Keep probability in view

Lest men suspect your tale untrue,

GEOPHYSICS,VOL.63,NO.4(JULY-AUGUST 1998);P .1122–1124,2FIGS.

What is noise?

John A.Scales ?and Roel Snieder ?

Lest men suspect your tale untrue,Keep probability in view

—John Gay

The concept of “noise”plays a crucial role in the statisti-cal analysis of data.As an example of a noisy record consider Figure 1that shows the ground motion of the seismological station NE51in St.Petersburg after an earthquake in Egypt.(In earthquake seismology,periods may be orders of magni-tude 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 cen-ter 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 prospect-ing,Dobrin and Savit (1988)de?ne noise as “spurious seismic signals from ground motion not associated with re?ections.”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 re-ferred to as random noise http://ually 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 scat-tering 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 dif?cult to give a precise mathematical de?nition of randomness,so we won’t try.(A brief perusal of randomness in Volume 2of 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)de?nition,which is as good as any:

?

Dept.of Geophysics and Center for Wave Phenomena,Colorado School of Mines,Golden,CO 80401.E-mail:jscales@dix.mines.edu.?Dept.of Geophysics,Utrecht University,P .O.Box 80.021,3508TA Utrecht,The Netherlands.E-mail:snieder@geof.ruu.nl.c 1998Society of Exploration Geophysicists.All rights reserved.

A random (or chance)phenomenon is an empirical phenomenon characterized by the property that its observation under a given set of circumstances does not always lead to the same observed outcomes (so that there is no deterministic regularity)but rather to different outcomes in such a way that there is statistical regularity.By this is meant that numbers exist between 0and 1that represent the relative fre-quency with which the different possible outcomes may be observed in a series of observations of in-dependent occurrences of the phenomenon....A random event is one whose relative frequency of oc-currence,in a very long sequence of observations of randomly selected situations in which the event may occur,approaches a stable limit value as the num-ber of observations is increased to in?nity;the limit value of the relative frequency is called the pro-bability of the random event.

It is precisely this lack of deterministic reproducibility that al-lows us to reduce random noise by averaging over many repeti-tions of the http://ing this de?nition,the “incoherent noise”of Dobrin and Savit (1988)is not random.1

But why should we care about the de?nition of noise?As geophysicists,the data at our disposal will always contain some features that we will not bother to explain.If we accepted our data as being absolutely precise and reproducible,then no model whose response disagreed with the observations even to the slightest degree could be correct.But we don’t believe that our data are exact and exactly reproducible.And further,because we cannot calculate the exact response of our Earth models (because we cannot afford to put all the physics on the computer)and because we have only approximate mod-els anyway (we cannot use an in?nite number of parameters),there are likely to be deterministic aspects of the data that we

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The term “coherency spectrum”was coined by Wiener to denote the absolute value of the cross covariance of two signals pided by the square root of the product of the respective autocovariance functions (cf.Priestley,1981,661).If two stationary processes are uncorrelated,for example,if they are independent,then the coherency spectrum is zero at all frequencies.

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Lest men suspect your tale untrue,

What Is Noise?

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cannot or do not want to explain.Keep in mind,however,that with enough degrees of freedom one can ?t any data,even if it’s not worth ?tting.And the resulting model might be excessively complicated or physically unreasonable.

In fact,in many situations “noise”is highly reproducible be-tween different experiments and corresponds therefore to a deterministic process.For example,let us return to the seismo-gram of Figure 1.In Figure 2the same seismogram is shown (on the same scale)but now the signal before the ?rst arriving P -wave around 400s is shown as well.The signal before the P -wave consists purely of ambient noise.It can be seen that this noise level is negligible compared to the later parts of the signal.This means that the signal shown in Figure 1is Earth response that corresponds to a multitude of different arrivals rather than random noise.Some of these arrivals can be ex-plained by a s …… 此处隐藏：10197字，全部文档内容请下载后查看。喜欢就下载吧 ……