At the heart of the migration process is the operation of downward continuing data. Given the input data on the plane of the earth's surface z=0, we must manufacture the data that could be recorded at depth z. This is most easily done in the Fourier domain. Mathematically shifting on the z-axis is like shifting on the time t-axis. To shift on time by an amount t_{0} we multiply by and to shift on depth by an amount z_{0} we multiply by .It is not quite this trivial however: the data field is not evidently a function of k_{z} because we do not have data given on a z-axis so we cannot simply Fourier transform it to k_{z}. Instead we have
(9) |
Downward continuation is a product relationship in both the -domain and the k_{x}-domain. What does the filter look like in the time and space domain? It turns out like a cone, that is, it is roughly an impulse function of .More precisely, it is the Huygens secondary wave source that was exemplified by ocean waves entering a gap through a storm barrier. Adding up the response of multiple gaps in the barrier would be convolution over x. Superposing many incident ocean waves would be convolution over t.
Now let us see why the downward continuation filter has the mathematical form stated. Every point in the -plane refers to a sinusoidal plane wave. The variation with depth will also be sinusoidal, namely .The value of k_{z} for the plane wave is found simply by solving equation (8):
(10) | ||
(11) | ||
(12) |
Mathematically we can imagine an arbitrary function of t, x, and z. Its three-dimensional Fourier transform fills a volume in the space of , k_{x}, and k_{z}. This arbitrary function is not a wavefield unless it vanishes everywhere except the cone-shaped surface given by the wave equation. Thus equation (9) makes sense even though we cannot Fourier transform data over the z-axis because we don't observe data along the z-axis. It makes sense because if we know the mathematical function is a wavefield and if we know its Fourier transform over and k_{x}, then we know its Fourier transform at any k_{z} and we can ``delay'' it in z (downward shift) by multiplying by .
The input-output filter, being of the form , appears to be a phase-shifting filter with no amplitude scaling. This bodes well for our plans to deconvolve. It means that signal-to-noise power considerations will be much less relevant for migration than for ordinary filtering.
(13) |
(14) |