Theory of multisource crosstalk reduction by phase-encoded statics

Formulas are derived that relate the strength of the crosstalk noise in supergather migration images to the variance of time, amplitude and polarity shifts in encoding functions. A supergather migration image is computed by migrating an encoded supergather, where the supergather is formed by stacking a large number of encoded shot gathers. Analysis reveals that for temporal source static shifts in each shot gather, the crosstalk noise is exponentially reduced with increasing variance of the static shift and the square of source frequency. This is not too surprising because larger time shifts lead to less correlation between traces in different shot gathers, and so should tend to reduce the crosstalk noise. Analysis also reveals that combining both polarity and time statics is a superior encoding strategy compared to using either polarity statics or time statics alone. Signal-to-noise (SNR) estimates show that for a standard migration image and for an image computed by migrating a phase-encoded supergather; here, G is the number of traces in a shot gather, I is the number of stacking iterations in the supergather and S is the number of encoded/blended shot gathers that comprise the supergather. If the supergather can be uniformly divided up into Q unique sub-supergathers, then the resulting SNR of the final image is, which means that we can enhance image quality but at the expense of Q times more cost. The importance of these formulas is that they provide a precise understanding between different phase encoding strategies and image quality. Finally, we show that iterative migration of phase-encoded supergathers is a special case of passive seismic interferometry. We suggest that the crosstalk noise formulas can be helpful in designing optimal strategies for passive seismic interferometry and efficient extraction of Green's functions from simulated supergathers. © 2011 The Authors Geophysical Journal International © 2011 RAS.