Theory of Large-Step Wavefield Depth Extrapolation

Abstract

Wavefield extrapolation is the foundation of seismic migration imaging. Finding a way of quick and object oriented wavefield extrapolation is significant not only for deep seismic exploration but also for imaging with super-tremendous seismic data acquired from non-combination receivers, both highly needed to develop in petroleum prospecting. To image target zones, as far as all present wavefield extrapolation methods are concerned, including travel-time based Dix-formula methods, ray tracing, small-step depth recursion, etc., the adaptability to complex media, precision and computing efficiency are not satisfying for practical needs in some respects. In this paper, we propose a new technique (so-called large-step wavefield depth extrapolation method), that is based on the Lie algebraic integral of operator's phase to compute depth extrapolation operator efficiently. In the method, the complex phase of object oriented wavefield extrapolation operator's symbol is expressed as a linear combination of wavenumbers, and the coefficients of linear combination are all in the form of the integral of interval velocity functions and their derivatives over depth. Moreover, the computation is convenient and needs less storage space. With phase shift plus correction, the wavefield extrapolation operator is implemented using FFT (Fast Fourier Transform) in spatial and wavenumber domains. Here, the kinetic equivalent relationship is resorted to derive the expression convenient for computation. We compare the precision of one-step scheme with that of two-step scheme for expanding an operator's primary symbol function, which illustrates the large-step scheme is more accurate than its recursive counterpart. Besides, in a linearly variant medium laterally and vertically, the point source pulse response of the large-step method is compared with that of depth recursion methods. The numerical examples indicate that travel-time precision of the large-step extrapolation operator mainly depends on the lateral velocity variation modification items in phase shift operator. In addition, in different approximation cases, the dispersion caused by the large-step operator is rather small.