THE PARAXIAL WAVE EQUATION

The scalar wave equation, unlike Fourier equations, allows arbitrary spatial variation in density and velocity. Because of this you might expect that it would be used directly in the manufacture of migrated sections. But it is used little for migration, and we will first review why this is so. Then we will meet the paraxial wave equation, which is the basis for most production migration.

Philosophically, the paraxial wave equation is an intermediary between the simple concepts of rays and plane waves and deeper concepts embodied in the wave equation. (The paraxial wave equation is also called the single-square-root equation, or a parabolic wave equation). The derivation of the parabolic wave equation does not proceed from simple concepts of classical physics. Its development is more circuitous, like the Schroedinger equation of quantum physics. You must study it for a while to see why it is needed. When I introduced the parabolic wave equation to seismic calculations in 1970, it met with considerable suspicion. Fortunately for you, years of experience have enabled me to do a better job of explaining it, and fortunately for me, its dominance of the industrial scene will give you the interest to persevere.

The paraxial equation will be introduced by means of Fourier methods. Fourier methods are incompatible with space-variable coefficients. Since we want to incorporate spatial variations in velocity, this limitation is ultimately to be avoided, so after getting the paraxial equation in the Fourier domain, ik_z is replaced by $\partial / \partial z$,and ik_x is replaced by $\partial / \partial x$.Then, being in the space domain, the velocity can be space-variable. The result is a partial differential equation often solved by the finite-differencing method. This procedure turns out to be valid, but new students of migration understandably regard it with misgiving. Thus, the final part of this section is a derivation of the paraxial wave equation which makes no use of Fourier methods.