Summary
We show an explicit scheme that separates seismic source and receiver wave-fields individually into up- and down- going components. The main challenge of the work is generating wavefields that correspond to positive or negative temporal frequencies in space-time 4D volume. This difficulty arises because the seismic propagators we typically employ store wave-fields with slowest dimension in time but the Fourier transform operates most efficiently on data that are stored contiguously. We solve this issue by a temporal Hilbert transform of the source term of the wave-equation followed by the conventional propagations. The pair of wave-fields, namely, the wave-field propagated without a Hilbert transformed source and the wave-field generated by the Hilbert transformed source, constitute the desired positive or negative temporal components in real and imaginary parts, respectively and separately. The down-going and up-going wave components can then be conveniently obtained by applying 1D Fourier filters in depth. We pursue a causal imaging condition that correlates the down-going source component with the up-going receiver component for subsurface imaging. We demonstrate that by applying the causal imaging condition certain strong near-salt imaging artifacts are removed.
Introduction
In the forward time axis, an incident source wave-field causally excites the reflected source wave-field at the boundary between two mediums. In this work, we refer to the causal reflection imaging condition as the correlation between the incident source wave-field and the wave-field that kinematically agrees with the reflected source wave-field. The latter can be produced as part of the receiver wave-field which propagates along the reversed time axis using the receiver data acquired on some boundary (Lions, 1972). Let the word, incident, be used to indicate the state "before" reflections occur in time, forward or reversed, in which source and receiver wave-field, respectively, is computationally propagated. It then makes sense to distinguish the incident receiver wave-field from the reflected receiver wave-field at the boundary of velocity contrast. With the terminologies laid above, we can restate the causal reflection imaging condition as the correlation between the incident source wave-field and the incident receiver wave-field, and this is consistent with the survey sinking concept introduced in Claerbout (1985).
An RTM, reverse-time migration, involves propagation of source wave-field, adjoint-state propagation of receiver wave-field, and correlation of source and receiver wave-field along the forward time axis at zero lag (Baysal et al., 1983; Loewenthal and Mufti, 1983; McMechan, 1983; Whitmore, 1983). The "two-way" propagator engineered in RTM does not separate incident from reflected, neither refractions from reflections. A conventional imaging condition implemented as a straight forward correlation between source and receiver wave-field thus produces many types of artifacts. A practical approach that removes most of the large angle imaging artifacts is to apply the Laplacian filter to the stacks (Zhang and Sun, 2009).