结合稀疏变换的稀疏约束反演一次波估计研究
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摘要
在基于波动方程的反馈迭代法去除多次波领域里，SRME方法在最近20年间已经发展成为工业界较为成熟的去除表面多次波的有效的方法。最近几年发展起来的EPSI（稀疏反演一次波估计）方法是继SRME方法以后发展出来的一种基于大尺度反演来直接对一次波估计的方法。该方法避免了SRME方法的预测减去过程，进一步提高了一次波的计算精度。原始的EPSI方法是对一次波脉冲响应进行迭代更新，它是基于L0范数约束的稀疏反演问题，采用了传统的最速下降算法每次对一次波脉冲响应的梯度进行更新时加上时窗和反演参数上的设定等约束。在进行迭代更新过程中，时窗包含有一次波信息，而不能包含多次波信息，反演参数也是要经过多次的调试。所以原始的EPSI问题存在着多种条件限制和稳定性等问题。
为了避免原始的EPSI方法中的基于L0范数约束的稀疏反演求解过程中所带来的问题，在足够稀疏的条件下，L0范数最优化问题可以转化为L1范数约束的凸优化问题，并且采用SPGL1（L1谱梯度投影）算法进行求解，由于它是一种稳健的最优化反演方法，即使在没有局部最小解的情况下仍然能够保持稳定，最后可以收敛于全局解，在反演一次波脉冲响应的过程由于采用了叠前多炮数据，所以引入了3D曲波变换作为稀疏约束条件，一次波脉冲响应在曲波域中表现的更为稀疏，反演得到的一次波通过曲波反变换结果与震源子波褶积得到。因此通过将原始的EPSI改进，不仅提高了一次波反演估计的计算精度，而且避免了在反演过程中的多种条件约束。基于L1范数约束的稀疏反演一次波估计方法不仅能够提高一次波估计的精度而且能够相对提高了计算效率。通过数值模拟数据和实际数据的测试体现了该方法的有效性和稳定性。
原始的EPSI方法通过一个大尺度的反演过程来实现一次波的估计，所以计算耗时较多，相当于100多次的SRME方法的运算量。同时由于是反演得到的一次波脉冲响应，对于深层的反射信息，不管是原始的EPSI还是经过改进的L1范数约束的稀疏约束反演一次波估计方法，反演的一次波脉冲响应结果都不够理想，很多深层的有效信息不够完整，会对后续的偏移成像和地震解释带来了一定的影响，为了在反演一次波脉冲响应效果和运算量之间得到一个更好的折中，我们可以采用SRME方法与结合曲波变换的基于L1范数约束的EPSI方法联合进行多次波衰减，通过理论数据的实验，得到了较为理想的效果，一次波估计精度不仅得到了提高，深层反射也得到了较大程度的改进，而且运算量也大大降低，得到了较为理想的预期。
结合稀疏变换的基于L1范数约束的稀疏约束反演方法不仅适合于海上常规拖缆采集技术，针对于海底电缆（OBC）数据采集方式，我们通过改进反演公式，将常规拖缆数据信息和海底电缆数据信息同时应用在其中，同样可以采用该方法进行海底电缆数据的稀疏约束反演一次波估计。为了提高稀疏变换速度，同时采用了二维曲波变换与一维小波变换结合的稀疏变换方法，采用了双凸优化方式，即反演一次波脉冲响应的过程中采用目标函数是一个凸函数以及应用的约束集合同样也是一个凸函数。对一次波脉冲响应运用SPGL1（L1谱梯度投影）算法求解，同时引入了pareto曲线作为收敛准则，提高了收敛速度，采用了最小平方QR分解算法进行子波估计，通过交替优化迭代过程，最后达到更加精确的反演一次波脉冲响应，进而得到一次波反射，同时得到震源子波，在OBC数据中应用了改进型的稀疏反演一次波估计。该方法适用于OBC数据中的各个分量。在模拟算例中我们采用了水检数据，即压力分量来进行一次波估计。通过理论数据得到验证，效果良好。
对于被动源地震数据，运用常规的地震干涉理论中的互相关算法得到的虚拟炮记录中通常含有表面相关多次波。然而通过被动源数据稀疏反演一次波估计(EPSI)方法，可以求得只含有一次波，不含表面相关多次波的虚拟炮记录。由于被动源数据中的时间窗口选择和反演参数设定更加困难，我们改进了原始的被动源数据稀疏反演一次波估计的求解方法，将基于脉冲震源的被动源稀疏反演一次波估计求解问题转化为L1范数约束的最优化反演问题，避免了在传统的稀疏反演一次波估计过程中用时窗防止反演陷入局部最优化的情况。在这种脉冲震源类型的被动源数据中采用最优化的求解过程中，又结合了二维曲波变换和一维小波变换联合方法，采用基于L1范数约束的SPGL1算法与基于L2范数能量约束的最小二乘QR分解算法分别进行反演，使被动源数据在稀疏变换域中反演得到，从而使求得的结果直接可以获得一次波反射信息，效果更加优越，成像质量得到了进一步改善。
对于上述结合稀疏变换的基于L1范数约束的稀疏反演一次波估计方法通过对数据矩阵公式的推导与改进。应用拓展到了常规的海上拖缆数据、海底电缆数据（OBC）以及被动源地震数据中。通过模型验证和实际数据验证，都达到了一定的应用效果。从最早提出的SRME方法，继而到传统的EPSI方法，最后我们通过进一步改进EPSI方法。不仅避免了SRME方法中的预测减去的过程，而且优化了反演条件，提高了反演速度，为多次波去除领域及其EPSI方法今后的大规模工业化应用提供了参考。
Feedback iteration method based on the wave equation in the filed of multipleattenuation, the SRME(Surface-related multiple elimination) has developed into theindustry is relatively mature efficient and effective approach to reduce the surfacemultiples in the recent20years, EPSI (estimation of primaries by sparse inversion)was developed following the SRME method based on a large scale inversion anddirectly to a wavelet estimation method in recent years, which not only avoids theSRME method prediction and subtraction the process but further improve thecalculation precision of primaries. Original EPSI method is the gradient of primariesimpulse response of the update, it is based on the sparse inversion problem of L0norm constraint, using the conventional steepest descent algorithm, each timeprimaries impulse response gradient for update with time window and set ofconstraints on the inversion parameters. When making gradient update process, thewindow contains primary information and cannot contain multiple imformation withinversion parameters should be debug many times. So the original EPSI haverestrictions in a variety of conditions and stability problems.
In order to avoid the original EPSI method of sparse inversion based on L0normconstraint in the process of solving the problem. In the condition of sparse enough, L0norm optimization problem can be converted into the L1norm constrained convexoptimization problem, we adopt a algorithm that called SPGL1(Spectrum ProjectedGradient L1) is used, because it is a kind of robust optimization inversion method,even there is not local minimum solution can converge to global solutions, in theprocess of inversion primaries impulse response since the prestack data, so the3Dcurvelet transform is introduced as a sparse constraint conditions, primaries impulse response in the curvelet domain performance more sparse, the inversion of primariesimpulse response result with the source wavelet convolution. So the original EPSIimprovement, not only primary inversion raised the calculation precision ofestimation, but avoids restrictions the variety of conditions in the inversion process.Sparse inversion based on L1norm constraint primary estimation method can not onlyimprove the accuracy of primary estimation but also improved the relative calculationefficiency. Through the numerical simulation data and real data test showed theeffectiveness of the proposed method and stability.
Original EPSI through a large scale inversion process to implement primaryestimation, so calculation time consuming, equivalent to more than100times theSRME method of computation. At the same time as a result of the inversion is to getprimary impulse response, for deep reflection information, whether it's original EPSIor improved L1norm constraint sparse constraint inversion of primary estimationmethod, primary impulse response of the inversion results are not ideal, a lot ofeffective information is not complete, deep for subsequent migration imaging andbrought certain influence seismic interpretation, in order to respond in primaryimpulse response inversion, multiples attenuation effect and get a better compromisebetween computational complexity, we can use SRME and combining curvelettransform EPSI method based on L1norm constraint joint multiple attenuation,through the theoretical data of the experiment, the ideal result is obtained through, notonly primary estimation precision is improved, the deep reflection have beenimproved largely, and the computation is greatly reduced, the ideal expectations.
Combined with sparse transformation based on L1norm constraint on the sparseconstraint inversion method is not only suitable for sea of conventional towingacquisition technology, in view of the ocean bottom cable (OBC) data acquisitionmethods, we improve the inversion formula, the conventional towing data informationand submarine cable data information application in it at the same time, also can usethis method to primary sparse constraint inversion of submarine cable data estimation,in order to increase the speed of sparse transform, and applied to two-dimensional curvelet transform and one dimensional wavelet transform in combination withmathematical sparse transform method, adopted double convex optimization method,the inversion of primary impulse response in the process of using the objectivefunction is a collection of convex function and application of the constraint is also aconvex function, in order to improve the speed and reduce memory footprint sparsetransformation, using the two-dimensional curvelet transform combined with wavelettransform, the method of impulse response of primary using SPGL1, at the same timeintroduced the pareto curve as a convergence criterion, improves the convergencespeed, using the least-squares QR decomposition algorithm for wavelet estimation,through alternating optimization iteration process, and finally achieve a more accurateinversion primary impluse response, and then get primary information, at the sametime to get the source wavelet, in OBC data of the application of an improved sparseinversion primary is estimated at a time. This method is suitable for the OBC eachcomponent in the data. We adopted in the simulation example hydrophone data,namely pressure component for another primary of estimates. It have been the goodeffect verified by theoretical data.
For passive source seismic data, using conventional seismic interference theory ofcross-correlation algorithm of virtual shot records contain of surface multiples. Yet bya passive source data estimation by primaries sparse inversion, contains only primary,can be obtained without surface multiples of the virtual shot record. Due to timewindow selection and inversion parameters of the passive source data set moredifficult, we improved the original passive data sparse inversion solution of primaryof estimates,it will be based on primary impulse response source of passive sparseinversion estimation to solve the problem is transformed into the L1norm constraintoptimization inversion problem, avoided the traditional sparse inversion of primaryestimation process with time window to prevent trapped in local optimum in inversion.In the passive source data of the impulse source type uses the solution of theoptimization process, and combined with two-dimensional curvelet transform and onedimensional wavelet transform method, the SPGL1algorithm based on L1norm constraint and energy of the L2norm constraint inversion with least-squares QRdecomposition algorithm respectively, the passive data in the sparse transform domaininversion, so that we can be obtained results can directly obtain primary reflectioninformation, the effect is more superior, imaging quality has been further improved.
For the combination of sparse transformation based on L1norm constraintestimation of primaries by sparse inversion based on the data matrix formula isderived and improvement of the application to the Marine towing data of conventionalocean bottom cable (OBC), and passive seismic data, through the model and realdata,it have reached a certain application effect. From the earliest SRME method isput forward, and then to traditional EPSI method, finally, we further improve EPSImethod, not only avoid the SRME method of prediction and subtraction, the processof inversion condition was optimized at the same time, improve the inversion speed,multiple removing areas and EPSI method for the future of the large-scale industrialapplication provides reference.
为了避免原始的EPSI方法中的基于L0范数约束的稀疏反演求解过程中所带来的问题，在足够稀疏的条件下，L0范数最优化问题可以转化为L1范数约束的凸优化问题，并且采用SPGL1（L1谱梯度投影）算法进行求解，由于它是一种稳健的最优化反演方法，即使在没有局部最小解的情况下仍然能够保持稳定，最后可以收敛于全局解，在反演一次波脉冲响应的过程由于采用了叠前多炮数据，所以引入了3D曲波变换作为稀疏约束条件，一次波脉冲响应在曲波域中表现的更为稀疏，反演得到的一次波通过曲波反变换结果与震源子波褶积得到。因此通过将原始的EPSI改进，不仅提高了一次波反演估计的计算精度，而且避免了在反演过程中的多种条件约束。基于L1范数约束的稀疏反演一次波估计方法不仅能够提高一次波估计的精度而且能够相对提高了计算效率。通过数值模拟数据和实际数据的测试体现了该方法的有效性和稳定性。
原始的EPSI方法通过一个大尺度的反演过程来实现一次波的估计，所以计算耗时较多，相当于100多次的SRME方法的运算量。同时由于是反演得到的一次波脉冲响应，对于深层的反射信息，不管是原始的EPSI还是经过改进的L1范数约束的稀疏约束反演一次波估计方法，反演的一次波脉冲响应结果都不够理想，很多深层的有效信息不够完整，会对后续的偏移成像和地震解释带来了一定的影响，为了在反演一次波脉冲响应效果和运算量之间得到一个更好的折中，我们可以采用SRME方法与结合曲波变换的基于L1范数约束的EPSI方法联合进行多次波衰减，通过理论数据的实验，得到了较为理想的效果，一次波估计精度不仅得到了提高，深层反射也得到了较大程度的改进，而且运算量也大大降低，得到了较为理想的预期。
结合稀疏变换的基于L1范数约束的稀疏约束反演方法不仅适合于海上常规拖缆采集技术，针对于海底电缆（OBC）数据采集方式，我们通过改进反演公式，将常规拖缆数据信息和海底电缆数据信息同时应用在其中，同样可以采用该方法进行海底电缆数据的稀疏约束反演一次波估计。为了提高稀疏变换速度，同时采用了二维曲波变换与一维小波变换结合的稀疏变换方法，采用了双凸优化方式，即反演一次波脉冲响应的过程中采用目标函数是一个凸函数以及应用的约束集合同样也是一个凸函数。对一次波脉冲响应运用SPGL1（L1谱梯度投影）算法求解，同时引入了pareto曲线作为收敛准则，提高了收敛速度，采用了最小平方QR分解算法进行子波估计，通过交替优化迭代过程，最后达到更加精确的反演一次波脉冲响应，进而得到一次波反射，同时得到震源子波，在OBC数据中应用了改进型的稀疏反演一次波估计。该方法适用于OBC数据中的各个分量。在模拟算例中我们采用了水检数据，即压力分量来进行一次波估计。通过理论数据得到验证，效果良好。
对于被动源地震数据，运用常规的地震干涉理论中的互相关算法得到的虚拟炮记录中通常含有表面相关多次波。然而通过被动源数据稀疏反演一次波估计(EPSI)方法，可以求得只含有一次波，不含表面相关多次波的虚拟炮记录。由于被动源数据中的时间窗口选择和反演参数设定更加困难，我们改进了原始的被动源数据稀疏反演一次波估计的求解方法，将基于脉冲震源的被动源稀疏反演一次波估计求解问题转化为L1范数约束的最优化反演问题，避免了在传统的稀疏反演一次波估计过程中用时窗防止反演陷入局部最优化的情况。在这种脉冲震源类型的被动源数据中采用最优化的求解过程中，又结合了二维曲波变换和一维小波变换联合方法，采用基于L1范数约束的SPGL1算法与基于L2范数能量约束的最小二乘QR分解算法分别进行反演，使被动源数据在稀疏变换域中反演得到，从而使求得的结果直接可以获得一次波反射信息，效果更加优越，成像质量得到了进一步改善。
对于上述结合稀疏变换的基于L1范数约束的稀疏反演一次波估计方法通过对数据矩阵公式的推导与改进。应用拓展到了常规的海上拖缆数据、海底电缆数据（OBC）以及被动源地震数据中。通过模型验证和实际数据验证，都达到了一定的应用效果。从最早提出的SRME方法，继而到传统的EPSI方法，最后我们通过进一步改进EPSI方法。不仅避免了SRME方法中的预测减去的过程，而且优化了反演条件，提高了反演速度，为多次波去除领域及其EPSI方法今后的大规模工业化应用提供了参考。
Feedback iteration method based on the wave equation in the filed of multipleattenuation, the SRME(Surface-related multiple elimination) has developed into theindustry is relatively mature efficient and effective approach to reduce the surfacemultiples in the recent20years, EPSI (estimation of primaries by sparse inversion)was developed following the SRME method based on a large scale inversion anddirectly to a wavelet estimation method in recent years, which not only avoids theSRME method prediction and subtraction the process but further improve thecalculation precision of primaries. Original EPSI method is the gradient of primariesimpulse response of the update, it is based on the sparse inversion problem of L0norm constraint, using the conventional steepest descent algorithm, each timeprimaries impulse response gradient for update with time window and set ofconstraints on the inversion parameters. When making gradient update process, thewindow contains primary information and cannot contain multiple imformation withinversion parameters should be debug many times. So the original EPSI haverestrictions in a variety of conditions and stability problems.
In order to avoid the original EPSI method of sparse inversion based on L0normconstraint in the process of solving the problem. In the condition of sparse enough, L0norm optimization problem can be converted into the L1norm constrained convexoptimization problem, we adopt a algorithm that called SPGL1(Spectrum ProjectedGradient L1) is used, because it is a kind of robust optimization inversion method,even there is not local minimum solution can converge to global solutions, in theprocess of inversion primaries impulse response since the prestack data, so the3Dcurvelet transform is introduced as a sparse constraint conditions, primaries impulse response in the curvelet domain performance more sparse, the inversion of primariesimpulse response result with the source wavelet convolution. So the original EPSIimprovement, not only primary inversion raised the calculation precision ofestimation, but avoids restrictions the variety of conditions in the inversion process.Sparse inversion based on L1norm constraint primary estimation method can not onlyimprove the accuracy of primary estimation but also improved the relative calculationefficiency. Through the numerical simulation data and real data test showed theeffectiveness of the proposed method and stability.
Original EPSI through a large scale inversion process to implement primaryestimation, so calculation time consuming, equivalent to more than100times theSRME method of computation. At the same time as a result of the inversion is to getprimary impulse response, for deep reflection information, whether it's original EPSIor improved L1norm constraint sparse constraint inversion of primary estimationmethod, primary impulse response of the inversion results are not ideal, a lot ofeffective information is not complete, deep for subsequent migration imaging andbrought certain influence seismic interpretation, in order to respond in primaryimpulse response inversion, multiples attenuation effect and get a better compromisebetween computational complexity, we can use SRME and combining curvelettransform EPSI method based on L1norm constraint joint multiple attenuation,through the theoretical data of the experiment, the ideal result is obtained through, notonly primary estimation precision is improved, the deep reflection have beenimproved largely, and the computation is greatly reduced, the ideal expectations.
Combined with sparse transformation based on L1norm constraint on the sparseconstraint inversion method is not only suitable for sea of conventional towingacquisition technology, in view of the ocean bottom cable (OBC) data acquisitionmethods, we improve the inversion formula, the conventional towing data informationand submarine cable data information application in it at the same time, also can usethis method to primary sparse constraint inversion of submarine cable data estimation,in order to increase the speed of sparse transform, and applied to two-dimensional curvelet transform and one dimensional wavelet transform in combination withmathematical sparse transform method, adopted double convex optimization method,the inversion of primary impulse response in the process of using the objectivefunction is a collection of convex function and application of the constraint is also aconvex function, in order to improve the speed and reduce memory footprint sparsetransformation, using the two-dimensional curvelet transform combined with wavelettransform, the method of impulse response of primary using SPGL1, at the same timeintroduced the pareto curve as a convergence criterion, improves the convergencespeed, using the least-squares QR decomposition algorithm for wavelet estimation,through alternating optimization iteration process, and finally achieve a more accurateinversion primary impluse response, and then get primary information, at the sametime to get the source wavelet, in OBC data of the application of an improved sparseinversion primary is estimated at a time. This method is suitable for the OBC eachcomponent in the data. We adopted in the simulation example hydrophone data,namely pressure component for another primary of estimates. It have been the goodeffect verified by theoretical data.
For passive source seismic data, using conventional seismic interference theory ofcross-correlation algorithm of virtual shot records contain of surface multiples. Yet bya passive source data estimation by primaries sparse inversion, contains only primary,can be obtained without surface multiples of the virtual shot record. Due to timewindow selection and inversion parameters of the passive source data set moredifficult, we improved the original passive data sparse inversion solution of primaryof estimates,it will be based on primary impulse response source of passive sparseinversion estimation to solve the problem is transformed into the L1norm constraintoptimization inversion problem, avoided the traditional sparse inversion of primaryestimation process with time window to prevent trapped in local optimum in inversion.In the passive source data of the impulse source type uses the solution of theoptimization process, and combined with two-dimensional curvelet transform and onedimensional wavelet transform method, the SPGL1algorithm based on L1norm constraint and energy of the L2norm constraint inversion with least-squares QRdecomposition algorithm respectively, the passive data in the sparse transform domaininversion, so that we can be obtained results can directly obtain primary reflectioninformation, the effect is more superior, imaging quality has been further improved.
For the combination of sparse transformation based on L1norm constraintestimation of primaries by sparse inversion based on the data matrix formula isderived and improvement of the application to the Marine towing data of conventionalocean bottom cable (OBC), and passive seismic data, through the model and realdata,it have reached a certain application effect. From the earliest SRME method isput forward, and then to traditional EPSI method, finally, we further improve EPSImethod, not only avoid the SRME method of prediction and subtraction, the processof inversion condition was optimized at the same time, improve the inversion speed,multiple removing areas and EPSI method for the future of the large-scale industrialapplication provides reference.
引文
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