While most of the modern seismic imaging methods per- form imaging by separating input data into parts (shot gathers), we develop a formulation that is able to incor- porate all available data at once while numerically prop- agating the recorded multidimensional wavefield back- ward in time. While computationally extensive, this approach has the potential of generating accurate im- ages, free of artifacts associated with conventional ap- proaches. We derive novel high-order partial diferential equations in source-receiver-time domain. The fourth order nature of the extrapolation in time has four solu- tions two of which correspond to the ingoing and out- going P-waves and reduces to the zero-of set exploding- reector solutions when the source coincides with the receiver. Using asymptotic approximations, we develop an approach to extrapolating the full prestack wavefield forward or backward in time.
Wave-equation depth migration methods are commonly divided into two types: one-way for wave extrapolation in depth and two-way for wave extrapolation in time or reverse-time migration (Biondi, 2006; Etgen et al., 2009). Conventionally, both methods are applied on individual shot gathers. With the one-way approach, it is also possible to combine all data (multiple shot gathers) into one wave-extrapolation procedure with the survey-sinking or DSR (double-square-root) formulation of the wave equation (Claerbout, 1985; Popovici, 1996; de Hoop et al., 2003). With modern 3-D data, DSR migration appears feasible only for single-azimuth data under the common-azimuth approximation (Biondi and Palacharla, 1996). However, it is important to preserve it in the arsenal of theoretically possible imaging meth- ods, because one day its computational cost may become affordable for a more general case. Another limitation of the DSR formulation is the one-way nature of wave ex- trapolation, which limits the imaging accuracy at large structural dips. Extended coordinate systems may help (Sava and Fomel, 2005a; Shragge, 2008) but do not solve the issue completely. In this paper, we extend the survey-sinking approach to extrapolation in time rather than depth. The possi- bility of such extension was first indicated by Duchkov and de Hoop (2009). We develop a constructive theory for two-way shot-receiver extrapolation in the form of a mixed-domain space-wavenumber operator, which leads to a second-order in time and fourth-order in space par- tial differential equation. In practice, a two-way source- receiver extrapolation operator can be implemented with either finite-difference or spectral operators (Fomel et al., 2010; Song and Fomel, 2010). Application of two-way extrapolators to modeling and migration follows the exploding reector concept (Loewen- thal et al., 1976; Claerbout, 1985). In the modeling mode, the source-receiver wavefield is initialized with signal at zero time and zero offset and corresponds to the reectivity strength. The image is ex- tracted then at zero time from the zero offset slice or, alternatively, image gathers are formed from non-zero time slices or non-zero offsets (Fomel, 2004; Sava and Fomel, 2005b, 2006; Sava and Vasconcelos, 2010). Un- like one-way survey sinking extrapolation, the two-way wave extrapolation does not impose any limits on the extrapolation direction.
