Convolution vs the Wave Equation

The Whole Course in One Figure

This closer answers, by experiment, the question the course opened with: when do you need the wave equation, and when is convolution enough? Take one earth and image it both ways, the cheap convolutional section beside the full finite-difference section, then take their difference and overlay the two traces at any column you like.

The Physics Is in the Residual

On the flat primaries the two engines put the same events in the same places, so the difference panel stays quiet there; the engines' wavelets and arrival details differ slightly, so quiet is not perfectly blank. It blazes where the wave equation adds what convolution cannot: the diffraction off the fault tip, the bowtie under the syncline, all the events Part 5 has been reintroducing. Drag the probe and the two traces line up on the primaries, then part company completely where those extra events live. The convolution trace is not wrong on the primaries; it is simply silent on everything else.

That difference panel is fit for purpose made literal. It is not a slogan or a rule of thumb; it is a measurement. If nothing your task needs shows in the difference, well ties, tuning, structural training data, then convolution was the correct, cheap engine and the wave equation would only have cost you compute. If the difference is where your answer lives, diffractions, multiples, subtle imaging, then only the wave equation reaches it and convolution would have quietly misled you. You have now built both engines from first principles, validated them, and seen precisely what separates them. Choosing between them with your eyes open is the entire craft of synthetic seismic modelling, and from here the course turns to the physics of amplitude, anisotropy, and the Modeling Lab where you put it all to work.