From: "James L. Gundersen" jgundie@.......

Date: Fri, 13 Jul 2012 13:59:48 -0700

Hi Randall, Enjoy reading you emails in the PSN group. With my EE background I found the seismologists density usage not very clear for understanding exactly what they defined. Trying to change their minds reminds me of the old joke about " don't confuse me with facts; my mind is made up":-) . It will probably take a generation or two for them to change. I've considered using magnetic levitation for a vertical seismograph but never figured out how to circumvent the relatively high temperature coefficients of the rare earth magnets from limiting my useable measurement resolution and stability. Lately I been looking at using electrostatic force. Good performance (1e-6 m/s^2) looks possible but would be no competition to the long period performance of a good seismograph. Jim On 7/13/2012 9:35 AM, Randall Peters wrote: > > Brett, > > What you mention has been presented to me by the professional > seismologists; however, it 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 > decade, such as 1/7^th (as in the celebrated work of Jon Berger, whose > units are correct at m^2/s^3 /( 1/7^th decade). Unfortunately, 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 plot) or > m^2/s^3/1/7^th 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 normalized) 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 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) = df/f. The log > transformation introduces a new 1/f term that causes the units to > become m^2/s^3/octave. The actual (specific) power spectral density > calculation requires that the single sided FFT spectrum of the > acceleration have every 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 no units, since no math function is capable of recognizing units > (and the log is a math function). > > It is incorrect to say that m^2/s^4/Hz is equivalent 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), you 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 is > equivalent to working with a quantity asd(f)df, where df = 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 differential there is no > rigorously meaningful means for summing (integrating) to obtain the > total power. > > The classic example of the critical importance of recognizing this > issue involves the density 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 of either frequency bin or wavelength bin. > The functions look entirely different because wavelength is inversely > proportional to the frequency, and the differential of 1/f yields > --df/f^2. In the case of seismology functions, the psd representation > in terms of period is just the mirror image 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 functions are > entirely different in appearance when plotted in terms of a linear > abscissa. 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 differential that is always part of a density function. To > do so becomes an exercise in meaninglessness. It is not therefore > permissible to simply replace Hz with 1/s and 'doctor' the units to > yield apparently equivalent results. > > In the event that my discussion above is difficult to follow, I > recommend that you take a serious look at the article. The self > consistency of all its parts has been validated by careful numerical > simulations. A highly regarded seismologist (whom I admire greatly) > has indicated that he found nothing wrong with my paper. > Unfortunately, he did not give me permission 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 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 course of > the Mississippi River." Nevertheless, as I mentioned to him, an event > in the early 1800's, at the New Madrid fault, did just that! > > Randall >Hi Randall,

Enjoy reading you emails in the PSN group.

With my EE background I found the seismologists density usage not very clear for understanding exactly what they defined.

Trying to change their minds reminds me of the old joke about " don't confuse me with facts; my mind is made up" :-) . It will probably take a generation or two for them to change.

I've considered using magnetic levitation for a vertical seismograph but never figured out how to circumvent the relatively high temperature coefficients of the rare earth magnets from limiting my useable measurement resolution and stability. Lately I been looking at using electrostatic force. Good performance (1e-6 m/s^2) looks possible but would be no competition to the long period performance of a good seismograph.

Jim

On 7/13/2012 9:35 AM, Randall Peters wrote:

Brett,

What you mention has been presented to me by the professional seismologists; however, it 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 decade, such as 1/7

^{th}(as in the celebrated work of Jon Berger, whose units are correct at m^2/s^3 /( 1/7^{th}decade). Unfortunately, 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 plot) or m^2/s^3/1/7^{th}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 normalized) 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 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) = df/f. The log transformation introduces a new 1/f term that causes the units to become m^2/s^3/octave. The actual (specific) power spectral density calculation requires that the single sided FFT spectrum of the acceleration have every 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 no units, since no math function is capable of recognizing units (and the log is a math function).

It is incorrect to say that m^2/s^4/Hz is equivalent 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), you 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 is equivalent to working with a quantity asd(f)df, where df = 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 differential there is no rigorously meaningful means for summing (integrating) to obtain the total power.

The classic example of the critical importance of recognizing this issue involves the density 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 of either frequency bin or wavelength bin. The functions look entirely different because wavelength is inversely proportional to the frequency, and the differential of 1/f yields –df/f^2. In the case of seismology functions, the psd representation in terms of period is just the mirror image 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 functions are entirely different in appearance when plotted in terms of a linear abscissa. 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 differential that is always part of a density function. To do so becomes an exercise in meaninglessness. It is not therefore permissible to simply replace Hz with 1/s and ‘doctor’ the units to yield apparently equivalent results.

In the event that my discussion above is difficult to follow, I recommend that you take a serious look at the article. The self consistency of all its parts has been validated by careful numerical simulations. A highly regarded seismologist (whom I admire greatly) has indicated that he found nothing wrong with my paper. Unfortunately, he did not give me permission 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 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 course of the Mississippi River.” Nevertheless, as I mentioned to him, an event in the early 1800’s, at the New Madrid fault, did just that!

Randall