Relating the pressure and displacement wavefields

The pressure wavefield $\phi(x,z,t)$ in a general heterogeneous, isotropic fluid medium, obeys the scalar wave equation

$$K(x,z) \; \nabla \cdot \; \{ {1 \over \rho(x,z)} \nabla \phi(x,z,t) \} \; = \; {\partial^2 \over \partial t^2} \phi(x,z,t)$$

(Claerbout, 1985), which in the frequency domain is represented by

 

$$K(x,z) \; \nabla \cdot \; \{ {1 \over \rho(x,z)} \nabla \phi(x,z,\omega) \} \; = \; -\omega^2 \phi(x,z,\omega).$$ (1)

In the equations above, K and $\rho$ represent the bulk modulus and the density of the media, respectively.

The P wave particle-displacement vector field ${\bf u}$ is related to the pressure by the following expression:

 

$$\phi(x,z,\omega) \; = \; - K(x,z) \: \nabla \! \cdot \! {\bf u}(x,z,\omega).$$ (2)

Substituting equation (2) into equation (1) leads to  

$$\nabla \! \cdot \! {\bf u}(x,z,\omega) \; = \; \nabla \cdot \; \{ {1 \over \omega^2 \rho(x,z)} \; \nabla \phi(x,z,\omega) \}.$$ (3)

Recalling that, in general, the vector field ${\bf u}$ can be represented by the sum of a gradient potential with a rotational potential, and that in the recording medium (water) the displacement field must be irrotational, equation (3) can be simplified to

 

$${\bf u}(x,z,\omega) \; = \; {1 \over \omega^2 \rho(x,z)} \; \nabla \phi(x,z,\omega).$$ (4)

This equation relates the displacement vector field to the scalar pressure field for general liquid media.

Difficulties arise in the computation of the pressure gradient when equation (4) is to be applied to standard offshore data. Because conventional marine datasets are collected with a single cable, positioned nearly parallel to the water surface, the horizontal derivative of the pressure field can be easily evaluated, while the absence of vertical sampling hinders the direct evaluation of the vertical derivative. In the next sections I derive a method for evaluation of the pressure gradient from data acquired with a standard geometry.