UPCOMING PRESSURE FIELD AT SEA FLOOR

As shown in Figure , the recorded pressure field can be decomposed into the downgoing and upcoming wavefields at the cable depth

$$\phi(x,z,t) = \acute{\phi}(x,z,t) + \grave{\phi}(x,z,t) ,$$ (1)

and, assuming a perfect reflection at the water's surface,

$$\phi(x,z=z_0,t) =\acute{\phi}(x,z=z_0,t) - \acute{\phi}(x,z=-z_0,t) .$$

Let ${\cal L}(z)$ be the extrapolation operator for upcoming waves

$$\acute{\phi}(x,z=-z_0,t) = {\cal L}(-2z_0) \acute{\phi}(x,z=z_0,t) ,$$

then

$$\acute{\phi}(x,z=z_0,t)= [1 - {\cal L}(-2z_0) ]^{-1} \phi(x,z=z_0,t) .$$

and since $\acute{\phi}(x,z,t)= {\cal L}(z-z_0) \acute{\phi}(x,z=z_0,t)$ we get

 

$$\acute{\phi}(x,z,t) = {\cal H}(z,z0) \: \phi(x,z=z_0,t) ,$$ (2)

where  

$${\cal H}(z,z_0) = {\cal L}(z-z_0) [ 1 - {\cal L}(-2z_0) ]^{-1} .$$ (3)

The operator defined in equation (3) performs two distinct processes: the conversion of the recorded pressure field into the upcoming pressure field, and its downward continuation to the sea bottom.

In the $\omega$-k_x domain, the downward extrapolation operator for upcoming waves can be expressed by

$${\cal L}(z) = e^{i k_z z} = e^{-i \sqrt{{\omega^2 \over v^2}-k_x^2} \: {\textstyle z}} ,$$

which can be substituted into the Fourier transform of equation (2) to obtain

$$\acute{\Phi}(k_x,z,\omega) = {\cal H} \; \Phi(k_x,z=z_0,\omega) ,$$ (4)

where  

$${\cal H}(z,z_0,k_x,\omega)={e^{i k_z z} \over 2 \; i \; \sin{k_z z_0}} .$$ (5)

It is evident from equations (3) and (5) that the operator ${\cal H}$ has poles at $k_z \: z_0 = n \pi$, corresponding to vertical wavelengths of $2 \: z_0 \over n$ .

For waves with these wavelengths, the downgoing and upcoming fields will cancel each other at the cable depth (assuming perfect reflection at the water's surface), and the result is a null recorded wavefield. Of course no information can be obtained for the upcoming field in this case, but since it represents only a couple of lines in the $\omega$-k_x plane we can neglect its contribution for the final, downward continued wavefield.