Muir square-root expansion

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 $\exp (-i \omega t \ +\ i k_x x \ +\ i k_z z )$into the two-dimensional scalar wave equation () yields the dispersion relation  

$$k_z^2 \ +\ k_x^2 \eq { \omega^2 \over v^2 }$$ (7)

Solve for k_z selecting the positive square root (thus for the moment selecting downgoing waves).  

$$k_z \ \ \ =\ \ \ {\omega \over v }\ \sqrt { 1 \ - \ { v^2 \, k_x^2 \over \omega^2 }}$$ (8)

To inverse transform the z-axis we only need to recognize that i k_z corresponds to $\partial / \partial z$.The resulting expression is a wavefield extrapolator, namely,  

$${\partial \ \ \over \partial z}\ P(\omega,k_x,z) \ \ \ =\ \... ...rt { 1 \ - \ {v^2 \, k_x^2 \over \omega^2 }} \ P(\omega,k_x,z)$$ (9)

Bringing equation (9) into the space domain is not simply a matter of substituting a second x derivative for k_x². 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$^\circ$ and 45$^\circ$ methods were just truncations of a continued fraction expansion. To see this, define  

$$X \eq {v k_x \over\omega} \quad\quad {\rm and} \quad\quad R \eq {v k_z \over\omega}$$ (10)

So equation (8) is more simply and abstractly written as  

$$R \eq \sqrt { 1 \ - \ X^2 }$$ (11)

which you recognize as meaning that cosine is the square root of one minus sine squared. The desired polynomial ratio of order n will be denoted R_n, and it will be determined by the recurrence  

$$R_{n+1} \eq 1 \ -\ { X^2 \over 1 \ +\ R_n }$$ (12)

The recurrence is a guess that we verify by seeing what it converges to (if it converges). Set $n \ =\ \infty$ in (12) and solve

$$\begin{eqnarray} R_{\infty} \ \ &=& \ \ 1 \ -\ { X^2 \over 1 \ +\ R_{\infty} } \... ...1 \ +\ R_{\infty} \ -\ X^2 \nonumber \\ R^2 \ \ &=& \ \ 1 \ -\ X^2\end{eqnarray}$$ (13)

The square root of (13) gives the required expression (11). Geometrically, (13) says that the cosine squared of the incident angle equals one minus the sine squared and truncating the expansion leads to angle errors. Muir said, and you can verify, that his recurrance relationship formalizes what I was doing by re-estimating the $\partial_{zz}$ term. Although it is pleasing to think of large values of n, in real life only the low-order terms in the expansion are used. Table .1 shows the result of three Muir iterations beginning from R₀ = 1

 

$$5^\circ R_0 \eq 1$$

$$15^\circ R_1 \eq 1 -\displaystyle {\strut X^2\over 2}$$

$$45^\circ R_2 \eq 1 -{\displaystyle{\strut X^2}\over \displaystyle 2 - {\strut X^2\over 2}}$$

$$60^\circ R_3 \eq 1 - {\displaystyle {\strut X^2} \over\displaystyle 2 - {\strut X^2 \over \displaystyle 2 - {\strut X^2\over 2}}}$$

For various historical reasons, the equations in Table .1 are often referred to as the 5$^\circ$, 15$^\circ$, and 45$^\circ$ 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$^\circ$ equation. It then turns out that a slightly wider range of angles can be accommodated if the recurrence is begun with something like $R_0 = \cos$ 45$^\circ$.Figure 1 shows some plots.

 

disper
Figure 1 Dispersion relation of Table .2. The curve labeled $45^\circ_+$ was constructed with $R_0 = \cos 45^\circ$ . It fits exactly at 0$^\circ$ and 45$^\circ$ .