Natural Seismic Signal Reconstruction Based On Low Rank Completion

Imaging for underground structures and inversion to target reservoirs place high demand on the regularity and integrity of natural seismic data.Due to the influence of geographical environment and collection cost,the natural seismic data collected in the field often present irregular and incomplete distribution,which will directly affect the processing and accurate interpretation of subsequent natural seismic data.Therefore,high-precision reconstruction of natural seismic data is of great practical significance.The complete 2D and 3D natural seismic data have a low rank structure after Hankel pre-transformation,while the random missing of the seismic trace increases the rank of the pre-transform matrix or tensor.Therefore,the low rank matrix or tensor completion method can be used.Natural seismic data is reconstructed.This paper is based on three different pre-transformation methods: Hankel transform,block Hankel transform and Hankel tensor transform.For 2D and 3D natural seismic data,the missing data is performed by low rank matrix completion and low rank tensor reconstruction respectively.For the reconstruction of 2D natural seismic data,this paper applies a orthogonal rank-one matrix pursuit(OR1MP)algorithm based on the reduced rank completion theory to the 2D natural earthquake in San Jacinto fault zone,California.The Hankel matrix pre-transformation is performed on the frequency slice after the Fourier transform of the seismic data to obtain the Hankel matrix with low rank structure.The traditional singular spectrum analysis(SSA)algorithm uses SVD decomposition to perform rank reduction processing on Hankel matrices.However,as the amount of seismic data increases,the efficiency of the algorithm decreases,and the reconstruction accuracy is not ideal.In order to overcome this difficulty,this paper use the fast OR1 MP algorithm instead of singular value decomposition(SVD)to perform high-precision reconstruction of Hankel matrix,and finally do anti-Hankel transform to obtain reconstruction data in the frequency domain.For the reconstruction of 3D natural seismic data,this paper applies OR1 MP to the natural seismic data reconstruction of 3D San Jacinto fault zone in California,the difference is the pre-transformation processing of frequency slices.For 3D natural seismic data,since the frequency slice after the Fourier transform is a 2D data volume,we need to perform block Hankel matrix pre-transform on the fixed frequency slice.The core steps of the traditional Multichannel Singular Spectrum Analysis(MSSA)algorithm are still using SVD decomposition for the rank reduction process.In order to improve the efficiency and reconstruction accuracy of the algorithm,the OR1 MP algorithm is also applied to the low rank complement of the block Hankel matrix.The OR1 MP algorithm for 2D and 3D natural seismic data of the San Jacinto fault zone in California shows that compared with the traditional SSA and MSSA algorithms,the OR1 MP algorithm can effectively increase the peak signal-to-noise ratio of seismic data and achieve better natural Reconstruction of seismic signals.Finally,for the reconstruction of 3D natural seismic data,the reconstruction based on Hankel tensor is more accurate and robust than the block-based Hankel matrix in the case of high missing.Thus,a Hankel tensor pre-transformation of each 2D frequency slice of 3D natural seismic data in the frequency-space domain can be represented as a 4D Hankel tensor with a low rank structure.The presence of missing seismic data and random noise increases the rank of the pre-transform tensor.Therefore,the 3D natural seismic data can be reconstructed using the tensor downgrade method.High-order singular value decomposition(HOSVD)is a generalized SVD decomposition in the tensor generalization,and has achieved good results in high-dimensional seismic data reconstruction.However,SVD decomposition is very time consuming and computationally inefficient.To this end,this paper proposes parallel matrix factorization hankel tensor reconstruction(PMFHTR).The core step of the algorithm is to use the low-rank tensor completion by parallel matrix factorization(TMac)to perform low-rank matrix decomposition on the pre-transformed Hankel tensor,which can avoid the calculation of SVD.The 3D natural seismic data from the San Jacinto fault zone in California shows that the PMFHTR algorithm can quickly reconstruct natural seismic data with high precision compared with the classical HOSVD algorithm.