Abstract

The units of acceleration spectral density (ASD), commonly employed by the seismic community, are m²/s⁴/Hz. It is here shown that a more meaningful (specific) power spectral density (PSD) is one having units m²/s³/Hz. The PSD can be simply generated from the ASD by dividing every spectral component by its associated frequency.

1  Background

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:

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:

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 m²/s³. 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:

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:

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

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.,

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 m²/s³ to m²/s⁴ 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 m²/s³ 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 m²/s³/Hz, and when in terms of a log plot, m²/s³/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 m²/s³/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 V²/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 m²/s⁴/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 m²/s⁴/one-6th-decade that are sometimes reported, are not consistent with specific power spectral density. The only acceptable set of units is m²/s³/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 = m²/s³/Hz results naturally from the use of the FFT of acceleration. The alteration from m²/s⁴/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× 10²⁴ 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.