Two efficient Fourier migration methods termed the extended local Born Fourier (ELBF) method and the extended Rytov Fourier (ELRF) method have been developed recently for imaging complex 3D structures. They are recursive methods based on local applications of Born and Rytov approximations within each extrapolation interval. The ELBF method becomes unreliable when the lateral slowness variations are large and/or the frequency is high, while the ELRF method is reliable for such cases. However, the ELRF method is approximately 30-40% slower than the ELBF method because the ELRF method requires one more computational step where exponentials of complex numbers are calculated than the ELBF method and propose an implementation scheme using variable extrapolation intervals to make the ELBF method reliable for all lateral slowness variations and frequencies. The size of the extrapolation interval depends on the lateral slowness variations within a given extrapolation region and the frequency, and consequently, the computational time of the ELBF method with variable extrapolation intervals increases with the lateral slowness variation and frequency. To take advantage of the faster computational speed of the ELBF method compared to the ELRF method and the better stability of the ELRF method compared to the ELBF method, we propose a hybrid local Born/Rytov Fourier migration method. In the hybrid method, the ELBF method is used for regions with small lateral slowness variations and/or low frequencies, otherwise, the ELRF method is used. Migrations of two synthetic datasets for complex structures using the ELBF method with variable extrapolation intervals and the hybrid method demonstrate that the quality of images obtained using these two methods is comparable to that of images obtained using the ELRF method. Comparison of computational times for migrations using different methods shows that the ELBF method with variable extrapolation intervals takes much more computational time than the ELRF method but the hybrid method saves more than 10% of the computational time required by the ELRF method.
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