What is noise ?

The concept of “noise” plays a crucial role in the statistical analysis of data. As an example of a noisy record consider Figure 1 that shows the ground motion of the seismological station NE51 in St. Petersburg after an earthquake in Egypt. (In earthquake seismology, periods may be orders of magnitude larger than in exploration seismology, but the principles are the same.) This time series shows no distinct arrivals or other apparent signatures of an organized nature. Given the proximity of the recording station to a major population center and to the coast, such a noisy record does not seem to be very surprising. But what is noise exactly? In the context of seismic prospecting, Dobrin and Savit (1988) define noise as “spurious seismic signals from ground motion not associated with reflections.” They have in mind such things as surface waves, near-surface reverberations and so on; coherent but uninteresting signal in other words. Fair enough. One might dispute the use of the term noise here, but these authors are certainly within their rights to identify certain signal as being uninteresting. But they go on to speak of “incoherent noise, sometimes referred to as random noise ... usually associated with scattering from near-surface irregularities.” (Emphasis in the original.) By identifying random noise with incoherency they have sailed into rough waters. For although the signal associated with scattering from near-surface irregularities may well be incoherent (though that is debatable), it is clearly reproducible, so does it make sense to call it random? And further, there is no law that says that random processes must be uncorrelated. (Just take an uncorrelated “white” process and apply a smoothing operator to it.) It turns out to be extraordinarily difficult to give a precise mathematical definition of randomness, so we won’t try. (A brief perusal of randomness in Volume 2 of Knuth’s great The Art of Computer Programming is edifying and frustrating in equal measures.) In any case, it is more satisfying undoubtedly to think in terms of observations of physical experiments. Here is Parzen’s (1960) definition, which is as good as any: