Convolution from scratch

A seismic trace is not what the earth reflects; it is what the earth reflects convolved with the wavelet your source emitted. Every processing step that tries to recover the earth — deconvolution, inversion, migration — has to undo this convolution. So we start by understanding what convolution actually does.

1. The convolutional model

where

• s(t) is the recorded seismic trace,
• w(t) is the source wavelet (plus receiver response, coupling, everything band-limited by acquisition),
• r(t) is the earth’s reflectivity — a spike train at every impedance contrast,
• n(t) is noise (ambient + coherent),
• the ∗ symbol is convolution.

This one line is the forward model for almost everything we do. Deconvolution is the inverse problem: given s, recover r.

2. Discrete convolution — the recipe

In the digital world, signals are sequences. The discrete convolution of an input x[n] with a kernel h[n] is defined as

That looks intimidating. In practice it is a four-step mechanical procedure you can always fall back on:

• Flip the kernel: h[k] becomes h[−k].
• Slide the flipped kernel to position τ.
• Multiply the flipped-and-shifted kernel, sample by sample, against x.
• Sum the products. That number is the output at lag τ.

Advance τ by one sample, repeat. The whole output trace is built one sample at a time. The widget below shows this, mechanical step by mechanical step.

3. What to notice in the widget

• Single spike: the output is simply a copy of the wavelet, centered on where the spike was. Convolution with a unit impulse returns the kernel itself — this is why the kernel is called the impulse response.
• Two close spikes (tuning): the two wavelet copies overlap. Depending on spacing and polarity, they add or cancel. This is the origin of tuning in thin-bed seismic interpretation.
• Dipole (+/−): an opposing-sign pair — a thin layer — produces a tuned doublet response. When the layer is thinner than a quarter wavelength, the two wavelets interfere destructively and the reservoir appears weaker than it is.
• Spike train (reflectivity): a full earth-like reflectivity. Notice that in the output, every spike has been smeared into the shape of the wavelet. Deconvolution is trying to undo this smearing.

4. Properties you will use constantly

Three properties make convolution pleasant to reason about:

• Commutative: x ∗ h = h ∗ x. You can think of either signal as “the kernel.”
• Associative: (x ∗ h₁) ∗ h₂ = x ∗ (h₁ ∗ h₂). Two filters in series are equivalent to one combined filter — the basis of wavelet estimation by cascading known operators.
• Distributive: x ∗ (h₁ + h₂) = x ∗ h₁ + x ∗ h₂. Linearity. A sum of reflectivities convolves independently.

5. Length arithmetic

If x has length N and h has length M, then x ∗ h has length N + M − 1. The wavelet smears each sample of the input across M output samples. This matters when you design filters: a long operator produces long edge effects (ringing, edge tapering).

6. Special cases worth memorizing

• Convolution with a delta δ[n] is the identity: x ∗ δ = x.
• Convolution with a shifted delta δ[n − k] shifts: x ∗ δ(⋅ − k) = x(⋅ − k).
• Convolution with a rectangular pulse of length L is a moving average of width L.
• Convolution in the time domain is multiplication in the frequency domain (proved in §0.4). This is the single fact that makes FFT-based processing fast enough to be practical.

Where this goes next

Section §0.4 proves that this “smearing” operation has a staggeringly simpler form once we pass to the frequency domain: convolution in time becomes multiplication in frequency. That one theorem is why the FFT sits at the heart of virtually every processing flow.

References

• Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill.
• Oppenheim, A. V., Schafer, R. W. (2009). Discrete-Time Signal Processing (3rd ed.). Prentice Hall.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.