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Randall D. Peters

Physics Department, Mercer University

Macon, Georgia

Copyright March 2008

The units of acceleration spectral density (ASD), commonly employed by
the seismic
community, are m^{2}/s^{4}/Hz. It is here shown that a more meaningful
(specific) power spectral density (PSD) is one having units m^{2}/s^{3}/Hz.
The PSD can be simply generated from the ASD by dividing every spectral
component by its associated frequency.

**PSD Physics**

The actual PSD in watts per Hz is rarely explicitly considered in science or engineering applications; rather a `specific' power is often employed. For example, in electrical engineering (a classic example in most every tutorial on the subject) the quantity labeled `power' is really power per unit conductance- with conductance being the reciprocal of resistance. We see this from the expression for the power, based on Ohm's law:

| (1) |

Since density always involves a `per' something, in the case of a PSD
graphed with a linear frequency scale this
is per Hz; i.e., W/Hz. It must be noted, however, that the commonly
plotted electrical PSD is really a
*specific* spectral power having units of watts/mho/Hz, because
the value of R is not used during the calculation.
The unit of conductance (mho) is universally omitted from the
specification but is recognized by everyone as being present by
inference only. The reason the result is a specific power is because
the modulus squared of the FFT values of the voltage delivered to the
load is what gets plotted versus frequency.

Note: For present purposes, the specific power, as opposed to the actual PSD (identified with upper case letters) will be labeled using lower cases letters; i.e., psd. In the material that follows, for purpose of demonstrating the difficulties encountered in the seismology case, a subscript s for seismology will also be employed to distinguish from the electrical subscript e.

With this convention, to calculate the total power P dissipated by a
load resistor R, consistent with
psd_{e}, the following equation is used:

| (2) |

where the integral is performed over the bandpass of frequencies
corresponding to psd_{e}.

**Seismometer case**

The power of type appropriate to the description
of a seismometer is mechanical rather than electrical.
For a seismometer, the only legitimate set of actual power
units is thus W = J/s = N· m/s = kg m^{2}/s^{3}.
A physically meaningful specific mechanical spectral power *cannot*
be based on any expression that modifies either the length unit (here m)
or the time unit (here s).
Therefore, the only component of this
set that can be removed from consideration to generate the specific
spectral power is the mass M in kg.
Consequently, the only acceptable units for the seismic (specific)
power spectral density psd_{s} are in system international units
m^2/s^3/Hz. In similar manner to the
electrical case one recognizes for the total power:

| (3) |

**Required change of the density when using the Log-scale**

A linear frequency scale is rarely employed because the psd_{s}
typically spans
several decades of frequency. When the psd is plotted versus log f,
a subtle change is required because of the following
mathematical property:

| (4) |

The log-plot represents a change of variable for which the following condition must be met:

| (5) |

where ASD is the acceleration spectral density, also sometimes called
acceleration power spectral density or even (incorrectly) power spectral
density. The absolute value signs imply dimensionless forms for the
quantities;
|psd_{s}| and |ASD| must differ by a frequency ratio
f / f_{min} for reason of the change of
variable; i.e.,

| (6) |

where f_{min} is the lowest frequency of the distribution.
Confusion with the units results from a use of the log-plot, for reason of
the fact that log(f) has no literal meaning if `Hz' is retained with
the number placed in the argument.
Like every other mathematical function, it must
be a dimensionless argument; thus it is necessary to use log| f |.

Another way to recognize the validity of Eq(6) is to consider the
nature of the FFT. Its discrete values are distributed with a separation
distance that is independent of frequency; i.e., there is
no frequency dependence to the Df between points. When psd_{s}
is plotted versus log |f|, the graph contains more points per octave at high
frequency than it does at low frequency. This `compression'
that is proportional to frequency requires a change that masquerades like
a conversion from m^{2}/s^{3}
to m^{2}/s^{4} if the shape of the graph versus log |f| is to be proper
in terms of power per octave. Keep in mind, however, that the units must
remain m^{2}/s^{3} per octave-due to the requirement on the log function
that it be dimensionless.

Further appreciation for the subtlety of the units in Eq(6) is to be
realized as follows.
It is seen that d(log |f|) is a dimensionless quantity
even if one were to erroneously associate `Hz' with its specification
of the frequency.
This results from the very nature
of the derivative of the log-function.
Thus the psd_{s} in terms of a linear plot must be m^{2}/s^{3}/Hz, and when
in terms of a log plot, m^{2}/s^{3}/octave. Of course the octave could
instead be a decade or fraction thereof. The subtlety of the units
causes the correct graph
in terms of m^{2}/s^{3}/octave (at least in terms of shape)
to result when one
simply plots the square of the magnitude of the Fourier transform components
of the acceleration.

Yet another clue to the validity of Eq(6) are the following comments by those who do acoustics and who long ago recognized the subtleties of density-calculation-type:

(i) White noise is flat per Hz, and (ii) Pink (1/|f|) noise is flat per octave.

**Need for more precise terminology**

What the above acoustics comments point out is the need in general for
more precise specification of the units associated with power spectral
densities. Although electrical engineers rarely would specify psd_{e}
as V^{2}/octave on a log-frequency plot, to do so would be cause for less
confusion. To do so, the numbers normally plotted should be first
multiplied by f/f_{min}. For example,
in the case of voltages governed
by a pink noise distribution,
the resulting graph would be a constant dB value, rather than
showing an increase as the frequency decreases. Such would be consistent
with the statement above that pink (1/|f|) noise is flat per octave.

**Getting the right answer for the total power**

To repeat-a plot of acceleration FFT components raised to
the 2nd power versus
log-frequency gives the correct shape for psd_{s}. As seen from the
discussion, however,
the units m^{2}/s^{4}/Hz cannot be correct.
A proper integration of the
psd_{s} over the complete frequency range (integral in terms of
df rather than d(log |f|) encompassed by the function, must
yield the total power contained within that range. If in doing
the integration one
fails to first
multiply the function by f_{min}/f (when summing all
the components as
a discrete approximation to the integral)-then the answer will not
be correct. It appears that seismologists have not paid serious
attention to the total power by doing the integral; otherwise they
would have discovered the problem with their units.

It should be noted that the direct integral
over |ASD| ·d(log |f|) would also yield the correct result;
however, it requires a consideration of the amount of power per octave
(or decade, or fraction of either) rather than per Hz. The units of
m^{2}/s^{4}/one-6th-decade that are sometimes reported, are not consistent with
specific power spectral density. The only acceptable set of units
is m^{2}/s^{3}/one-6th-decade.

**Physics of the matter**

It is easy to show that the actual power in watts delivered to a
seismometer amounts to work done against friction (damping) of the
instrument. If there
were no damping, then an instrument set into motion by an impulse
would oscillate forever, without further external
influence. In other words, no drive-power would be required to sustain
the motion. For an external harmonic drive whose frequency matches that of
the seismometer (corresponding to resonance), the velocity-dependent
(viscous) friction force
and the driving force are 180 degrees out of phase. Thus the product
of drive force and inertial-mass velocity yields the power supplied to
maintain the motion at steady state, while doing work against the
friction. It is thus readily seen that the specific power spectral density
in W/kg = m^{2}/s^{3}/Hz results naturally from the use of the FFT
of acceleration. The alteration from m^{2}/s^{4}/Hz is due to the w
multiplier associated with a = w v.

Because of Newton's 3rd law involving action/reaction forces, the
work done per unit mass of external ground force
that causes a response must equal the internal work done per instrument
mass. If the quality
factor of ground oscillations were the same at all frequencies as that of
the seismometer (ideally 0.707), and if the Earth's vibrations were the same
everywhere on its surface; then one could estimate the total vibrational
power of
the Earth. It would be the integral over the specific power
spectral density multiplied by the mass of the Earth (6× 10^{24} kg),
times a `form factor'.
A data set for which this makes some sense in terms of a global `equality'
of vibrations is the seismic background noise. By correcting the
low-noise model for
Q as a function of frequency (if known), one could
estimate the total quiescent seismic background noise power.

File translated from T

On 15 Oct 2008, 13:46.