In order to overcome the problem that wavelets of each scale data are unknown in multi-scale FWI, we define a frequency shift (FS) operator to decompose seismic data to multi-scale data corresponding to arbitrary theoretical wavelet as we expect. Application of FS operator includes two steps: first, source wavelet is obtained by wavefield reconstruction, in which the direct wave in near offset is set as the boundary condition; second, we apply FS operator to seismic data and the source wavelet extracted above, and construct multi-scale data corresponding to a set of theoretical wavelets. Compared with digital filter, multi-scale data computed by our FS operator can overcome the wavelet mismatch problem in FWI. A layered model is used to demonstrate the feasibility and accuracy of FS operator, in which stable multi-scale results is consistent with the forward modeling results using theoretical wavelet. The successful application for actual data proves that FS operator is stable, but the existence of low frequency noise indicates the high requirements for SNR.


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