## Least-squares reverse-time migration of surface-related multiples

**LIU Xue-Jian;LIU Yi-Ke;Key Laboratory of Engineering Geomechanics,Institute of Geology and Geophysics, Chinese Academy of Sciences;University of Chinese Academy of Sciences;**

Surface-related multiples are traditionally treated as noise and are attenuated using surface-related multiples elimination(SRME)and/or radon-based multiple-elimination methods.Multiples penetrate into the subsurface several times and contain abundant reflection information of small angles.Compared with migrating of primaries,migrating of multiples extends all the receivers as second sources and sometimes provides additional subsurface illumination.For reverse-time migration(RTM)of all-order multiples,however,the main challenge is that undesired crosscorrelations between forward and backward propagated seismic waves generate so many crosstalk artifacts.The crosstalks may distribute in the whole image profile,which can destruct the true image of reflectors and mislead the interpreting result of a migrated image.Compared with conventional RTM,least-squares reverse-time migration(LSRTM)can invert recorded primaries as an image with more balanced amplitude and higher resolution.Moreover,we develop the conventional LSRTM to invert multiples as an image while iteratively suppressing crosstalk artifacts.However,LSRTM of multiples can′t totally attenuate the artifacts in the image of multiples,and usually many iterations are required to invert a well-accepted image.Alternatively,if first-order multiples can be separated from all-order multiples in advance,LSRTM of first-order multiples can be developed to reduce the iteration number.With the same iterations used,compared with LSRTM of multiples,LSRTM of first-order multiples can provide a much cleaner image section and a similar true image of reflectors.The motivation to develop LSRTM of first-order multiples can be further summarized as:(1)conventional migration of first-order multiples can avoid the most undesired crosscorrelations between forward and backward propagated wavefields and can maintain some advantages of imaging multiples at the same time;although the subsurface information contributed by higher-order multiples is neglected,RTM of first-order multiples have already avoided most artifacts.(2)There are still some crosstalk artifacts in the RTM image of first-order multiples;then,compared with RTM of first-order multiples,LSRTM of first-order multiples can further enhance the image in detail by suppressing the crosstalk artifacts,balancing the amplitude,and improving the resolution.In order to invert primaries as an image,LSRTM iteratively solves a misfit function that is the L2 norm of the amplitude residual between the modeled and observed primaries.Born modeling is a linear two-step procedure and synthesizes primaries perturbed by an image,which bases the LSRTM.Conventional RTM is the adjoint of Born modeling,whereas the analytical solution of the misfit function is the generalized-inverse of the Born modeling.The analytical solution is hard to be obtained because the Hessian matrix is so large,so a nonlinear optimal scheme,e.g.,the steepest-descent method,can be used to iteratively solve the misfit function.Taking the released Sigsbee2 bdata as an example,we can intuitively conclude that LSRTM provides an image with higher resolution and more balanced amplitude and suppresses the migration artifacts compared with conventional RTM.Different with the misfit function for the conventional LSRTM,the misfit function for the LSRTM of multiples is the L2 norm of the amplitude residual between the modeled multiples and estimated multiples during the regular seismic data processing.The accurate calculation for the modeling of multiples is crucial for the success of this method,where a modified Born modeling procedure and an accurate background velocity are utilized.Instead of a point source,the recorded data including primaries and multiples are forward propagated and stacked as the downgoing wavefield.Each discrete point of the image is seen as a scatter.The two-order time derivative of downgoing wavefield is scattered by the RTM image of multiples,and upgoing wavefield is the stack of scattered waves.Surfacerelated multiples are modeled by recording the upgoing wavefield at receivers.Similar to the conventional LSRTM,LSRTM of multiples also can iteratively seek the reflectivity model using a nonlinear optimal method.Moreover,to invert first-order multiples as an image,the misfit function based on the L2 norm of the amplitude residual between the observed and Born modeled first-order multiples should be built.Compared with the Born modeling of all-order multiples,instead of total recorded data,only primaries are forward propagated for the Born modeling of first-order multiples.The observed first-order multiples are estimated by a modified SRME,which includes two steps:(1)predicting higher-order multiples by the convolution of primaries and multiples;(2)adaptively subtracting higher-order multiples from all multiples.RTM of all-order multiples,LSRTM of all-order multiples,RTM of first-order multiples and LSRTM of first-order multiples have been tested on a three-layer and the Marmousi2 model.Only 16 shot gathers are used for imaging on the three-layer model.RTM image of multiples provide wider illumination and higher fold for subsurface,whereas there are a lot of artifacts in the image of multiples.After 10 iterations,LSRTM attenuate most of the artifacts in the image of multiples,except the artifacts at bottom.Moreover,LSRTM of first-order multiples provide amore cleaner section than LSRTM of all-order multiples.There are artifacts in the modeled data using the RTM image of multiples,whereas the modeled data using LSRTM image of multiples have a good match with the estimated multiples using SRME and avoid most artifacts.On the Marmousi2 model,there are many artifacts in the RTM image of multiples,which are mostly attenuated by LSRTM after 5iterations.However,after 5iterations,there are still residual artifacts in the LSRTM image of multiples,which disappear in LSRTM image of first-order multiples.On above two experiments,LSRTM of multiples and LSRTM of first-order multiples both converge very fast and robust.Compared with RTM,LSRTM provides image with more balanced amplitude and better resolution and suppresses the migration artifacts.RTM of multiples can provide a wider illumination and higher fold for subsurface.However,there are many crosstalk artifacts in the RTM image of all-order multiples.LSRTM can attenuate most of the crosstalk artifacts in the image of multiples but costs huge computation of many iterations.A modified SRME is proposed to filter first-order multiples.With the same iterations used,LSRTM of first-order multiples provide a much cleaner section,and provider a similar true image of reflectors compared with LSRTM of all-order multiples.Prior to LSRTM of first-order multiples,first-order multiples are needed to be estimated by a modified SRME.

【Fund】：
国家自然科学基金项目(41430321 41374138);;
中国科学院战略性先导科技专项(B类)(XDB01020300)联合资助

【CateGory Index】： P631.4

【CateGory Index】： P631.4

CAJViewer7.0 supports all the CNKI file formats; AdobeReader only supports the PDF format.