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

Date: Wed, 06 Feb 2008 16:11:33 -0500

Brett, Probably the most important thing about the differences among the transfer functions for the different state variables is their differing functional dependence of SNR. The multiplication by angular frequency, when working with the derivative ('velocity sensor')--causes the power spectral density in that case to approach the electronics noise level more rapidly at low frequencies than is true for the position sensor case. In other words the signal goes below noise more rapidly for the velocity sensor than for the position sensor, as the frequency decreases. A sensor that is 'flat to velocity' is not immune to this limitation; since acceleration, being fundamental (and not velocity) is what regulates the frequency dependence of the signal to noise ratio. A way to understand the importance of the electronic noise in this matter is as follows. Nobody should question the fact that the only thing that allows any sensor to function is the transfer of power to it. In the case of a seismometer, the specific power (power divided by the magnitude of the inertial mass) is given by the product of velocity and acceleration. Students of physics should remember the expression for mechanical power as the dot product of force and velocity. In terms of acceleration, the specific power is given by the square of the peak acceleration divided by the angular frequency--having units of meters squared per second cubed. When the spectral density of the power is graphed Log-Log (or dB-Log), the logarithmic linear 'compression' in frequency of the FFT values (which are equispaced for a linear scale) causes the reciprocal omega term to disappear. In other words, for a position sensor, the mechanical specific power, in a spectral density sense, is constant for frequencies below the corner frequency. Those familiar with Jon Berger's well known paper on earth noise will remember that the ordinate of his power spectral density (PSD) graphs is specified in terms of meters squared per second cubed per one-seventh decade (expressed in dB). (Note: his graphs do not specifically mention the bin-width of one-seventh decade; this must be understood from written descriptions in the paper.) His units are consistent with what I have just indicated, but the common (erroneous) meters squared per second to the fourth per Hz are not! In fact, these common units cannot be a proper power spectral density statement, because they are dimensionally unacceptable. Whereas the PSD is flat below the corner when calculated with data from a position sensor, the same is not true in the case of a velocity sensor. In the velocity case, the power is given by omega times the square of the peak value of the velocity. The PSD is in turn (because of the compression mentioned in the usual Log-Log representation) given by omega squared times the square of the peak velocity. Thus, as omega (two pi times the frequency) decreases below the corner value, the PSD expression decreases with the square of the frequency--falling off 20 dB per decade. Just from the electronics noise alone, we see that with the velocity sensor--as frequency decreases--a point is reached where the mechanical PSD falls below the power spectral density of electronics noise. Thereafter, unless some noise reduction scheme is employed the signal responsible for mechanical motion below those frequencies--cannot be seen with the velocity sensor. They can, however, still be seen with the position sensor. Randall