Abstract：The reconstruction of missing seismic data traces plays an important role in seismic data processing. However, the majority of the existing reconstruction algorithms, held back by their low convergence speed and high computational cost, can barely meet the needs of mass data processing. This paper proposed a rapid reconstruction method combining the accelerated linearized Bregman method (ALBM) with a threshold iteration method, namely the iterative shrinkage-thresholding al-gorithm (ISTA). Besides, multi-scale and multi-directional curvelet transform was adopted as the sparse basis. ALBM converges rapidly in the early stage of iteration as it can obtain more effective signals from unthresholded curvelet coefficients. Nevertheless, its reconstruction accuracy is weakened by the increasing noise brought by the unthresholded curvelet coefficients in the later stage. In contrast, ISTA needs to threshold the curvelet coefficients all along. Although its convergence speed is slow in the early stage of iteration as most of the effective coefficients are filtered out, its reconstruction accuracy is high in the later stage owing to the restoration of weak effective signals. To maximize the advantages of the two algorithms, this paper presented two weighting parameter formulas (linear and exponential) in the range of 0~1. In this way, ALBM and ISTA were effectively combined linearly to ensure that ALBM played a major role in the early stage of iteration while ISTA dominated the later stage of iteration and thereby to enable this joint algorithm to iterate rapidly and accurately. In the combination process, a soft thresholding formula was adopted, and an exponential thresholding parameter formula was introduced. The theoretical simulation results demonstrate that compared with ALBM, ISTA, and traditional joint methods, the proposed accelerated joint method delivers fast calculation and an evident reconstruction effect.
唐刚. 基于压缩感知和稀疏表示的地震数据重建与去噪[D]. 北京：清华大学，2010.TANG Gang. Seismic Data Reconstruction and Denoising Based on Compressive Sensing and Sparse Representation[D]. Tsinghua University，Beijing，2010.
[2]
张华，陈小宏. 地震数据重建理论与方法[M]. 北京：科学出版社，2017.ZHANG Hua，CHEN Xiaohong. Theory and Method of Seismic Data Reconstruction[M]. Science Press，Beijing，2017.
[3]
霍志周，熊登，张剑锋. 地震数据重建方法综述[J]. 地球物理学进展，2013，28(4)：1749-1756.HUO Zhizhou，XIONG Deng，ZHANG Jianfeng. The overview of seismic data reconstruction methods[J]. Progress in Geophysics，2013，28(4)：1749-1756.
NAGHIZADEH M，SACCHI M D. Beyond alias hie-rarchical scale curvelet interpolation of regularly and irregularly sampled seismic data[J]. Geophysics，2010，75(6)：WB189-WB202.
[6]
MA J W. Three-dimensional irregular seismic data Reconstruction via low-rank matrix completion[J]. Geophysics，2013，78(5)：V181-V192.
[7]
GAO J J，SACCHI M D，CHEN X H. A fast reduced-rank interpolation method for prestack seismic volumes that depend on four spatial dimensions[J]. Geophysics，2013，78(1)：V21-V30.
[8]
薛亚茹，唐欢欢，陈小宏，等. 高阶高分辨率Radon变换地震数据重建方法[J]. 石油地球物理勘探，2014，49(1)：95-100，131.XUE Yaru，TANG Huanhuan，CHEN Xiaohong，et al. Seismic data Reconstruction based on high order high resolution Radon transform[J]. Oil Geophysical Prospecting，2014，49(1)：95-100，131.
[9]
唐欢欢，毛伟建，詹毅. 3D高阶抛物Radon变换在不规则地震数据保幅重建中的应用[J]. 地球物理学报，2020，63(9)：3452-3464.TANG Huanhuan，MAO Weijian，ZHAN Yi. Reconstruction of 3D irregular seismic data with amplitude preserved by high-order parabolic Radon transform[J]. Chinese Journal of Geophysics，2020，63(9)：3452-3464.
[10]
罗腾. 基于压缩感知理论的地震数据重构方法研究[D]. 吉林长春：吉林大学，2015.LUO Teng. Study on the Reconstruction of Seismic Data Based on Compressive Sensing Theory[D]. Jilin University，Changchun，Jilin，2015.
[11]
温睿，刘国昌，冉扬. 压缩感知地震数据重建中的三个关键因素分析[J]. 石油地球物理勘探，2018，53(4)：682-693.WEN Rui，LIU Guochang，RAN Yang. Three key factors in seismic data reconstruction based on compressive sensing[J]. Oil Geophysical Prospecting，2018，53(4)：682-693.
韩良良. 基于Curvelet变换的地震数据重建方法研究[D]. 山东青岛：中国石油大学(华东)，2018.HAN Liangliang. The Study on Seismic Data Reconstruction Method Based on Curvelet Transform[D]. China University of Petroleum (East China)，Qing-dao，Shandong，2018.
[14]
王本锋，陆文凯，陈小宏，等. 基于3D Curvelet变换的频率域高效地震数据插值方法研究[J]. 石油物探，2018，57(1)：65-71.WANG Benfeng，LU Wenkai，CHEN Xiaohong，et al. Efficient seismic data interpolation using three-dimensional Curvelet transform in the frequency domain[J]. Geophysical Prospecting for Petroleum，2018，57(1)：65-71.
[15]
张华，陈小宏. 基于jitter采样和曲波变换的三维地震数据重建[J]. 地球物理学报，2013，56(5)：1637-1649.ZHANG Hua，CHEN Xiaohong. Seismic data reconstruction based on jittered sampling and curvelet transform[J]. Chinese Journal of Geophysics，2013，56(5)：1637-1649.
[16]
刘财，李鹏，刘洋，等. 基于seislet变换的反假频迭代数据插值方法[J]. 地球物理学报，2013，56(5)：1619-1627.LIU Cai，LI Peng，LIU Yang，et al. Iterative data interpolation beyond aliasing using seislet transform[J]. Chinese Journal of Geophysics，2013，56(5)：1619-1627.
[17]
温睿. 基于Seislet变换的地震数据重建方法研究[D]. 北京：中国石油大学(北京)，2018.WEN Rui. Study on Seismic Data Reconstruction Methods Based on Seislet Transform[D]. China University of Petroleum (Beijing)，Beijing，2018.
[18]
ZHANG H，CHEN X H，WU X M. Seismic data reconstruction based on CS and Fourier theory[J]. Applied Geophysics，2013，10(2)：170-180.
[19]
梁东辉. 基于傅里叶变换的地震数据规则化和插值[D]. 浙江杭州：浙江大学，2015.LIANG Donghui. Seismic Data Normalization and Interpolation Based on Fourier Transform[D]. Zhejiang University，Hangzhou，Zhejiang，2015.
[20]
HENNENFENT G，HERRMANN F J. Seismic denoising with nonuniformly sampled curvelets[J]. Computing in Science & Engineering，2006，8(3)：16-25.
[21]
孔丽云，于四伟，程琳，等. 压缩感知技术在地震数据重建中的应用[J]. 地震学报，2012，34(5)：659-666.KONG Liyun，YU Siwei，CHENG Lin，et al. Application of compressive sensing to seismic data reconstruction[J]. Acta Seismologica Sinica，2012，34(5)：659-666.
[22]
ZHANG H，CHEN X H，ZHANG L Y. 3D simultaneous seismic data reconstruction and noise suppression based on the curvelet transform[J]. Applied Geo-physics，2017，14(1)：87-95.
[23]
张岩，任伟建，唐国维. 基于曲波变换的地震数据压缩感知重构算法[J]. 吉林大学学报(信息科学版)，2015，33(5)：570-577.ZHANG Yan，REN Weijian，TANG Guowei. Algorithm of compressed sensing reconstruction of seismic data based on curvelet transform[J]. Journal of Jilin University (Information Science Edition)，2015，33(5)：570-577.
[24]
OSHER S，BURGER M，GOLDFARB D，et al. An iterative regularization method for total variation-based image restoration[J]. Multiscale Modeling & Simulation，2005，4(2)：460-489.
[25]
YIN W T，OSHER S，GOLDFARB D，et al. Bregman iterative algorithms for l1-minimization，with applications to compressed sensing[J]. SIAM Journal on Imaging Sciences，2008，1(1)：142-168.
[26]
刘洋，张鹏，刘财，等. 地震数据Bregman整形迭代插值方法[J]. 地球物理学报，2018，61(4)：1400-1412.LIU Yang，ZHANG Peng，LIU Cai，et al. Seismic data interpolation based on Bregman shaping iteration[J]. Chinese Journal of Geophysics，2018，61(4)：1400-1412.
[27]
郭萌，张会星，刘明珠. 基于双重Bregman迭代的地震数据重构与去噪[J]. 石油物探，2020，59(5)：804-814.GUO Meng，ZHANG Huixing，LIU Mingzhu. Seismic data reconstruction and denoising based on dual Bregman iteration[J]. Geophysical Prospecting for Petroleum，2020，59(5)：804-814.
[28]
CAI J F，OSHER S，SHEN Z W，et al. Linearized Bregman iterations for compressed sensing[J]. Mathe-matics of Computation，2009，78(267)：1515-1536.
[29]
CAI J F，OSHER S，SHEN Z. Convergence of the linearized Bregman iterations for l1-norm minimization[J]. Mathematics of Computation，2009，78(268)：2127-2136.
[30]
CANDES E J，ROMBERG J K. Signal recovery from random projections[C]. Electronic Imaging 2005，2005：76-86.
[31]
张华，杨会，刁塑，等. 基于指数阈值迭代法的高精度地震数据重建[J]. 东华理工大学学报(自然科学版)，2017，40(3)：253-260.ZHANG Hua，YANG Hui，DIAO Su，et al. High precision reconstruction of seismic data based on exponential threshold iteration method[J]. Journal of East China Institute of Technology (Natural Science Edition)，2017，40(3)：253-260.
[32]
HERRMANN F J，HENNENFENT G. Non-parametric seismic data recovery with curvelet frames[J]. Geophysical Journal International，2008，173(1)：233-248.
[33]
白兰淑，刘伊克，卢回忆，等.基于压缩感知的Curvelet域联合迭代地震数据重建[J]. 地球物理学报，2014，57(9)：2937-2945.BAI Lanshu，LIU Yike，LU Huiyi，et al. Curvelet-domain joint iterative seismic data reconstruction based on compressed sensing[J]. Chinese Journal of Geophysics，2014，57(9)：2937-2945.
[34]
BAI L S，LU H Y，LIU Y K，et al. A fast joint seismic data reconstruction by sparsity-promoting inversion[J]. Geophysical Prospecting，2017，65(4)：926-940.
[35]
HUANG B，MA S Q，GOLDFARB D. Accelerated linearized bregman method[J]. Journal of Scientific Computing，2013，54(2)：428-453.
[36]
ABMA R，KABIR N. 3D interpolation of irregular data with a POCS algorithm[J]. Geophysics，2006，71(6)：E91-E97.
