## PSN-L Email List Message

Subject: psd units
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=
"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
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-->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 1/7th (as in the celebrated work of Jon Berger, w=
hose units are correct at m^2/s^3 /( 1/7th 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 1/7th 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=
```