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Imaging - Principle of wave field extrapolation
Imagine a box, like the one to the left. This box contains the reflector surfaces of the Earth interior, and also contains the waves, going through the media. The box has a horizontal x coordinate and a vertical z coordinate. The vertical x-z plain represents the earth interior, as a vertical slice below the x line.
The other horizontal coordinate is the time. Our shot record is the top cover slice of the box. It was recorded at z=0, for all x and for all t.
If we cut an x-z slice out of this box at different times, we will find the same reflectors. The Earth interior is independent of our recording time. Of course, looking the waves at different time slices, we will find different picture, because the waves are moving by the time. Our task is to investigate the possible reflector surfaces.
One shot record represents the data of one physical experiment. One shot record contains the data of some kind of waves. Therefore a shot record is fully described by the wave equation.
At this point you don't have to bother yourself with the mathematics of differential equations; we will come to it later on. Also, there are several versions of the wave equation: acoustic, elastic, skalar, paraxial, ... etc. Leave them now, you will see them enough times reading the literature. You should understand: seismic data processing is an industrial tool. It is based on solid, simple, understandable ideas. Usually the fancy mathematical background only follows the practical solutions.
Some well educated people used to ask at this moment the crucial question: "Did you find the solution of your wave equation ?"
The fact is: the wave equation has milliards of solutions. Any function, what you can derivate at least two times; might be a solution to this quadratic differential equation. For example, our shot record is a good candidate. Our shot record is a function of the x horizontal coordinate and the t time. The z vertical coordinate is fixed at zero.
Our shot record is a frozen image of the total solution at x = receiver positions, z = 0 (the surface) and t = the recorded time.
We can write it as P = P(x, z=0, t), where x = x1, x2 ... xN and t = 0, dt, 2*dt, ... M*dt...
Suppose, we would like to see what kind of reflector surfaces (layer boundaries) are at dz depth.
A clever idea is: dig down to dz depth and place the receivers there. This way we obtain a series of frozen wave images: P(x, z=dz, t), P(x, z=2dz, t) ... etc.
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