The Same Earth, Different Engines

One Model, Two Engines

Everything in Part 0 comes together here. We take a single earth model and image it two ways. The first engine is convolution: build each trace on its own by placing the reflectivity under a wavelet, with no communication between neighbouring traces. The second is the wave equation, solved by finite difference as an exploding-reflector simulation, so the wavefield actually propagates across the whole model. The third panel subtracts one from the other. It holds everything the wave equation produces that convolution structurally cannot.

Reading the Difference

Choose the faulted layer and the difference lights up with a diffraction: a hyperbola fanning out from the fault edge. Convolution truncates the reflector and stops there; a real wavefield lets the sharp edge scatter energy sideways, and migration is the process that later collapses those hyperbolas back to the edge. Choose the syncline and the wave equation focuses the field into a bowtie, imaging the same trough three times at its centre, an effect no trace-by-trace method can make. The pinchout adds a tip diffraction where the wedge closes. Switch on a free surface and surface multiples join the difference, energy bounced back down off the top and repeated at multiples of its time.

The Point of Part 0

None of this makes convolution wrong. For a well tie, a tuning study, or a bulk library of training sections, the flat primaries are the whole answer and the difference panel is a distraction you happily skip. But when the question is a diffraction, a subtle syncline, or a multiple, that difference panel is the answer, and only the wave equation reaches it. That is fit for purpose stated as an image. You have now seen the forward problem, the ladder of engines, the jobs synthetics do, the inverse crime that keeps them honest, and exactly what the cheapest engine leaves on the table. Part 1 starts building the ladder for real, beginning with the wavelet.