One-way propagation (PDKO)

Let us consider the equation:  

$$\left( {d \over dz} - ik_{z}\right) \tilde{u} =0,$$ (95)

at

k_z=k_z(z).

For homogeneous medium this equation can be obtained by splitting the wave equation operator ${d^{2} \over dz^{2}} + k_{z}^{2}$. Equation (95) has the solution:

$$\tilde{u}_{0} \; \exp \left\{ i \int^{z}_{0} k_{z} (\xi) d \xi \right\}$$

which describes upgoing waves. The operator  

$$u ( {\bf r},t ) = {\bf F}^{-1}_{x,y,t} \left\{ \tilde{u}_{0} e^{i \int _{0}^{z}k_{z} (\xi) d \xi } \right\}$$ (96)

is kinematically equivalent (in the sense of the classical eikonal equation), but there is no partial derivative equation in space-time domain that has equation (96) as a solution. This operator is an example of PDKO (Pseudo-Differential K-Operator). But we can find a sequence of equations

$${\bf L}_{j}^{(-)} u =0$$

for which:

• each equation permits only one-way propagation along the z-axis.
• the characteristic equation for ${\bf L}_{j}^{(-)}$ approximates the classical eikonal equation.
• The accuracy of the approximation increases with the number j.

The way to perform it is the expansion

$$ik_{z}= {i \omega \over v} \left( 1- {{(ik_{x})}^{2} + {(ik_... ... \over 2 {{(i \omega )}^{2} \over v^{2}}} + \ldots \right)$$

If we substitute the nth approximation of ik_z into equation (95) and return to the (r,t) domain, remembering the correspondence

$$\begin{array} {l} {(i \omega )}^{r} \leftrightarrow {\partia... ...\leftrightarrow {\partial ^{r} \over \partial x^{r}}\end{array}$$

we get the equation which we looked for. The first approximation:  

$${\partial ^{2}u \over \partial z\partial t} + {1\over v}{\pa... ...rtial t^{2}}-{v \over 2}{\partial ^{2}u \over \partial x^{2}}=0$$ (97)

yields to Jon Claerbout's famous 15-degree migration algorithm. With accordance to the formula (24) (in Chapter 3), the characteristic equation which corresponds to equation (97) is as follows:  

$$\tau_{{x}}^{2}-{2\over v}\tau _{z}={1\over v^{2}}.$$ (98)

It differs from classical eiconal's equation for isotropic medium.