摘要
Seismic data typically contain random missing traces because of obstacles and economic restrictions,influencing subsequent processing and interpretation.Seismic data recovery can be expressed as a low-rank matrix approximation problem by assuming a low-rank structure for the complete seismic data in the frequency–space(f–x)domain.The nuclear norm minimization(NNM)(sum of singular values)approach treats singular values equally,yielding a solution deviating from the optimal.Further,the log-sum majorization–minimization(LSMM)approach uses the nonconvex log-sum function as a rank substitution for seismic data interpolation,which is highly accurate but time-consuming.Therefore,this study proposes an efficient nonconvex reconstruction model based on the nonconvex Geman function(the nonconvex Geman low-rank(NCGL)model),involving a tighter approximation of the original rank function.Without introducing additional parameters,the nonconvex problem is solved using the Karush–Kuhn–Tucker condition theory.Experiments using synthetic and field data demonstrate that the proposed NCGL approach achieves a higher signal-to-noise ratio than the singular value thresholding method based on NNM and the projection onto convex sets method based on the data-driven threshold model.The proposed approach achieves higher reconstruction efficiency than the singular value thresholding and LSMM methods.
受采集环境和经济因素的影响,地震数据在空间上往往存在道缺失的现象,严重影响后续资料解释的准确性。缺失的地震道破坏了完整数据的低秩性,因此,地震数据重建问题可以转化为秩最小化问题。核范数最小化(nuclear norm minimization,NNM)是经典的基于低秩约束的地震数据重建方法。但是,NNM是秩最小化的凸松弛,得到的只是原始秩最小化问题的次优解。基于log-sum函数(log-sum majorization minimization,LSMM)的方法使用非凸的log-sum函数代替秩函数用于地震数据重建,精度较高,但时间消耗较大。基于此,本文提出高效的非凸重建模型:基于非凸Geman函数的地震数据重建方法(nonconvex Geman low rank,NCGL),利用更近似秩函数的Geman函数代替核范数。根据Karush–Kuhn–Tucker(KKT)条件理论求解非凸问题,无需引入正则化参数。仿真与真实实验结果表明,非凸NCGL方法重建精度显著高于基于核范数最小化的奇异值阈值方法(singular value thresholding,SVT)和基于数据阈值驱动的凸集投影方法(projection onto convex sets,POCS),且NCGL方法具有较快的收敛速度,重建效率显著高于SVT和LSMM方法。
基金
financially supported by the National Key R&D Program of China(No.2018YFC1503705)
the Science and Technology Research Project of Hubei Provincial Department of Education(No.B2017597)
the Hubei Subsurface Multiscale Imaging Key Laboratory(China University of Geosciences)(No.SMIL-2018-06)
the Fundamental Research Funds for the Central Universities(No.CCNU19TS020).
