The convolutional model

What convolution actually does

Convolution is the mathematical operation that answers: "if every spike in $r(t)$ is replaced by a wavelet scaled by that spike’s amplitude, what does the sum look like?" In other words:

• Start with the spike train $r(t)$. Each spike sits at a particular time and has a particular height.
• At every spike, draw a copy of the wavelet scaled by that spike’s height. Positive spikes give upright wavelets; negative spikes give flipped wavelets.
• Add all those scaled wavelets together, sample by sample.
• The sum is your seismic trace.

If that sounds abstract, the interactive below makes it concrete. Try each of the five scenarios and watch how the synthetic trace on the right is always just "a wavelet, scaled and shifted and summed, one copy per spike on the left."

Lesson 1: A spike becomes a wavelet

In the Single spike scenario, the trace on the right is just the wavelet itself. This is the foundational observation: on seismic, you do not see reflectors — you see wavelets placed where reflectors are. The peak of the wavelet is where the spike sits; the flanking troughs are the "side effects" of using a band-limited pulse instead of an infinitely-sharp source.

Switch to a 15 Hz wavelet. Notice how much fatter the wavelet is — a lower-frequency pulse smears the reflector out over a larger time window. Now try 50 Hz. Sharper, more localized. The width of the pulse is what limits how precisely we can locate a reflector in time.

Lesson 2: Close reflectors interfere

In the Two close spikes (tuning) scenario, both reflectors have the same polarity. At lower frequencies, the two wavelets overlap so much that they merge into a single fat peak — you cannot tell there are two reflectors underneath. At higher frequencies, the wavelets separate and you can finally see the two events as distinct. This is tuning, and the interplay between bed thickness, wavelet frequency, and visibility is the subject of §1.7. Tuning also makes thin beds brighter than they "should be" at the tuning frequency — an effect that can help or deceive an interpreter depending on whether they recognize it.

Lesson 3: Thin beds have a characteristic signature

The Polarity reversal (thin bed) scenario shows the top of a bed (positive R, where impedance goes up entering the bed) followed closely by the base of the bed (negative R, where impedance goes down exiting). Watch what happens on the synthetic: a "peak above trough" pair, with the pair’s appearance changing dramatically with frequency. Below a critical thickness the pair merges into a single peak or trough whose polarity depends on the dominant effect — the top-of-bed pulse and the base-of-bed pulse cancel in the middle.

The tell-tale peak-over-trough signature is how interpreters recognize a thin bed on seismic. The absence of a clear base-of-bed event below a top-of-bed event is how you know the bed is thinner than seismic can resolve at this frequency — even though you still know where it starts.

Bandwidth sets resolution

The width of the wavelet (in time) is inversely related to the width of its frequency spectrum (in Hz). A narrow-band 20 Hz wavelet is fat and blurs events; a broadband 5–80 Hz wavelet is narrow and crisp. Everything that seismic processing does to "broaden bandwidth" (deconvolution, Q-compensation, zero-phasing) is ultimately trying to make the wavelet sharper so the synthetic trace looks more like the underlying spike train.

Formally, the resolution limit is on the order of $\lambda/4$, where $\lambda$ is the wavelength at the dominant frequency. For a 30 Hz wavelet at 3000 m/s velocity, that is 25 m vertical resolution — two reflectors closer than that blend into one event on the trace.

The inverse problem is deconvolution

If we have a seismic trace $s(t)$ and we know the wavelet $w(t)$, can we recover the reflectivity $r(t)$? In principle yes: it is called deconvolution (or seismic inversion when we want the impedance directly rather than reflectivity). In practice the operation is unstable and imperfect because we never know the wavelet exactly, and noise amplifies during deconvolution. Parts 6 and 7 return to this.

What the convolutional model assumes — and when it breaks

• 1D earth. It treats each trace independently, as if the earth were a stack of horizontal layers under that trace. In reality, events dip, diffract, and scatter — problems that migration (§1.5) solves.
• Stationary wavelet. It assumes the wavelet is the same at every depth. In reality the wavelet changes as it travels — it loses high frequencies (attenuation, or Q-effect), and the phase can shift. Q-compensation in processing tries to undo this.
• Linearity. It ignores multiples (reflections off already-reflected events) and mode conversions (P to S at non-normal incidence). For most stacked seismic these are second-order; for AVO work (Part 5) we care about mode conversions explicitly.
• Small reflectivity. The derivation requires |R| « 1, valid for typical sedimentary contrasts but breaking down at huge boundaries like the top of salt, where multiples become loud and "transparency" gets questionable.

The assumptions are violated all the time, but the convolutional model remains the best first-order description of what a seismic trace is. It is the frame on which everything in Part 2 and beyond hangs.