Analytic example

In 1D, let a trace be represented in the Fourier domain by the expression  

$$R(\omega)= \mbox{e}^{-i\phi_{w}}+ \mbox{e}^{-i\phi_{e}}-... ...^{-i(\phi_{w}+\phi_{e})}- \mbox{e}^{-i2\phi_{e}} = P + M_1.$$ (18)

The trace has primary reflections, P, at phase delays, $\phi$, representing a water-bottom and a subsurface event. Also included are first order multiples, M₁, which are the water-bottom multiple at $2\phi_{w}$, two peg-leg multiples at $\phi_{w}+\phi_{e}$, and the event multiple at $2\phi_{e}$. SRMP dictates the autoconvolution of P to derive M₁, which is clearly true. Including the events M₁ in the autoconvolution will derive the second order multiples as well.

Extrapolating trace R to a deeper depth applies a common phase shift, say $\mbox{e}^{-i\phi_z}$, to all terms in equation 18. The trace then becomes  

$$R_z(\omega)=e^{-i\phi_z}( \mbox{e}^{-i\phi_{w}}+ \mbox{e... ... \mbox{e}^{-i2\phi_{e}}) = e^{-i\phi_z}P + e^{-i\phi_z}M_1.$$ (19)

This equation shows that the extrapolation of data without multiple subtraction produces the superposition of the redatumed primaries and the redatumed multiples. The extraction of the zero-time lag in the imaging condition of migration states that energy in the wavefield should be mapped into the image domain only when the extrapolation phasor $\phi_z$, is equal to the time delay of the event in the data. Thus the water bottom primary is imaged when $\phi_z=\phi_w$, the water-bottom multiple is imaged when $\phi_z=2\phi_w$, etc. Whether the data R is first separated into its constituent parts, P and M_i, or not, it can be seen where in the image domain the various events in the above example will be placed.

However, by first squaring the trace, as dictated by the multiple prediction imaging condition (equation 9), the water-bottom primary is mapped into the image domain when $\phi_z=2\phi_w$. This is the same phasor that maps the recorded water-bottom multiple into the image domain using the conventional imaging condition. Thus, the multiple prediction in the image space can be directly calculated with a multiple-only imaging condition due to the commutability of convolution and extrapolation.

For prestack data in higher dimensions, the phasor $e^{-i\phi_z}$ is the combination of the down-going energy D and the exponential delay operators in equations 1 and 2.