Snell Waves in Fourier space

The chain rule for partial differentiation says that  

$$\left[ \ \matrix { \partial_t \cr \partial_x \cr \partial_z... ...tial_{ t' } \cr \partial_{{x}' } \cr \partial_{{z}' } } \right]$$ (35)

In Fourier space, the top two rows of the above matrix may be interpreted as

$$\begin{eqnarray} -\,i\,\omega \ \ \ &=&\ \ \ -\,i\,\omega' \\ i\, k_x \ \ \ &=&\ \ \ +\,p\,\omega' \ \ +\ \ i\, {{ k' }_{ \kern-0.25em x }}\end{eqnarray}$$ (36) (37)

Of particular interest is the energy that is flat after linear moveout (constant with x'). For such energy $\partial / \partial x' = i{{ k' }_{ \kern-0.25em x }} = 0$.Combining (36) and (37) gives the familiar equation  

$$p\ \eq \ {k \over \omega }$$ (38)

EXERCISES:

1. Explain the choice of sign of the s-axis in Figure 11.
2. Equations (30), (31) and (32) are for upgoing Snell waves. What coordinate system would be appropriate for downgoing Snell waves?
3. Express the scalar wave equation in the coordinate system (30), (31) and (32). Neglect first derivatives.
4. Express the dispersion relation of the scalar wave equation in terms of the Fourier variables $( \omega' , {{ k' }_{ \kern-0.25em x }} , {{ k' }_{ \kern-0.25em z }} )$.