WAVE EQUATIONS

The key equation is the dispersion relation, the equation that links the temporal and spatial frequencies of an eigenfield in the Fourier domain. The four variable parameters all have units of velocities squared. W_x and W_z are the actual horizontal and vertical square velocities that apply equally well to waves and rays, and W_xnmo and W_znmo are the apparent square velocities as determined from move-out measurements. For example, W_xnmo is the square NMO velocity that we would use every day in surface-to-surface data processing. If the x subscript seems unusual for a paraxial measurement about the vertical, remember that in an elliptic world it is the horizontal velocity that move-out measures.

$$\omega^{2} = \frac {W_{x}^3 k_{x}^6 + W_{x}^2 (2 W_{z} + W_{... ...2 k_{z}^4 + W_{z}^6 k_{z}^6}{(W_{x} k_{x}^2 + W_{z} k_{z}^2)^2}$$ (1)

With minor modifications that are discussed in Muir & Dellinger (1985), there are precisely equivalent plane wave and partial differential equations.