The convolutional model

What convolution actually does

Convolution answers one question: "if every spike in $r(t)$ is replaced by a copy of the wavelet scaled by that spike’s height, what does the sum look like?" In steps:

1. Start with the spike train $r(t)$. Each spike sits at the two-way time of an interface, and its height is the reflection coefficient $R = (Z_2 - Z_1)/(Z_2 + Z_1)$ of Section 1.1.
2. At every spike, place a copy of the wavelet scaled by that spike’s height. Positive spikes give upright copies; negative spikes give inverted copies.
3. Add all the copies together, sample by sample.
4. The sum is your seismic trace.

Figure 1.2 builds a trace this way from an earth. Pick an earth in its model table, from one interface to a gradational boundary, and change the wavelet’s peak frequency $f_{\mathrm p}$, the spacing $d$ of the interfaces and the noise. Watch each spike in (b) become a scaled copy of the wavelet (c) in (d), and read the tuning thickness, the bed thickness at which the top and base reflections interfere most strongly, in the table.

Lesson 1: A spike becomes a wavelet

One interface, $R$ = +0.091, becomes one copy of the wavelet in (d): a peak of 0.091 at 200 ms between two troughs, 26.0 ms from trough to trough. This is the foundational observation: on seismic, you do not see reflectors, you see wavelets placed where reflectors are. The peak marks the interface; the troughs on either side are the price of a band-limited pulse instead of an infinitely sharp one.

Lower $f_{\mathrm p}$ to 12 Hz (exercise 1). The same interface now becomes a pulse 65.0 ms from trough to trough, and the tuning thickness grows from 13.0 to 32.5 ms. The interface has not moved; only the pulse that marks it has grown, and with it the smallest spacing the trace can separate.

Lesson 2: Close interfaces interfere

Choose two interfaces of the same sign and close them from 40 ms (exercise 3). At 40 ms the trace shows two peaks. Near 26 ms their side lobes add into a trough between them where there is no interface at all. Near 14 ms each copy’s trough lands on the other’s peak, and the two peaks read at only 57 % of either interface alone. By 6 ms the copies have merged into one peak 1.6 times as tall as either: the trace no longer shows that there are two interfaces. This interplay of spacing, wavelet and visibility is tuning, the subject of Section 1.7.

Lesson 3: Thin beds brighten, then fade

The Thin bed earth is a hard bed in shale: its top is a positive coefficient (impedance rises into the bed) and its base an equal negative one. Thick, it gives a peak over a trough, each at its own amplitude. Close it to 13 ms at 30 Hz (exercise 2) and the top’s peak and the base’s trough reinforce each other’s side lobes: the bed reads 1.4 times brighter than either reflection alone. Thinner than that it fades, to 85 % of one interface at 5 ms and 18 % at 1 ms, because top and base cancel.

Below the tuning thickness the peak and trough also stop moving apart as the bed thins: at 30 Hz they stay about 11 ms apart whether the bed is 5 ms or 1 ms thick. The time between them is set by the wavelet, not by the bed. So an interpreter cannot read the thickness of a thin bed from its peak-to-trough time; its amplitude is what still changes with thickness.

Lesson 4: Seismic sees contrasts, not trends

The Eight layers earth is a sand-shale sequence with beds 9 to 22 ms thick. At 30 Hz the trace has eight lobes, but their heights are made by interference: the largest is 1.4 times the strongest interface alone. At 10 Hz only four lobes are left. A layered interval appears on seismic as an interference pattern, not a picture of its layers.

The Gradational earth has the same total contrast as the single interface, spread over a ramp. Thicken the ramp to 40 ms (exercise 4) and the trace peak falls to 13 % of a sharp step’s. Plate (e) shows why: in frequency, convolution is multiplication, $|S| = |R|\,|W|$, and the ramp’s reflectivity $|R|$ has little left at the frequencies where the wavelet’s $|W|$ lives. A gradual change in impedance is nearly invisible; an abrupt one of the same size is not.

Bandwidth sets resolution

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

For a Ricker wavelet the tuning thickness is half its trough-to-trough length, $\sqrt{6}/(2\pi f_{\mathrm p})$, about $1/(2.6 f_{\mathrm p})$ in two-way time. That is the familiar $\lambda/4$, with $\lambda$ the wavelength at the wavelet’s dominant frequency, about $1.3 f_{\mathrm p}$. For a 30 Hz Ricker at 3000 m/s: 13.0 ms two-way, or 19.5 m of rock. Closer than that, two interfaces can no longer be told apart; their reflections begin to interfere at about twice that spacing.

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: that is deconvolution (or seismic inversion, when we want the impedance itself rather than the reflectivity). In practice it is unstable and imperfect, because we never know the wavelet exactly, and noise is amplified at the frequencies where the wavelet is weak. Section 1.3 places deconvolution in the processing sequence, and Part 7 returns to inversion.

What the convolutional model assumes, and when it breaks

• 1D earth. It treats each trace independently, as if the earth were a stack of flat layers under that trace. In reality events dip, diffract and scatter, which migration (Section 1.4) corrects.
• Stationary wavelet. It assumes the wavelet is the same at every time. In reality the wavelet changes as it travels: it loses high frequencies (attenuation, the Q effect) and its phase can shift. Q-compensation in processing tries to undo this.
• Primaries only. It ignores multiples (reflections of already reflected energy) 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 the angle dependence explicitly.
• Small reflectivity. It needs $|R| \ll 1$, so that transmission losses and multiples can be neglected. That holds for typical sedimentary contrasts (every coefficient in Figure 1.2 is 0.11 or less) and fails at strong boundaries such as the top of salt or a hard seafloor.
• Additive noise. It treats noise as something added to the trace. Raise the noise in Figure 1.2 (exercise 5) and weak interference lobes become indistinguishable from it well before the strongest event does.

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.