From: Randall Peters PETERS_RD@..........

Date: Fri, 13 Jul 2012 12:35:31 -0400

Brett, What you mention has been presented to me by the professional seismo= logists; however, it is not correct, for the following reason: Any density function always has a 'per' quantity associated with it-for exa= mple per Hz or per octave or per some fraction of a decade, such as 1/7th (= as in the celebrated work of Jon Berger, whose units are correct at m^2/s^3= /( 1/7th decade). Unfortunately, he does not in his earth background nois= e papers directly specify the differential that I've shown in parentheses. = One must look in the body of his paper(s) to discover that his bin width i= s 1/7th decade. Incidently the proper set is either m^2/s^3/Hz (linear plo= t) or m^2/s^3/1/7th decade (log plot). It is worth noting that a plot of asd in terms of a log abscissa of = frequency (or period) gives the correct shape (though not properly normaliz= ed) as what is obtained in terms of a proper accounting of the power. The = reason can be readily understood on the basis of nothing other than the mat= h; i.e., let asd(f)df be the acceleration spectral amount in the range from= f to f+df. To plot in log form we must deal formally with asd(ln f)d(ln f= ). The integral over the log form must yield the same result as the integr= al over the linear form, asd(f)df. But notice that d (ln f) =3D df/f. The= log transformation introduces a new 1/f term that causes the units to beco= me m^2/s^3/octave. The actual (specific) power spectral density calculati= on requires that the single sided FFT spectrum of the acceleration have eve= ry component divided by its associated frequency. That is where the change= from s^4 to s^3 comes from-not from a replacement of Hz with 1/s. There is indeed a profound difference between per Hz and per octave-as has = been addressed in considerable detail in the tutorial that I referenced. The difference derives from the fact that Hz has units 1/s and octave has n= o units, since no math function is capable of recognizing units (and the lo= g is a math function). It is incorrect to say that m^2/s^4/Hz is equivalent to m^2/s^3. To r= emove the Hz from the equivalent to g^2/Hz by replacing Hz with 1/s yields = a function that is no longer a density. As the highly esteemed professor R= ichard Present of Tennessee Physics that I had as a student used to greatly= admonish us: "Any time you work with a density function, such as p(x), you should descri= be the quantity represented by p (having variability with x) as the amount = of p that is found in the range from x to x+dx; given by p(x)dx. To repr= esent asd (f) as having units of m^2/s^4/Hz is equivalent to working with a= quantity asd(f)df, where df =3D 1 Hz. If you remove the df-which is what = you have suggested, the density function has been tampered with (gutted to = yield something no longer having a density meaning). Without the different= ial there is no rigorously meaningful means for summing (integrating) to ob= tain the total power. The classic example of the critical importance of recognizing this issu= e involves the density function derived by Einstein to explain the quantum = nature of blackbody radiation. It is possible to express this Planck distr= ibution, which describes radiated energy density within a given bin width-i= n terms of either frequency bin or wavelength bin. The functions look enti= rely different because wavelength is inversely proportional to the frequenc= y, and the differential of 1/f yields -df/f^2. In the case of seismology f= unctions, the psd representation in terms of period is just the mirror imag= e of the representation in terms of frequency-but only when the abscissa is= log scaled. The frequency form and the period form of the density functio= ns are entirely different in appearance when plotted in terms of a linear a= bscissa. This also is discussed in detail in the paper. And so to repeat= , there is no math self-consistency to be realized, if one takes away the d= ifferential that is always part of a density function. To do so becomes an= exercise in meaninglessness. It is not therefore permissible to simply re= place Hz with 1/s and 'doctor' the units to yield apparently equivalent res= ults. In the event that my discussion above is difficult to follow, I reco= mmend that you take a serious look at the article. The self consistency of= all its parts has been validated by careful numerical simulations. A high= ly regarded seismologist (whom I admire greatly) has indicated that he foun= d nothing wrong with my paper. Unfortunately, he did not give me permissio= n to quote him. The tradition of erroneously associating 'power' with m^2/= s^4/Hz, as though it represents an actual physical (specific) power is at l= east three decades old. One engineer with whom I spoke about the matter sa= id, "even if you are absolutely correct, Peters, trying to change such a lo= ng held tradition will be like trying to change the course of the Mississip= pi River." Nevertheless, as I mentioned to him, an event in the early 1800= 's, at the New Madrid fault, did just that! Randall=Brett,

What you men= tion has been presented to me by the professional seismologists; however, i= t is not correct, for the following reason:

Any density function always has a ‘per’ quantity associated= with it—for example per Hz or per octave or per some fraction of a d= ecade, such as 1/7

^{th}(as in the celebrated work of Jon Berger, w= hose units are correct at m^2/s^3 /( 1/7^{th}decade). Unfort= unately, he does not in his earth background noise papers directly specify = the differential that I’ve shown in parentheses. One must look = in the body of his paper(s) to discover that his bin width is 1/7^{th decade. Incidently the proper set is either m^2/s^3/Hz (linear pl= ot) or m^2/s^3/1/7th decade (log plot). }It is worth not= ing that a plot of asd in terms of a log abscissa of frequency (or period) = gives the correct shape (though not properly normalized) as what is obtaine= d in terms of a proper accounting of the power. The reason can be rea= dily understood on the basis of nothing other than the math; i.e., let asd(= f)df be the acceleration spectral amount in the range from f to f+df. = To plot in log form we must deal formally with asd(ln f)d(ln f). The= integral over the log form must yield the same result as the integral over= the linear form, asd(f)df. But notice that d (ln f) =3D df/f. = The log transformation introduces a new 1/f term that causes the units to b= ecome m^2/s^3/octave. The actual (specific) power spectral dens= ity calculation requires that the single sided FFT spectrum of the accelera= tion have every component divided by its associated frequency. That i= s where the change from s^4 to s^3 comes from—not from a replacement = of Hz with 1/s.

There is indeed a= profound difference between per Hz and per octave—as has been addres= sed in considerable detail in the tutorial that I referenced.

The difference derives from the fact that Hz has unit= s 1/s and octave has no units, since no math function is capable of recogni= zing units (and the log is a math function).

It is incorrect to say that m^2/s^4/Hz is equ= ivalent to m^2/s^3. To remove the Hz from the equivalent to g^2/Hz by= replacing Hz with 1/s yields a function that is no longer a density. = As the highly esteemed professor Richard Present of Tennessee Physics that= I had as a student used to greatly admonish us:

“Any time you work with a density function, such as p(x), yo= u should describe the quantity represented by p (having variability with x)= as the amount of p that is found in the range from x to x+dx; given = by p(x)dx. To represent asd (f) as having units of m^2/s^4/Hz i= s equivalent to working with a quantity asd(f)df, where df =3D 1 Hz. = If you remove the df—which is what you have suggested, the density fu= nction has been tampered with (gutted to yield something no longer having a= density meaning). Without the differential there is no rigorously me= aningful means for summing (integrating) to obtain the total power. <= o:p>

The classic exam= ple of the critical importance of recognizing this issue involves the densi= ty function derived by Einstein to explain the quantum nature of blackbody = radiation. It is possible to express this Planck distribution, which = describes radiated energy density within a given bin width—in terms o= f either frequency bin or wavelength bin. The functions look entirely= different because wavelength is inversely proportional to the frequency, a= nd the differential of 1/f yields –df/f^2. In the case of seism= ology functions, the psd representation in terms of period is just the mirr= or image of the representation in terms of frequency—but only when th= e abscissa is log scaled. The frequency form and the period form of t= he density functions are entirely different in appearance when plotted in t= erms of a linear abscissa. This also is discussed in detail in the pa= per. And so to repeat, there is no math self-consistency to be = realized, if one takes away the differential that is always part of a densi= ty function. To do so becomes an exercise in meaninglessness. I= t is not therefore permissible to simply replace Hz with 1/s and ‘doc= tor’ the units to yield apparently equivalent results.

=In the event that= my discussion above is difficult to follow, I recommend that you take a se= rious look at the article. The self consistency of all its parts has = been validated by careful numerical simulations. A highly regarded se= ismologist (whom I admire greatly) has indicated that he found nothing wron= g with my paper. Unfortunately, he did not give me permission to quot= e him. The tradition of erroneously associating ‘power’ w= ith m^2/s^4/Hz, as though it represents an actual physical (specific) power= is at least three decades old. One engineer with whom I spoke about = the matter said, “even if you are absolutely correct, Peters, trying = to change such a long held tradition will be like trying to change the cour= se of the Mississippi River.” Nevertheless, as I mentioned to h= im, an event in the early 1800’s, at the New Madrid fault, did just t= hat!

= ; Randall