Muir's method of finding wave extrapolators seeks polynomial ratio approximations to a square-root dispersion relation. Then fractions are cleared and the approximate dispersion relation is inverse transformed into a differential equation.
Substitution of the plane wave into the two-dimensional scalar wave equation () yields the dispersion relation
To inverse transform the z-axis we only need to recognize that i kz corresponds to .The resulting expression is a wavefield extrapolator, namely,
Bringing equation (9) into the space domain is not simply a matter of substituting a second x derivative for kx2. The problem is the meaning of the square root of a differential operator. The square root of a differential operator is not defined in undergraduate calculus courses and there is no straightforward finite difference representation. The square root becomes meaningful only when the square root is regarded as some kind of truncated series expansion. It will be shown in chapter that the Taylor series is a poor choice. Francis Muir showed that my original 15 and 45 methods were just truncations of a continued fraction expansion. To see this, define
For various historical reasons, the equations in Table .1 are often referred to as the 5, 15, and 45 equations, respectively, the names giving a reasonable qualitative (but poor quantitative) guide to the range of angles that are adequately handled. A trade-off between complexity and accuracy frequently dictates choice of the 45 equation. It then turns out that a slightly wider range of angles can be accommodated if the recurrence is begun with something like 45.Figure 1 shows some plots.