Continuation of a dipping plane wave.

Next consider a plane wave dipping at some angle $\theta$.It is natural to imagine continuing such a wave back along a ray. Instead, we will continue the wave straight down. This requires the assumption that the plane wave is a perfect one, namely that the same waveform is observed at all x. Imagine two sensors in a vertical well bore. They should record the same signal except for a time shift that depends on the angle of the wave. Notice that the arrival time difference between sensors at two different depths is greatest for vertically propagating waves, and the time difference drops to zero for horizontally propagating waves. So the time shift $\Delta t$ is $v^{-1} \cos\theta\,\Delta z$where $\theta$ is the angle between the wavefront and the earth's surface (or the angle between the well bore and the ray). Thus an equation to downward continue the wave is

$$\begin{eqnarray} U( \omega , \theta ,z+\Delta z) &=& U( \omega , \theta ,z) \ ... ...p \left( \, -i \omega \, {\Delta z \cos\theta \over v}\ \right)\end{eqnarray}$$ (5) (6)

Equation (6) is a downward continuation formula for any angle $\theta$.Following methods of chapter  we can generalize the method to media where the velocity is a function of depth. Evidently we can apply equation (6) for each layer of thickness $\Delta z$,and allow the velocity vary with z. This is a well known approximation that handles the timing correctly but keeps the amplitude constant (since $\vert e^{i\phi}\vert=1$)when in real life, the amplitude should vary because of reflection and transmission coefficients. Suffice it to say that in practical earth imaging, this approximation is almost universally satisfactory.

In a stratified earth, it is customary to eliminate the angle $\theta$ which is depth variable, and change it to the Snell's parameter p which is constant for all depths. Thus the downward continuation equation for any Snell's parameter is  

$$U( \omega , p,z+\Delta z) \quad =\quad U( \omega , p,z) \ ... ...-\ {i \omega \Delta z \over v(z) } \ \sqrt{1-p^2v(z)^2} \right)$$ (7)

It is natural to wonder where in real life we would encounter a Snell wave that we could downward continue with equation (7). The answer is that any wave from real life can be regarded as a sum of waves propagating in all angles. Thus a field data set should first be decomposed into Snell waves of all values of p, and then equation (7) can be used to downward continue each p, and finally the components for each p could be added. This process akin to Fourier analysis. We now turn to Fourier analysis as a method of downward continuation which is the same idea but the task of decomposing data into Snell waves becomes the task of decomposing data into sinusoids along the x-axis.