The Ricker Wavelet

The Default Seismic Wavelet

Every convolutional synthetic needs a wavelet, the shape the source and the earth stamp onto each reflection. The workhorse is the Ricker wavelet: a single central peak with a negative side lobe on each side, mathematically the second derivative of a Gaussian. It is popular for one reason above all: it has a single knob. In time,

$w(t) = \left(1 - 2\pi^2 f^2 t^2\right)\, e^{-\pi^2 f^2 t^2}$

where $f$ is the peak frequency, the frequency at which its amplitude spectrum is largest. That one number sets everything about the wavelet.

Two Faces of One Knob

A wavelet has two equivalent descriptions: its shape in time and its amplitude spectrum in frequency, connected by the Fourier transform. The widget shows both, and the point is what happens when you turn the single knob.

Raise the peak frequency and the wavelet narrows in time while its spectrum widens and shifts toward higher frequencies. This is the time-bandwidth tradeoff, and it is not a quirk of the Ricker but a property of the Fourier transform itself: you cannot be both compact in time and narrow in band. A signal short in time must be broad in frequency.

Why This Sets Resolution

Seismic resolution, the ability to see two close reflectors as two events rather than one blur, is set by the wavelet width in time, and therefore by the bandwidth. A higher peak frequency and a wider band give a shorter wavelet and thinner resolvable beds. That is why acquisition and processing fight so hard for bandwidth, and it is the thread from this wavelet to the tuning and resolution sections ahead. The next section widens the toolbox beyond the Ricker to the other wavelets you will meet in real data.