few iterative calculations. There are three key steps in LSM, (1) calculate data residuals between observed data and demigrated data using the inverted reflectivity model; (2) migrate data residuals to form reflectivity gradient and (3) update reflectivity model using optimization methods. In order to obtain an accurate and high-resolution inversion result, the good estimation of inverse Hessian matrix plays a crucial role. However, due to the large size of Hessian matrix, the inverse matrix calculation is always a tough task. The limited-memory BFGS (L-BFGS) method can evaluate the Hessian matrix indirectly using a limited amount of computer memory which only maintains a history of the past m gradients (often m < 10). We combine the L-BFGS method with least-squares pre-stack Kirchhoff depth migration. Then, we validate the introduced approach by the 2-D Marmousi synthetic data set and a 2-D marine data set. The results show that the introduced method can effectively obtain reflectivity model and has a faster convergence rate with two comparison gradient methods. It might be significant for general complex subsurface imaging. Key words: Image processing; Numerical solutions; Inverse theory. with optimization methods. The good estimation of inverse Hessian matrix is required to obtain an accurate and high-resolution inversion result. But, due to the large size of Hessian matrix, it is always a tough task to calculate the inverse matrix. The widely used gradient methods such as the steepest descent method and the conjugate gradient method only assume the Hessian matrix as an identity matrix. To aim for more reasonable computational accuracy, the Hessian matrix of second derivatives is evaluated indirectly by a series of gradient values in some quasi-newton methods. In this study, we introduce a quasi-newton method called L-BFGS (Nocedal 1980) to approximate to the inverse Hessian matrix. L-BFGS modifies the BFGS algorithm using a limited amount of computer memory which only maintains a history of the past m gradients (often m < 10). Then, we combine L-BFGS with the least-squares pre-stack Kirchhoff depth migration (LSKDM). This paper first presents the theory of LSKDM and two numerical optimization methods including the L-BFGS method and typical preconditioned gradient methods. Then, we validate the L-BFGS based least-squares pre-stack Kirchhoff depth migration by the 2-D synthetic Marmosi model and a 2-D marine data set. At the end, a short summary is provided. 738 C The Authors Published by Oxford University Press on behalf of The Royal Astronomical Society.

2 2 LEAST-SQUARES KIRCHHOFF PRE-STACK DEPTH MIGRATION Kirchhoff forward modelling in exploration seismology can be formulated as algebraic linear problems, which is represented by a matrix vector multiplication as: d = Lm, (1) where d is the modelled scattered seismic data and m is the true subsurface reflectivity distribution, L is the Kirchhoff forwardmodelling operator (or demigration operator in the LSM algorithm). A standard migrated image is obtained by applying the transpose of the forward-modelling operator (Claerbout 1992): m mig = L T d, (2) where m mig is the migration result. Substituting the expression of d in eq. (1) into eq. (2) yields: m mig = L T Lm, (3) where L T L is Hessian matrix and is also known as the resolution matrix which reflects the amount of blurring the seismic image (Duquet et al. 2000). In the physical, the model resolution matrix is that the standard migrated image is a linear combination of the true reflectivity model. The Kirchhoff forward operator and its corresponding adjoint operator in their explicit discrete form are given as: d (s, g, t) = [Lm] s,g,t, (4) and m (x) = [ L T m ] x, (5) where s, g, x and t are the source position, receiver position, trial image point and time sample, respectively. Due to geometric spreading and uneven illumination, Kirchhoff migration produces some artefacts in the migration image. These migration artefacts can be suppressed by the least-squares migration (Nemeth et al. 1999; Dai et al. 2012). Mathematically, the objective function can be defined as: f (m) = Lm d 2 + λrm 2 = [ L λr ] m [ ] d, (6) 0 2 where R is a spatial regularization operator and λ is the regularization weight parameter which is relative to the data-fitting goal. An optimal m can be sought to minimize the unconstrained objective function. Here, the regularization is used with a linear filter operator R = / x in above equation. The regularization parameter λ is exponentially decreased with the iterations proceed. Therefore, the regularization term dominates at the early iterations while the misfit term is dominant at later iterations (Nemeth et al. 1999). 3 NUMERICAL OPTIMIZATION METHODS The minimum of the misfit function f (m) can be obtained near the initial model m 0. The updated or the target model m can be set as the sum of the starting model m 0 and a perturbation model m : m = m 0 + m. So, the misfit equation can be expressed with the second-order Taylor series expansion (Tarantola 1984; Virieux &Operto2009): Limited-memory BFGS based least-squares 739 f (m 0 + m) = f (m 0 ) + + M M j=1 k=i M j=i f (m 0 ) m j m j 2 f (m 0 ) m j m k m j m k + o 2 (m 0 ). (7) Then, taking the derivative with respect to the model m: f (m) m = f (m M 0) m + 2 f (m 0 ) m j=i j m m j, (8) where the operator 2 f (m 0 ) is the Hessian matrix and f (m 0) m j is the m m gradient. So, the normal function gives the expression: [ ] 2 1 f (m) f (m) m = m 2 m=m0 m m=m 0. (9) There are some factoring methods for the unconstrained problem for solving the normal eq. (9), that is Cholesky factorization, QR factorization, singular-value decomposition (SVD, Nocedal & Wright 2006). The geophysical problems rise to thousands or millions of variables in practice, especially migration and inversion problem, therefore, it may be efficient to use iterative techniques. So, the normal function can be expressed with the iterative style at the kth iteration: m = α k p k = α k [ B 1 k f (m k ) ], (10) where p k is the searching direction, α k is the step length at iterations, the B k is short for 2 f (m).theb m 2 k is simply set as the identity matrix I in the steepest descent method and the conjugate gradient methods, When in the Newton s method, B k is the exact Hessian matrix 2 f (m), m 2 and it is replaced by an approximation for Hessian matrix in some quasi-newton methods such L-BFGS below. The step length α k is chosen to satisfy the Wolfe conditions, which can be measured by the following inequality (Nocedal & Wright 2006): f (m k + α k p k ) f (m k ) + c 1 α k f (m k ) T p k and f (m k + α k p k ) T p k f (m k ) T p c 2. (11) k For the constant parameter c 1 (0, 1) and constant parameterc 2 (c 1, 1), and in practice, the parameters are chosen as c 1 = 10 4 and c 2 = 0.9. In the beginning line search algorithm, the step α 0 = 1is always be used as the initial trial step length. With a sequence of trial step lengths {α i }, the best value will be accepted if it satisfies the above Wolfe conditions. Additionally, an interpolation along the search direction p k may be needed. When the problem is very large, the optimization algorithm needs to be solved efficiently with a tolerable level both for calculation accuracy and the costs of the storage and computation. So, the current developed optimization algorithm methods may be particularly effective for certain problem types. In this section, one gradientbased methods, the nonlinear conjugate gradient method, and one quasi-newton method, the L-BFGS are tested for the geophysical least-squares optimization problem. 3.1 Pre-conditioned conjugate gradient (PCG) method The linear conjugate gradient method is aimed to solve the optimization problem by successively minimizing objective function

3 740 S. Wu et al. Figure 1. (a) Shows Marmousi velocity model contains many complex small faults and structures and (b) shows theoretical reflectivity model calculated from the velocity model by using vertical rays and constant density assumptions. Both (a) and (b) are also used as a reference model and migration image, respectively. Figure 2. Kirchhoff migration result has distorted with unbalanced reflectivity amplitude. Due to the insufficient illuminations, it shows weaken amplitude and the low resolution especially in the bottom region. In the LSM algorithm, we set it as the initial model by multiplying a scaled factor. f (m) along its individual direction at every step within a conjugate set. In generating its set of conjugate vectors, the new conjugate vector p k can be computed only using the last vector p k 1,which is automatically conjugate to all the previous vectors. So, the linear algorithm can be expressed formally as follows: p k = f (x k ) + β k p k 1 (12) where β k is a scalar to make p k and p k 1 conjugate. We choose the first search direction p 0 to be the current direction at the initial point x 0. Aimed for minimizing general nonlinear functions or solving large-scale optimization problems, the nonlinear variants of the conjugate gradient are introduced by Fletcher & Reeves (1964). The nonlinear functions can be extended from the conjugate gradient method by making two simple changes in above formula (Nocedal & Wright 2006). First, the step length α k is computed by a line search with an approximate minimum of the non-linear function f along p k. Secondly, the residual r is the gradient of the nonlinear objective f. There are many choices for the important parameter β k, and the Fletcher Reeves method is used in this study (Fletcher &Reeves1964): β k = f (x k+1) T f (x k+1 ). (13) f (x k ) T f (x k ) The convergence rate depends on the properties of the matrix L, in particular on the condition number, which is the ratio between the largest and smallest eigenvalue. The precondition can accelerate the convergence rates associated with high condition numbers when a matrix is ill-conditioned or its eigenvalues are not clustered. Numerically, a preconditioned system can be obtained by multiplication with the inverse of the matrix C, C 1 Lm = C 1 d. (14) Normally, the matrix C should be easy to invert, and the condition number of the preconditioned matrix C 1 L smaller than the original condition number of L. An approximate choice for C is chosen as a diagonal matrix (Nemeth et al. 1999): 1 C = diag (L T L) + λdiag(r T R). (15) So, the PCG algorithm is described in Algorithm 1 (Hager & Zhang 2006) Algorithm 1 (preconditioned CG) Given preconditioner C; P k = CC T, Replace f (m k ) with C T f (m k ), p k with C 1 p k, β k = f (m k+1) T P k f (m k+1 ) f (m k ) T P k f (m k ) p k = P k f (m k ) + β k p k The preconditioned L-BFGS method The Newton method uses the quadratic direction, which has a fast rate of local convergence. But, the computation of this matrix of the second derivatives Hessian B k usually can be error-prone and expensive, especially for a large-scale optimization problem. Meanwhile, the Hessian matrix B k is not always an asymmetric and positive define matrix in the practice problem, so additional modification and extra steps are needed to satisfy the assumptions. p k = B 1 k f (x k ). (16) Quasi-Newton methods provide an attractive alternative to Newton s method without requiring the computation of the true Hessian matrix B k. The updates at each step take account of changes in the gradient f (x k ), which can provide additional information about the second derivative of f along the search direction. We focus

4 Limited-memory BFGS based least-squares 741 Figure 3. (a) Is the theoretical reflectivity model. The least squares Kirchhoff migration results after 50 iterations are shown in (b) using the SD method, (c) using the CG method and (d) using L-BFGS method. Compared with conventional Kirchhoff migration result (as shown in Fig. 2), the LSM can compensate uneven illumination and provide more balanced reflectivity amplitude in migration images. With the increasing of iteration numbers, CG and L-BFGS show improving resolutions indicated by rectangular windows a1 d1 and a2 d2. mainly on the BFGS updating formula, as a rank-two update formulae method, which can guarantee that the updated matrix maintains positive definiteness. The BFGS is named for its discoverers Broyden, Fletcher, Goldfarb and Shanno, can be expressed as: p k = H k f (x k ) (17) and H k+1 = ( I ρ k s k (y k ) T ) H k ( I ρk y k (s k ) T ) + ρ k s k (s k ) T H k+1 s k = x k+1 x k, y k = f (x k+1 ) f (x k ),ρ k = 1 yk T s, (18) k where H k is the approximation of the inverse of the Hessian matrix B 1 k. The updated approximation H k+1 will be positive definite whenever that H k is positive definite. Therefore, the initial approximation H 0 usually is simply set as an identity matrix, which can guarantee that the updated matrix maintains positive definiteness. Normally, the BFGS method updates and stores the entire approximate inverse Hessian at per iteration. So, the computational cost and the memory usage are too expensive when the matrices and the number of variables are large. To circumvent this problem, the L-BFSG method (Nocedal 1980; Liu & Nocedal 1989)is introduced to compact and store the approximations of Hessian matrices. The L-BFGS method constructs the Hessian approximation using the curvature information from only the most past m iterations, and the curvature information from earlier iterations is discarded which is less likely to be relevant to the current behaviour of the Hessian. H k+1 = k i=k m+1 + k n=k m+2 V T i H 0 ( k k V i i=k m+1 i=n V T i ρ n 1 s n 1 s T n 1 ) k V i + ρ k s k s T k. (19) There are many approaches to using a history of updates to form the current direction vector. The two loop recursion is recommended for that the multiplication by the initial matrix H 0 is isolated from the rest of the computations, which is easy and free to be chosen freely between iterations. The two-loop recursion scheme is described in Algorithm 2 (Nocedal & Wright 2006). As a preconditioned L-BFGS, the initial estimation H 0 is computed with a preconditioned gradient P 0 f (m 0 ) at the first iteration (Nocedal & Wright 2006). i=n

5 742 S. Wu et al. Figure 5. Panels (a) (d) magnified views of rectangular window (a2 d2) in Fig. 5, respectively. Compared to the true reflectivity model (a), (d) has improving resolution and small faults and structures have been recognized clearly where the arrows point to locations than (b) and (c). Figure 4. Panels (a) (d) magnified the views of rectangular window (a1 d1) in Fig. 5, respectively. Compared to the true reflectivity model (a), (b) show poorer migration image, (d) shows better resolution where artefactartefacts have been suppressed significantly in the region indicated by the arrows than (c).

6 Limited-memory BFGS based least-squares 743 Figure 6. Normalized data residual plotted against iteration number. Both L-BFGS method and CG method obtain almost the same convergence after 35 iterations and show higher convergence precision than SD method with the same iteration. But, L-BFGS provides a faster convergence rate than CG method in the same accuracy of solution case. Algorithm 2 (preconditioned L-BFGS two-loop scheme) Given preconditioner P 0 ; Replace f (m 0 ) with P 0 f (m 0 ) at the first iteration q = f (m k ) ; for i = k 1, k 2,...,k m α i = ρ i si T q q = q α i y i r = H 0 q for i = k m, k m + 1,...,k 1 β = ρ i yi T r r = r s i (α i β) H k f (m k ) = r; p k = k f (m k ). 4 COST AND CONVERGENCE ANALYSIS The matrix-free regime within an optimization algorithm is defined that the approach does not require the explicit storage or factorization of any matrix. Since we rely on an iterative method instead of factoring methods and the explicit problem formulation is avoided, the used PCG is matrix-free (Fountoulakis et al. 2014). Some fur- Figure 8. The velocity model mainly contains a shallow water layer (the blue part with a velocity 1500 m s 1 ) and some weak reflectors near the seafloor (the mainly white part). The model is estimated from the observed data, which contains velocity errors. ther matrix-free GPU preconditioned CG method is implemented with recalculate in every local grid cell instead of storing explicitly, which can reduce access to global memory for kernels were ported to the GPU but increase data throughput and some logical operations (Müller et al. 2013). Therefore, the GPU implemented CG does not well for the migration problem in this study, because the additional logical operations outweight the saving storage requirements. The L-BFGS method approximates the inverse Hessian matrix only using some previous vectors without forming explicitly the Hessian operator, which is a matrix-free solver. In PCG, both the measurement matrix L and the preconditioning matrix C need to be stored explicitly in the algorithm for solving the preconditioning equation. The vector p k is stored, which requires n locations per iteration. As the preconditioned CG show in Algorithm 1, there require 9n multiplications for updating the searching direction with a given preconditioner, including 6n multiplications for the multiplication β k. In general quasi-newton method like the BFGS, the approximate matrix of the inverse Hessian should be constructed sequentially, which usually is symmetric. Therefore, it requires n (n + 1) /2 Figure 7. Reflectivity curve plotted against depth on the offset 3000 m of the least squares Kirchhoff migration result after 50 iterations. All three methods perform well to estimate the reflectivity value in shallow region. But, L-BFGS method is closer to the true reflectivity value especially in bottom region compared to CG method and SD method.

7 744 S. Wu et al. Figure 9. (a) A raw CSG of the marine gather and its frequency of traces (b). Panel (c) is the same CSG with applying the steps and its frequency of traces (d). The free-surface-related multiples (black arrows) in the recorded data must been attenuated. Then, some alias frequency (green arrows) like strong imprint of multiple reflections at high frequency and some far offset seismic data (green boxes) should be muted. The processed data is with a higher signal-to-noise ratio. Figure 10. The Kirchhoff migration result has unbalanced reflectivity amplitude and the unfocused reflectors especially in the deep part. locations per iteration, in which the number n is the model parameters (Byrd et al. 1995). In the L-BFGS, some vector pairs { s k, y k } from the most past m iterations is used and stored, which needs most 2mn locations [during its first m 1 iterations, only about 2 (m 1) n]. As the two-loop recursion scheme showed in Algorithm 2, there require 4 (n + 1) m multiplications for updating the searching direction, including 2 nm multiplications in every loop (Nocedal 1980; Zheng et al. 2013), m multiplications for the multiplication H 0 q. In theory, the preconditioned nonlinear conjugate gradient method and the L-BFGS method can reduce O(n 2 ) locations of quasi-newton methods like BFGS to O (n), which are meaningful for some realistic large-size problems with n m. For updating the searching direction, about O (n) multiplications are required with both methods. Next, there is additional forward modelling calculation and some other multiplications in every line search algorithm. So, the numbers of tries before selecting the best steps length should not be negligible for considerations of the computation, which depend on the condition numbers of the objective matrix, the choice for the iterative optimization methods and the choice for the provide proper constant parameters c 1 and c 2 in Wolfe conditions. In practice, the first trial step for the estimation α k can also use the selected step length α k+1 at the previous iteration, which can satisfies the Wolfe conditions with few additional linesearch iterations most of the time. Theoretically, the conjugate gradient method modifies the steepest descent direction with a conjugate direction. Moreover, the precondition implemented can make the distribution of the eigenvalues of the coefficient matrix more favourable and accelerate the convergence improve the convergence of the method significantly. The L-BFGS method construct the Hessian approximation only using the most recent iterations, which are dominant relevant to the actual behaviour of the Hessian. The limited-memory quasi-newton

8 Limited-memory BFGS based least-squares 745 Figure 11. The least squares Kirchhoff migration results after 10 iterations are shown in (a) using the preconditioned CG method, (b) using the preconditioned L-BFGS method. Compared with conventional Kirchhoff migration result (as shown in Fig. 10), the LSM can provide more balanced reflectivity amplitude and improving resolution reflectors in migration images, especially in the rectangular windows a1, b1. method provides an attractive alternative to Newton s method and may still attain a superlinear rate of convergence, typically quadratic (Nocedal & Wright 2006). Under the reasonable assumptions, the L-BFGS has the potential for converging more rapidly than the preconditioned CG for using the improving information related to Hessian (Nocedal 1980). 5 EXAMPLES 5.1 Synthetic data In the first example, the above two numerical optimization methods based on the least squares Kirchhoff depth migration are tested on the synthetic Marmousi data set. The Marmousi velocity model has 1401 gridpoints in the horizontal direction and 341 gridpoints in the vertical direction, both directions with a grid interval of 10 m. We set 320 shots and 400 receivers per shot both deployed on the surface. The trace interval and shot interval are 10 and 20 m, respectively. The recording length for each trace is 5 s, with a temporal sampling interval of s. A Ricker wavelet with a 30 Hz peak frequency is used as the source wavelet. Fig. 1(a) shows the true velocity model contains many complex small faults and structures, which is used for modelling and migration or demigration in LSM algorithm. The reflectivity model in Fig. 3(a) calculated from the velocity model by using vertical rays and constant density assumptions. Fig. 2 shows the conventional Kirchhoff migration, which has distorted with unbalanced reflectivity amplitude due to geometric spreading and insufficient illuminations. The Kirchhoff migration can be set as the initial model by multiplying a scaled factor in the LSM algorithm. The least squares Kirchhoff migration results using the steepest descent method, the conjugate gradient method and L-BFGS method after 50 iterations are shown in Figs 3(b) (d), respectively. Compared with conventional Kirchhoff migration result (as shown in Fig. 2), the LSM using all three methods can compensate uneven illumination and provide more balanced reflectivity amplitude in migration images. With the increasing of iteration numbers, L-BFGS method show improving resolutions and clearer small faults and structures than the pre-conditioned CG method and the SD method. To demonstrate the advantages of the L-BFGS, we magnify the rectangular windows of Figs 4 and 5 contains zoomed views of rectangular windows 1 and 2, respectively in Fig. 3. Figs4(b) and 5(b) using SD method show worst migration image with the SD method; Figs 4(d) and 5(d) using L-BFGS method show that faults and structures are more clearly delineated where the arrows point to the locations and have a better migration resolution than Figs 4(b), (c) and 5(b), (c) using the SD method and the CG method, respectively. Fig. 6 is the reflectivity depth curve using the three methods on the offset 3000 m of the least squares Kirchhoff depth migration result after 50 iterations. In the comparison, all three methods perform well to estimate the reflectivity value in shallow region. But, L-BFGS is closer to the true reflectivity curve especially in bottom compared to CG method and SD method. Thus, it gives a better resolution in the migration images. Fig. 7 show normalized data residual curve against iteration number. Both L-BFGS method and CG method obtain almost the same convergence after 35 iterations and show higher convergence precision than SD method with the same iteration. But, L-BFGS provides faster convergence rate than CG method in the same accuracy of solution case. It is reasonable as that L-BFGS is a second-order optimization algorithm which evaluates the Hessian matrix indirectly using a history of the past m gradients, while both the SD method and the CG method only assume the Hessian matrix as an identity matrix. 5.2 Field data In this example, the above two numerical optimization methods based on the least squares Kirchhoff depth migration are tested on a 2-D marine data set from the South China Sea. The estimated velocity model has 700 gridpoints in the horizontal direction and 450 gridpoints in the vertical direction, both directions with a grid interval of 12.5 m. There are 240 shots with a shot interval of 50 m, and each shot is recorded with 200 receivers and a 25 m interval. The nearest offset is 250 m. The recording length for each trace is 10 s, with a temporal sampling interval of s. The velocity model used for LSM is showed in Fig. 8, which contains a shallow water layer and some weak reflectors near the seafloor. The estimation of the velocity model from the observed seismic data is often ambiguous and inaccurate by the current methods, and contains some velocity errors, which can mainly lead to an incoherent, defocused image. More complicated than the synthetic data example, as showed in Figs 9(a) and (b), some processing steps should be applied to the raw marine data prior to migration. First, the free-surface-related multiples in the recorded data have been attenuated, and the remained part contains mainly the primaries satisfy the assumptions.

9 746 S. Wu et al. Figure 13. Normalized data residual plotted against iteration number. The L-BFGS provides a faster convergence rate than CG method in the same accuracy of solution case. However, the decrease in residual is approximately per cent for the both methods and underperforms the synthetic data set examples, which may due to velocity errors in the estimated background model and strong attenuation and anisotropy of some soft sediments below the sea level for the wave propagation. The KM result and the 10th iteration least squares Kirchhoff migration results by two optimization schemes are presented in Figs 10 and 11, respectively. Compared to standard KM results, the LSM provides a better quality image with fewer artefacts, especially in the deep region and higher resolution for the shallow reflectors. Comparing these LSM images, we observe that the two optimization methods perform similarly, but there still some apparent differences in the areas enclosed by the boxes. Zoomed views of the areas in Figs 10 and 11(a) and (b) are shown in Figs 12(a) and (c), respectively. The corresponding reflectors with the L-BFGS method are more continuous and better focused. Meanwhile, some noise also be attenuated. The convergence profile for the marine case is showed in Fig. 13 with a normalized data residual curve. The L-BFGS method offers a higher convergence rate than the non-linear conjugate gradient method. But, the decrease in residual is approximately per cent within dozen times of iteration (it is actually meaningless with too many iteration for real data) for the both methods, which underperforms the synthetic data set example. In practice, the velocity errors in the estimated background model can lead to unexact paths of wave propagating, and imprecise computation of seismic traveltime in Kirchhoff migration. Further, some soft shale sediments below the sea level can lead to strong attenuation and anisotropy of the wave propagation, which is also a challenge for seismic imaging. Figure 12. Panels (b) and (c) are the zoomed views of the areas enclosed by yellow boxes in Figs 11(a) and (b), respectively. (a) is the corresponding parts in the Fig. 10. Compared these details parts, the reflectors with the L-BFGS is more continuous and better focused (black arrows), also, some noise also be attenuated (green circle). Then, a bandpass filter is applied for mainly removing the noise due to the limits of the actual acquisition, and some alias frequency like strong imprint of multiple reflections at high frequency. This bandwidth is chosen based on the frequency content of the data in which most of the signal is concentrated between 10 and 55 Hz. With the Kirchhoff migration, some far offset seismic data also need be cut. The processed common shot gather (CSG) and its frequency of traces are showed in Figs 9(c) and (d), and the signal-to-noise ratio (SNR) is effectively improved. 6 CONCLUSION The numerical experiments validate that L-BFGS based the least squares Kirchhoff depth migration is a computationally efficient method for suppressing migration artefacts and modifying spatial resolution than standard migration. The introduced approach has three major advantages: (1) compared to the conventional migration, it can compensate for uneven illumination and amplitude distortion after a few iterative calculations iterations, (2) compared to the SD method and the CG method, it can effectively obtain higher resolution reflectivity models and improve the ability for recognizing small faults and structures. Because of the above-mentioned advantages, the introduced approach might be a significant tool for general complex subsurface imaging.

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