Solving the wave equation
There are a number of ways to derive the wave equation, and many forms in which it is expressed. The approach taken in these notes will be based on the works by Claerbout , , Yilmaz , Stolt , and Brysk . The main objective is to define the paths to the solutions used in seismic processing for modelling and migration. The main emphasis will be on the downward continuation method of seismic migration. Since this is a collection from the above authors, the notations used will reflect that used by the specific author to enable comparison to with their results. Consequently some parameters may change, specifically when moving from the solutions of the wave equations to the finite difference solutions.
It is assumed that the density is constant, and that the velocity varies in a vertical (z) and horizontal (x) manner. The 2-D seismic model is assumed to be 3-D with axis x, z, and time (t), with the pressure amplitude defined at any point in the volume as P(x, z, t). The zero offset or stacked section is defined by the surface P(x, z=0, t), and the geological cross-section or desired depth migration as P(x, z, t=0). The interval velocity v(x, z) is assumed to be isotropic (independent of direction). Note that some solutions use modified forms of the velocities such as RMS velocities (i.e. Kirchhoff time) and other use velocities defined in time as v(x, t).