Marchenko imaging obtains the subsurface reflectors using one-way Green’s functions,which are retrieved by solving the Marchenko equation.This method generates an image that is free of spurious artifacts due to inter...Marchenko imaging obtains the subsurface reflectors using one-way Green’s functions,which are retrieved by solving the Marchenko equation.This method generates an image that is free of spurious artifacts due to internal multiples.The Marchenko imaging method is a target-oriented technique;thus,it can image a user specified area.In the traditional Marchenko method,an accurate velocity model is critical for estimating direct waves from imaging points to the surface.An error in the velocity model results in the inaccurate estimation of direct waves.In turn,this leads to errors in computation of one-way Green’s functions,which then affects the final Marchenko images.To solve this problem,in this paper,we propose a self-adaptive traveltime updating technique based on the principle of equal traveltime to improve the Marchenko imaging method.The proposed method calculates the time shift of direct waves caused by the error in the velocity model,and corrects the wrong direct wave according to the time shift and reconstructs the correct Green’s functions.The proposed method improves the results of imaging using an inaccurate velocity model.By comparing the results from traditional Marchenko and the new method using synthetic data experiments,we demonstrated that the adaptive traveltime updating Marchenko imaging method could restore the image of geological structures to their true positions.展开更多
The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rect...The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect regards “rectangular” versions of these theorems. Comparing the properties of Rayleigh Quotient matrices with those of Orthogonal Quotient matrices illuminates the subject in a new light. The Orthogonal Quotients Equality is a recent result that converts Eckart-Young’s minimum norm problem into an equivalent maximum norm problem. This exposes a surprising link between the Eckart-Young theorem and Ky Fan’s maximum principle. We see that the two theorems reflect two sides of the same coin: there exists a more general maximum principle from which both theorems are easily derived. Ky Fan has used his extremum principle (on traces of matrices) to derive analog results on determinants of positive definite Rayleigh Quotients matrices. The new extremum principle extends these results to Rectangular Quotients matrices. Bringing all these topics under one roof provides new insight into the fascinating relations between eigenvalues and singular values.展开更多
基金supported by the National Natural Science Foundation of China(No.41874167)the National Science and Technology Major Project of China(No.2017YFB0202904)the National Natural Science Foundation of China(No.41904130)。
文摘Marchenko imaging obtains the subsurface reflectors using one-way Green’s functions,which are retrieved by solving the Marchenko equation.This method generates an image that is free of spurious artifacts due to internal multiples.The Marchenko imaging method is a target-oriented technique;thus,it can image a user specified area.In the traditional Marchenko method,an accurate velocity model is critical for estimating direct waves from imaging points to the surface.An error in the velocity model results in the inaccurate estimation of direct waves.In turn,this leads to errors in computation of one-way Green’s functions,which then affects the final Marchenko images.To solve this problem,in this paper,we propose a self-adaptive traveltime updating technique based on the principle of equal traveltime to improve the Marchenko imaging method.The proposed method calculates the time shift of direct waves caused by the error in the velocity model,and corrects the wrong direct wave according to the time shift and reconstructs the correct Green’s functions.The proposed method improves the results of imaging using an inaccurate velocity model.By comparing the results from traditional Marchenko and the new method using synthetic data experiments,we demonstrated that the adaptive traveltime updating Marchenko imaging method could restore the image of geological structures to their true positions.
文摘The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect regards “rectangular” versions of these theorems. Comparing the properties of Rayleigh Quotient matrices with those of Orthogonal Quotient matrices illuminates the subject in a new light. The Orthogonal Quotients Equality is a recent result that converts Eckart-Young’s minimum norm problem into an equivalent maximum norm problem. This exposes a surprising link between the Eckart-Young theorem and Ky Fan’s maximum principle. We see that the two theorems reflect two sides of the same coin: there exists a more general maximum principle from which both theorems are easily derived. Ky Fan has used his extremum principle (on traces of matrices) to derive analog results on determinants of positive definite Rayleigh Quotients matrices. The new extremum principle extends these results to Rectangular Quotients matrices. Bringing all these topics under one roof provides new insight into the fascinating relations between eigenvalues and singular values.
