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11 Imaging, Two Dimensions

The partial differential equation for modeling, the wave equation, assumes a quiet background as the initial condition. What would happen if the recorded seismic data were introduced into the wave equation as a boundary condition. The data would propagate into the earth from the receiver positions very much like the exploding reflector model process if the reflector was at the surface. If the wave equation is formulated as a backward in time equation, the boundary condition data can be moved down in depth until time reaches zero. This zero time is called the imaging condition, t = 0 .

We shall see that this method works and propagates the surface recorded data back into depth. What is missing is the energy which left the sides and bottom of the image field. It is possible to record along the sides of an image field, but this is not done very often. The bottom data are quite difficult and expensive to obtain. Essentially, the only data routinely available are the surface seismic shot record data, which is stacked to improve the signal to noise ratio. This stacked data can then be migrated. Sounds so simple, but what is the earth velocity model to be used? If we examine Equation 53, we see that the velocity term is essential. We shall assume the velocities are known.