Convolution, the Machine

Part 2, Part 2: The Convolutional Model

Learning objectives

State convolution in one sentence: scale a wavelet copy by each spike and sum
Build a trace by depositing one scaled wavelet per reflection
See that overlapping wavelet copies interfere
Connect the convolutional model to the trace equation d = r * w

What Convolution Actually Does

You now have a reflectivity series from the last section and a wavelet from Part 1. The convolutional model is the rule that turns them into a trace, and despite its reputation it is simple to state:

$d(t) = r(t) * w(t) = \sum_k r_k\, w(t - t_k).$ krk,w(t−tk).

In words: replace every reflectivity spike with a copy of the wavelet, scaled by that spike's coefficient, and add all the copies together. A positive spike deposits an upright wavelet, a negative spike deposits a flipped one, a big spike a tall one. The trace is nothing more than their superposition.

Where Interference Comes From

Sweep the build bar and watch the trace assemble one reflection at a time. As long as the reflections are far apart, each wavelet copy sits in clear space and the trace shows one tidy loop per reflection. But bring two reflections close, as with the pair in the middle, and their copies overlap and add. The trace there is no longer two clean events; it is a single interfered shape whose amplitude and timing depend on the spacing.

That interference is not a flaw in the model; it is the physics of a band-limited wavelet meeting closely spaced reflectors, and it is the reason a thin bed does not look like two spikes. The next section turns that pair into a wedge and reads the interference systematically. Everything downstream in this course, from noise to 2D sections, is built on this one machine: scale, shift, and sum.