Shot-profile wavefield continuation

Assuming applicability of the scalar wave equation given by equation (1), the first step in GPR shot-profile migration is the extrapolation of surface-recorded data to depth. This is done by applying a one-way wave-equation operator to an arbitrary wavefield, as a function of space and frequency, to yield the wavefield at a deeper level:  

$$W(z+\Delta z, x, \omega) = W(z, x, \omega) \; {\rm e}^{\pm k_z \Delta z}.$$ (3)

Here, the positive or negative exponent corresponds to causal or acausal propagation, respectively, and $\Delta z$ is the size of the downward continuation step. The vertical wavenumber, k_z, is calculated from the scalar wave-equation dispersion relation,  

$$k_z = \sqrt{s^2 \omega^2 - k_x^2},$$ (4)

where k_x is the horizontal Fourier wavenumber component of the data wavefield.

Surface-recorded wavefields are extrapolated to all depths within the model through successive applications of equation (3) using a vertical wave-number given by equation (4). Although equation (4) is strictly valid only for vertically stratified media, techniques exist to extend it to laterally varying media. We employ a split-step Fourier approach Stoffa et al. (1990) that involves approximating k_z in equation (4) using a Taylor series expansion about a reference slowness, s₀:  

$$k_z \approx \omega \left(s - s_0\right) + \sqrt{s_0^2 \omega^2 - k_x^2}.$$ (5)

The first, mixed-domain term in equation (5) acts as a local correction to the second term that handles the bulk of the propagation. Increased accuracy can be achieved by summing the results of multiple reference velocity steps in order to minimize the quantity s(x)- s₀.

Shot-profile migration directly mimics the data collection process by migrating individual shot records. Receiver wavefields are comprised of individual shot profiles and are propagated acausally. Source wavefields have the same geometry, but are initially zero except for an appropriate source function at the transmitter location, and are propagated causally.