GENERATION OF SURFACE VALUES OF P AND $\partial P / \partial z$

The initiation of migration based on equation (4) requires the specification of both P and $\partial P / \partial z$ on the surface. Since only one of these fields is recorded in practice, the remaining field must be generated from mathematical assumptions. In a region in which the acoustic velocity c is constant, the acoustic wave equation can be doubly transformed in x and t to give

$${\partial^2 P \over \partial z^2} = -({\omega^2 \over c^2}-k_x^2)P(k_x,z,\omega)$$ (20)

where P is the two dimensionally transformed pressure and k_x is the horizontal wavenumber. The solution to equation (20) are given by

$$P(k_x,z,\omega) = \exp\left (i\sqrt{{\omega^2 \over c^2}-k_x^2}\right )P(k_x,z,\omega)$$ (21)

the solution (21) includes only upgoing waves under the convention that z increases with depth. The doubly transformed pressure gradients can be obtained from equation (21) by differentiation

$${\partial P \over \partial z} = i\sqrt{{\omega^2 \over c^2}-... ...0,\omega) = i\sqrt{{\omega^2 \over c^2}-k_x^2} P(k_x,z,\omega)$$ (22)

The generated vertical pressure gradient on the surface are obtained from equation (22) by setting z=0 and by an inverse Fourier transformation with respect to x.