Target-oriented wavefront synthesis

Rietveld et al. (1992) and Berkhout (1992) proposed a target-oriented wavefront synthesis for illuminating reflectors under a complex velocity overburden more clearly. Using the unitary property of the wavefield extrapolation operator, they generate a synthesis operator for a wave stack whose wavefront has a predefined shape at a certain subsurface location.

Suppose that we want to have a downgoing wavefield $D(\omega,x,z_n)$ at depth z_n. This wavefield can be obtained by propagating a certain wavefield at the surface, $D(\omega,x,z_0)$,to the depth level z_n.

$$D(\omega,x,z_n) = W(z_n,z_0) D(\omega,x,z_0)$$ (4)

where W(z_n,z₀) represents a wave propagation operator from z₀ to z_n in forward time. If we know $D(\omega,x,z_n)$,then $D(\omega,x,z_0)$ can be found approximately using the unitary property of the propagation operator,

$$\tilde{D}(\omega,x,z_0) \sim W^\ast(z_0,z_n) D(\omega,x,z_n)$$ (5)

where $W^\ast(z_0,z_n)$ represents a wave propagation operator from z_n to z₀ in backward time. Equation (6) tells us that the areal shot (1992, Berkhout), $\tilde{D}(\omega,x,z_0)$ will approximately generate the predefined wavefield, $D(\omega,x,z_n)$, at depth z_n.

If we assume each shot gather, U(w,x,z₀|x_s), is due to an impulse at x_s, the response of the areal shot can be synthesized by using the convolution formula as

$$\tilde{U}(\omega,x,z_0) = \sum_{x_s} \tilde{D}(\omega,x_s,z_0)U(w,x,z_0\vert x_s)$$ (6)

The imaging is performed in the same manner as profile imaging with $\tilde{U}(\omega,x,z_0)$ as the upcoming wave and $\tilde{D}(\omega,x,z_0)$ as the downgoing wave.