An Effective Correction Interpolation Method for Controllable Marine Sources Based on Frequency-Wavenumber Domain

Abstract

This invention provides an effective correction interpolation method for marine mobile controllable seismic sources based on the frequency-wavenumber domain. It primarily addresses the Doppler effect and insufficient spatial sampling rate issues in the correction process of existing marine mobile controllable seismic source data. The method constructs an objective function comprising a sampling operator, a masking operator, a convolution correction operator, and a dictionary mapping seismic data to basis functions. Within a sparse inversion framework, a fast threshold iteration algorithm is used to interpolate and correct marine mobile controllable seismic source data of different types of frequency sweep signals. The proposed method can compensate for the spatial aliasing problem during Doppler effect correction, ensuring the accuracy of subsequent structural imaging. Furthermore, the masking operator obtained based on the principal dip angle energy scanning method effectively solves the aliasing problem caused by regular interpolation in the reconstruction stage. Finally, the inversion process utilizes the conjugate symmetry of real signals in the Fourier transform process, significantly reducing the workload of the inversion process.

Description

Technical Field

[0001] This invention relates to the field of seismic signal processing technology, specifically to an effective correction interpolation method for marine controllable seismic sources based on the frequency-wavenumber domain, which simultaneously removes the Doppler effect and spatial aliasing of correction results from marine mobile controllable seismic source data. Background Technology

[0002] Compared to traditional airgun seismic sources, marine controlled seismic sources offer numerous advantages in seismic exploration, including environmental friendliness, good waveform control, and repeatability. However, they also introduce a series of new challenges, a prime example being the Doppler effect. Classical Doppler theory indicates that during the excitation of a frequency sweep signal by a marine controlled seismic source, Doppler distortion occurs during the movement of both the source and the detector, affecting the reliability of subsequent seismic data processing and reducing the matching degree of seismic data. Hampson points out that Doppler distortion caused by the movement of marine controlled seismic sources can be corrected using deconvolution techniques. However, when the spatial sampling rate at the shot point is insufficient, a simple deconvolution process can cause spatial aliasing, resulting in corrected data that fails to meet data quality requirements. Antoine Guitton et al.A

[0003] The deconvolution-interpolation approach with sparse inversion tomitigate the Doppler effect in marine vibrators data [C] / / First International Meeting for Applied Geoscience & Energy. Society of Exploration Geophysicists, 2021:2555-2559 proposes a temporal convolution operator to implement the above process in a sparse Rado domain. However, this method faces the problems of huge computational cost of temporal convolution and interference from spurious frequencies similar to the true spectrum caused by regular interpolation in the reconstruction stage. This often results in low efficiency and difficulty in convergence of the inversion results. Summary of the Invention

[0004] To address the efficiency issues of traditional time-domain deconvolution correction processes and the problem of spurious frequencies resembling the true spectrum caused by the lack of rules during reconstruction, this invention proposes an effective correction interpolation method to eliminate the Doppler effect in marine controllable seismic source data, simultaneously interpolating and correcting marine mobile controllable seismic source data.

[0005] The present invention adopts the following technical solution:

[0006] An effective correction interpolation method for marine controllable seismic sources based on the frequency-wavenumber domain, the method comprising:

[0007] The sampling operator Φ, the masking operator W, the convolution correction operator C in the FK domain, and the dictionary F that maps seismic data to basis functions are combined to form a linear composite operator A, i.e., A = Φ * F. T *W*C constructs the corresponding objective function based on the inversion relation Y = AX, where Y is the inversion relation of the data to be reconstructed and X is the correction data:

[0008]

[0009] In the formula, m0 is the controllable source data to be reconstructed and corrected, A is the linear composite operator, F is the forward Fourier transform operator, and σ is the threshold. The objective function is inverted to obtain the final reconstruction and correction result.

[0010] Furthermore, the sampling operator Φ is based on the ideal shot spacing that does not produce spatial aliasing in the shot sampling interval of the marine controllable source data, thus obtaining the corresponding ideal shot spatial sampling rate.

[0011] Furthermore, the convolution correction operator C in the FK domain is an arbitrary type of swept frequency signal for obtaining data from the controlled ocean seismic source, and the corresponding Doppler effect convolution correction operator is obtained according to the ship speed and the sampling interval of the shot point, and the Doppler effect convolution correction operator is converted into the convolution correction operator C in the FK domain.

[0012] Furthermore, the specific calculation process of the convolution correction operator C in the FK domain is as follows: the motion process of the ocean-controlled seismic source is equivalent to adding an extra dimension to the stationary controlled seismic source signal, resulting in the following effective expression for the controlled seismic source signal:

[0013] S eff (t, s)=S(t)σ(s0+u s t)

[0014] In the formula, s0 is the position of the fixed source, S(t) is the sweep frequency signal, and u s Let S be the velocity of the seismic source. When the seismic source is stationary, S eff (t, s) is a one-dimensional signal and there is no Doppler effect;

[0015] Based on the effective expression of the controllable source signal, the Doppler data generated by the source motion is transformed into a 2D convolution of the design time t and the source position s0. The convolution correction operator for the source motion is expressed as:

[0016] D=S(t)σ(s0-u s t)

[0017] The convolution correction operator for source motion is transformed by FK transformation to obtain the convolution correction operator C in the FK domain.

[0018] Furthermore, the calculation process of the masking operator W specifically includes:

[0019] Based on the original shot point sampling interval obtained from marine mobile controllable source data, an ideal spatial shot point sampling interval is set to obtain a regular missing data matrix. This regular missing data matrix is ​​formed by interpolating zero vectors from empty seismic gathers of the original shot point sampling intervals according to the ideal spatial shot point sampling interval. The seismic matrix is ​​then subjected to a principal dip energy scan in the normalized frequency-wavenumber domain to obtain the scan function M(p), and the dip values ​​p1, p2, p3...p corresponding to the main energy peaks in the energy spectrum are identified. n and indices L1, L2, L3...L n ,in:

[0020]

[0021] Where M(p) is the peak scan result, ω n Here, p represents the frequency, p is the tilt slope, and k is the wave number.

[0022] For a linear in-phase axis whose true spectrum often exhibits a "linear" distribution, for several major peaks obtained from the scan, initialize a zero matrix H with the same dimension as the data spectrum. Then, assign the obtained tilt angle values ​​p1, p2, p3...p... n The energy of the reflected wave from the ray is accurately distributed into the zero matrix H:

[0023]

[0024] Where n = 1, 2, ..., N ω j = 1, 2, ..., n, the resulting zero matrix H is combined with the one-dimensional square impulse function B(1, L). b Perform convolution, widen the ray at each angle along the wavenumber axis, to obtain the masking operator W:

[0025]

[0026] Among them, L b It is the length of function B along the wavenumber axis, representing the width of the ray scan at each angle, and function B has no value along the frequency axis.

[0027] For nonlinear in-phase axes whose true spectra are often distributed in a "planar" manner, what we obtain are the p values ​​corresponding to the left and right boundaries of the true spectrum distribution. min p max The corresponding index is L min L maxThen, the angle masking operator W, based on the identified principal tilt angle, is designed as follows:

[0028]

[0029] Where n = 1, 2, ..., N ω j = L min -L a ,L min -L a +1,...,L min +L b Set parameter L a ,L b Increase the slope scan range.

[0030] Compared with traditional Doppler correction methods, the advantages of this invention are as follows:

[0031] This invention provides an effective correction interpolation method for eliminating Doppler effects in marine controlled-source seismic data based on the frequency-wavenumber (FK) domain. This method utilizes the concept of sparse inversion, reconstructing missing shot points needed to mitigate spatial aliasing during the deconvolution stage through sparse FK transform. This ensures the reliability and matching accuracy of subsequent seismic data processing. Compared to traditional deconvolution methods for correcting Doppler effects, the method proposed in this invention considers the spatial aliasing problem caused by coarse shot point sampling intervals in the controlled-source data to be corrected during the deconvolution stage. Through a sparse inversion framework, it simultaneously interpolates and corrects the controlled-source data to be reconstructed and corrected, generating accurate seismic gathers. Compared to traditional deconvolution interpolation methods that use time-domain convolution correction operators to eliminate Doppler effects in marine controlled-source data, the method introduced in this paper uses a hidden operator based on tilt scanning, which effectively solves the interference of spurious frequencies similar to the true spectrum caused by the lack of rules in the reconstruction stage, enabling stable convergence of the inversion process and improving the accuracy of the inversion. Furthermore, by utilizing the conjugate symmetry property of the Fourier transform of real signals, the method proposed in this invention can reduce the workload by more than 50%, greatly improving the efficiency of the inversion.

[0032] To overcome the problems of traditional methods, this approach is carried out within the framework of sparse inversion. It deconvolves ocean-moving controllable source data with insufficient spatial sampling rate at the shot points and reconstructs the trajectory with the optimal (ideal) shot point spacing to compensate for the spatial aliasing phenomenon in the correction process caused by insufficient spatial sampling rate at the shot points.

[0033] The inversion objective function consists of a sampling operator, a masking operator, a convolution correction operator in the FK domain, and a dictionary mapping seismic data to basis functions. The introduction of a masking operator based on the principal dip energy scanning method effectively solves the problem of aliasing in the spectrum of the data to be reconstructed during interpolation, where the spectrum resembles the true spectrum. Simultaneously, the conjugate symmetry of the Fourier transform based on real signals significantly reduces the workload of the inversion process. Attached Figure Description

[0034] Figure 1 The flowchart shows an effective correction interpolation method for eliminating the Doppler effect in marine controllable seismic source data based on the frequency-wavenumber (FK) domain, according to the present invention.

[0035] Figure 2 To correct the spatial aliasing caused by the Doppler effect using the conventional deconvolution method at a coarse sampling rate (24m) of the shot points; (a) stratigraphic model parameters; (b) Doppler distortion data of the original shot point sampling interval; (c) seismic records obtained by correcting the Doppler shift using the deconvolution method; (d) for Figure 2 (b) and Figure 2 The residual panel obtained by subtracting the data in (c);

[0036] Figure 3 This is due to the spurious frequency phenomenon introduced by the missing rules in the interpolation data to be reconstructed; (a) the seismic record to be reconstructed and corrected; (b) the normalized frequency-wavenumber spectrum corresponding to (a); (c) the ideal seismic record; (d) is... Figure 3 (c) The corresponding normalized frequency-wavenumber spectrum;

[0037] Figure 4 To determine the concealment operator using the principal tilt scan method; (a) the M(p) function and (b) the corresponding calculated selection function W;

[0038] Figure 5 The residual records are shown for the inversion results and comparison of the inversion effects. (a) Seismic records obtained by inversion using the method in this paper. (b) Residual panel of ideal seismic records and inversion results. Detailed Implementation

[0039] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0040] Please refer to Figure 1 This invention provides an effective correction interpolation method for marine controllable seismic sources based on the frequency-wavenumber domain (FK), comprising:

[0041] The sampling operator Φ, the masking operator W, the convolution correction operator C in the FK domain, and the dictionary F that maps seismic data to basis functions are combined to form a linear composite operator A, i.e., A = Φ * F. T*W*C, based on the inversion relation Y=AX, where Y is the inversion relation of the data to be reconstructed and X is the correction data, construct the corresponding objective function:

[0042]

[0043] In the formula, m0 is the controllable source data to be reconstructed and corrected, A is the linear composite operator, F is the forward Fourier transform operator, and σ is the threshold. The objective function is inverted to obtain the final reconstruction and correction result.

[0044] For the objective function, the fast thresholding method is chosen for inversion. The fast thresholding method is a commonly used algorithm in the field of compressed sensing for solving underdetermined linear equations. It accelerates convergence by linearly combining the results of the first two iterations as the initial value for the next iteration and discarding redundant information during the iteration process.

[0045] This method simultaneously performs deconvolution correction on the moving, controllable source data and reconstructs the optimal spatial sampling interval of the shot points, thereby mitigating spatial artifacts caused by insufficient shot point sampling rate during the deconvolution correction stage. A fast threshold iteration algorithm is used to solve the above inversion process. The fast threshold iteration method is a commonly used algorithm in compressed sensing for solving underdetermined linear equation systems. By linearly combining the results of the first two iterations as the initial value for the next iteration, redundant information during the iteration process is discarded, thus accelerating the convergence speed.

[0046] The sampling operator Φ is based on the ideal shot spacing that does not produce spatial aliasing in the shot sampling interval of the marine controllable source data, and the corresponding ideal shot spatial sampling rate is obtained.

[0047] The system acquires sweep signals of any type excited by a controllable source method, while simultaneously determining the source velocity and the spatial sampling interval of the shot points. From these parameters, the corresponding convolution correction operator C (1D or 2D) can be obtained.

[0048] The specific calculation process of the convolution correction operator C in the FK domain is as follows: The motion process of the ocean-controlled seismic source is equivalent to adding an extra dimension to the stationary controlled seismic source signal, resulting in the following effective expression for the controlled seismic source signal:

[0049] S eff (t, s)=S(t)σ(s0+u s t)

[0050] In the formula, s0 is the position of the fixed source, S(t) is the sweep frequency signal, and u s Let S be the velocity of the seismic source. When the seismic source is stationary, S eff (t, s) is a one-dimensional signal and there is no Doppler effect;

[0051] Based on the effective expression of the controllable source signal, the Doppler data generated by the source motion is transformed into a 2D convolution of the design time t and the source position s0. The convolution correction operator for the source motion is expressed as:

[0052] D=S(t)σ(s0-u s t)

[0053] The convolution correction operator for source motion is transformed by FK transformation to obtain the convolution correction operator C in the FK domain.

[0054] Determine the optimal reconstructed shot sampling interval (the ideal spatial shot sampling interval). The selection of the optimal reconstructed shot sampling interval should avoid spatial aliasing problems that occur during the deconvolution stage, and maximize the maximum sampling interval to ensure that spatial artifacts are resolved while reducing the inversion workload.

[0055] The raw shot point sampling intervals obtained from ocean mobile controllable source data were analyzed. An ideal spatial shot point sampling interval was set to obtain a regularly missing data matrix. Then, in the normalized frequency-wavenumber (FK) domain, a principal dip angle energy scan was performed using a formula to obtain the scan function M(P), and the dip angle values ​​p1, p2, p3...p corresponding to the main energy peaks in the energy spectrum were identified. n And therefore L1, L2, L3...L n .

[0056]

[0057] For linear in-phase axes, their true spectrum often exhibits a "linear" distribution. For the main peaks obtained from the above scan, initialize a zero matrix H with the same dimension as the data spectrum. Then, assign the obtained p1, p2, p3...p... n The energy of the reflected wave from the ray is accurately distributed into matrix H:

[0058]

[0059] Where n = 1, 2, ..., N ω j = 1, 2, ..., n. Finally, considering that the energy on both sides of the rays at each adjacent angle should be the energy of the in-phase axis, in order to pick up more complete in-phase axis information, the obtained H is compared with the one-dimensional square pulse function B(1, L). b Perform convolution to broaden the ray at each angle along the wavenumber axis, ultimately yielding the masking operator W:

[0060]

[0061] Among them, L bIt is the length of function B along the wavenumber axis, representing the width of the ray scan at each angle, and function B has no value along the frequency axis.

[0062] For nonlinear in-phase axes, their true spectrum often exhibits a "planar" distribution. Therefore, the obtained spectrum is the p corresponding to the left and right boundaries of the true spectrum. min p max The corresponding index is L min L max Then, the principal tilt angle based on identification is used to design the angle masking operator:

[0063]

[0064] Where n = 1, 2, ..., N ω j = L min -L a ,L min -L a +1,...,L min +L b Considering that extreme value boundaries often contain true spectral information on both sides, the parameter L is set... a ,L b Increase the slope scanning range appropriately.

[0065] For data with severe spatial aliasing, a time-segmentation method is used to divide the data into several parts. For each seismic data segment, a principal dip energy scan method is used to obtain corresponding masking operators to increase the sparsity of the wavenumber domain and mitigate aliasing. Furthermore, for linear and nonlinear phase axes, different calculation methods are used to obtain masking operators W for different data types, which are then incorporated into the inversion objective function to remove spurious frequency interference caused by missing rules in the interpolation stage during the inversion process.

[0066] Example:

[0067] This embodiment utilizes sparse FK transform to reconstruct the shot points required to reduce aliasing during the deconvolution stage in the FK domain. By constructing a linear composite operator including a sampling operator Φ, a masking operator W, a convolution correction operator C in the FK domain, and a dictionary F that maps seismic data to basis functions, the above inversion problem is transformed into a basis pursuit denoising problem. The sparse inversion algorithm is then used to solve the objective function to generate accurate seismic gathers that are free from Doppler effects and spatial aliasing interference, ensuring the reliability and matching degree of subsequent seismic data processing.

[0068] The finite difference method (second-order time, fourth-order space) for acoustic waves was used to simulate ocean-moving controllable source data. To compare the spatial aliasing caused by the coarse spatial sampling rate during deconvolution correction, a relatively coarse spatial sampling interval for shot points was selected in the forward modeling process.

[0069] refer to Figure 2 Forward modeling was used to generate common receiver point domain data (CRGs) with a source velocity of -10 m / s and a shot point sampling interval of 24 m. The Doppler distortion data with the coarse sampling interval was corrected using deconvolution, and the corresponding correction results were obtained.

[0070] To highlight the impact of shot point spatial sampling on the correction results, the above correction results were subtracted from the original source static data to obtain residual records. The residual records show that significant spatial aliasing occurred during the deconvolution correction stage due to the coarse spatial sampling rate of the shot points.

[0071] refer to Figure 3 The ideal reconstructed shot sampling interval of 12m was determined to obtain the sampling operator Φ. The corresponding controllable source data to be reconstructed and interpolated was generated, and the difference between the FK spectrum of the controllable source data to be reconstructed and interpolated and the ideal data (source stationary, ideal shot sampling interval) was compared.

[0072] To compare the interference of false frequencies caused by the lack of rules, the ideal data (source stationary, ideal shot sampling interval) and the interpolation data to be reconstructed were subjected to FK transformation respectively, and their coordinate axis frequencies and wavenumbers were normalized to generate the corresponding normalized frequency-wavenumber (FK) spectrum.

[0073] Comparing the normalized FK spectrum reveals a significant spurious frequency problem caused by the lack of rules. This makes it difficult for the subsequent inversion process to converge, affecting the accuracy of the inversion.

[0074] refer to Figure 4 The normalized frequency-wavenumber spectrum of the generated data to be reconstructed is applied using an energy scanning method based on the principal tilt angle. The scanning function M(p) displays the energy values ​​corresponding to different tilt angles. We identify the true spectral boundaries of the data to be corrected and reconstructed based on the extreme points of M(p). Spatial aliasing similar to the true spectrum is generated to eliminate the missing rules mentioned above.

[0075] An energy scan based on the principal tilt angle was performed on the generated normalized frequency-wavenumber spectrum. Since the in-phase axis is nonlinear, the dispersed energy exhibits a "planar" distribution. Therefore, from the scan results, we identified the main energy peaks representing the boundary L of the true spectrum. mon ,L max ;

[0076] Considering that extreme boundary values ​​often have true spectral information on both sides, the parameter L is set... a ,L b Increase the slope scan range appropriately. After determining the boundary parameters, the slope scan range L can be obtained. min -L a Lmax +L b The spectrum within is the true spectrum, from which the masking operator W can be obtained.

[0077] The above operators are combined to form a linear composite operator, and an objective function based on a sparse inversion framework is constructed. refer to Figure 5 The above objective function is inverted using a fast threshold iteration algorithm to obtain accurate reconstruction and correction results. To compare the accuracy of the inversion results, the ideal data (stationary source, ideal shot sampling interval) and the inversion results are subtracted to obtain residual records. It can be seen that the residual records show that aliasing and phase errors are almost completely attenuated, which indicates the effectiveness of the method of the present invention.

[0078] This invention analyzes raw marine controlled seismic source data (data to be reconstructed and corrected) and the source-excited swept frequency signal S(t) to obtain a tilted two-dimensional convolution correction operator C with a swept frequency signal (2D) in the FK domain. A sampling operator is generated based on an ideal shot point spatial sampling rate, serving as a mapping operator between the acquired data and the reconstruction network. The marine mobile controlled seismic source data to be reconstructed and corrected is scanned at the principal dip angle in the FK domain to obtain a masking operator used to suppress spatial aliasing during interpolation. An objective function is constructed, including the sampling operator, masking operator, convolution correction operator, and a dictionary mapping seismic data to basis functions. Within the framework of sparse inversion, a fast threshold iteration algorithm is used to interpolate and correct marine mobile controlled seismic source data (data to be reconstructed and corrected) with different types of swept frequency signals. The method proposed in this invention can compensate for the spatial aliasing problem during Doppler effect correction, ensuring the accuracy of subsequent structural imaging. Furthermore, the masking operator obtained based on the principal dip angle energy scanning method can effectively solve the aliasing problem caused by regular interpolation in the reconstruction stage. Finally, the inversion process utilizes the conjugate symmetry of the real signal in the Fourier transform process, which greatly reduces the workload of the inversion process.

[0079] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. An effective correction interpolation method for marine controllable seismic sources based on the frequency-wavenumber domain, characterized in that, The method includes: sampling operator The linear composite operator consists of the masking operator W, the convolution correction operator C in the FK domain, and the dictionary F that maps seismic data to basis functions. ,Right now According to the inversion relationship Where Y is the inversion relationship of the data to be reconstructed and X is the correction data, the corresponding objective function is constructed as follows: ， In the formula, in the formula, For controllable seismic source data to be reconstructed and corrected, For linear composite operators, It is the forward Fourier transform operator. Using the threshold value, the objective function is inverted to obtain the final reconstruction and correction result; The convolution correction operator C in the FK domain is an arbitrary type of swept frequency signal used to acquire data from controlled ocean seismic sources. Based on ship speed and shot sampling interval, a corresponding Doppler effect convolution correction operator is obtained, which is then converted into a convolution correction operator in the FK domain. ; The calculation process of the masking operator W specifically includes: Based on the original shot point sampling intervals obtained from marine mobile controllable source data and the ideal spatial shot point sampling intervals, a regular missing data matrix is ​​obtained. This regular missing data matrix is ​​formed by interpolating zero vectors from empty seismic gathers of the original shot point sampling intervals according to the ideal spatial shot point sampling intervals. The seismic matrix is ​​then subjected to a principal dip energy scan in the normalized frequency-wavenumber domain to obtain the scan function M(p), and the dip angle values ​​corresponding to the main energy peaks in the energy spectrum are identified. , , ... and index , , ... ,in: ， in, For peak scan results, Represents frequency, The slope is the angle of inclination. It is the wave number; For a linear in-phase axis whose true spectrum often exhibits a "linear" distribution, for several major peaks obtained from the scan, initialize a zero matrix H with the same dimension as the data spectrum, and then use the obtained tilt angle values... , , ... The energy of the reflected wave from the ray is accurately distributed into the zero matrix H: =1， Where n=1,2,..., j=1,2,...,n, the resulting zero matrix H is compared with the one-dimensional square impulse function. Perform convolution, widening each angle ray along the wavenumber axis to obtain the masking operator W: , in, It is the length of function B along the wavenumber axis, representing the width of the ray scan at each angle, and function B has no value along the frequency axis. For nonlinear in-phase axes whose true spectra are often distributed in a "planar" manner, what we obtain are the left and right boundaries of the true spectrum distribution. The corresponding index is Then, the angle masking operator W, based on the identified principal tilt angle, is designed as follows: ， Where n=1,2,..., j= , ,..., Set parameters , Increase the slope scan range.

2. The effective correction interpolation method for marine controllable seismic sources based on the frequency-wavenumber domain as described in claim 1, characterized in that, The sampling operator To determine the ideal spatial sampling rate of the shot points, the ideal shot point spatial sampling interval is determined by using the shot spacing that does not produce spatial aliasing in the shot point sampling interval of marine controllable source data.

3. The effective correction interpolation method for marine controllable seismic sources based on the frequency-wavenumber domain as described in claim 1, characterized in that, Convolution correction operator in the FK domain The specific calculation process is as follows: The motion process of a controlled ocean seismic source is equivalent to adding an extra dimension to a stationary controlled seismic source signal, resulting in the following effective expression for the controlled seismic source signal: ， In the formula, Given a fixed source position, S(t) is a frequency sweep signal. Let be the velocity of the seismic source when the seismic source is stationary. It is a one-dimensional signal and there is no Doppler effect; Based on the effective expression of the controllable source signal, the Doppler data generated by the source motion is transformed into the relationship between the design time t and the source location. The 2D convolution, the convolution correction operator for source motion is expressed as: ， The convolution correction operator for source motion is transformed by FK transformation to obtain the convolution correction operator C in the FK domain.

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