Physics of a Seismic Instrument

Randall D. Peters

Department of Physics
Mercer University
Macon, Georgia 31207


Generic requirements for producing a functional seismic instrument are indicated. The performance limit of any such instrument is primarily determined by friction. Although a commonly treated form of friction, viscosity of air, can be important; it is the author's belief, from years of studying mesoanelastic complexity-that internal friction of test mass support structures is the primary culprit when it comes to looking for low-level long-periodic motions of the earth. These mesodynamic properties of seismic instruments are presently unknown to the geoscience community. Thus a proposal is offered to the geoscience community-a simple test of their instruments to prove or disprove some of the claims of this paper.

1  Introduction

A seismometer is fundamentally a very simple instrument to understand. One need ``only'' to suspend a test mass and look for relative motion between it and the earth. It is the inertial property of the test mass that is used to advantage; i.e., Newton's 1st law, which states that an object at rest wants to remain at rest in the absence of a net external force acting on it. Thus motion of the earth's surface gives rise to relative motion between the earth and the test mass. [NOTE: Most modern seismometers use a transducer and a feedback network to provide force-balance on the test mass. The error signal provided to the feedback network is a measure of mass motion were it permitted to occur. The system is effectively the same in many respects, although more will be said about this technique later.]

In the absence of friction, design of a seismometer would be trivial as compared to the many challenges posed by the real world. One form of friction that influences these instruments, especially ones with a small test mass, is that of air friction. Most professionals in the geoscience community know that the viscosity of air is great enough to require that small, sensitive instruments be evacuated.

1.1  Internal Friction

An equally important source of friction, the importance of which is appreciated by very few, is that of internal friction. Internal friction results from stress/strain hysteresis in the support structures of a seismometer, and it can never be completely eliminated. Moreover, it poses the fundamental limit on the sensitivity of an instrument designed to sense very slow changes in the earth.

1.2  What is measured?

In effect, every seismometer serves as a strain gauge; i.e., relative motion between the earth (case of the instrument), and the test mass is necessary for the instrument to function. Although the strain employed by the instrument can take many possible forms, it is the signal to noise ratio (SNR) in the detection of some strain that ultimately determines the smallest measurable acceleration of the test mass.

1.3  Conventional Seismometer

One of the simplest strains to understand is that of a vertically oriented coiled spring holding a mass in near-equilibrium, the top of the mass being attached to the case of this ``vertical seismometer''. Vertical earth motion results in harmonic oscillation of the test mass which is supported at the bottom of the spring. The harmonic strain of the spring in response to test mass motion in this case is simply described as the fractional change in the length of the spring. What is it that limits the sensitivity of such an instrument? Why use a weak spring rather than a strong spring for great sensitivity? The answer to these questions requires that we consider properties of the material which comprise (i) the spring, and (ii) the electronics. Although there is no such thing as a perfectly elastic spring, let's first assume such and consider the limitations imposed by the electronics.

The electronics responsible for producing an output voltage in response to the strain of the spring can exist in one of many possible forms: (i) optical, (ii) inductive, and (iii) capacitive just to name a few. In any case there must be a change in some characteristic dimension DL of the sensor if a voltage change DV is to occur at the output of the sensor. Thus there is a calibration constant, k that relates the two; i.e., DV  =  kDL. If the sensor is not linear, then k is not really constant, but takes on different values as L varies.
[NOTE: Conventional wisdom says that nonlinearity is most important when motions are large. The author's work has shown that the other extreme is even more important and interesting; i.e., ``corrugations'' in the potential well resulting in ``stiction'' at low levels and long periods. The same problem is encountered with barometers. The work of the author and others (personnel at NOAA) have demonstrated that ``gravity waves'' (not quantum gravitons) are present in the atmosphere. Particularly in windy regions, these long period pressure oscillations can be present at the parts per million (ppm) level. They do not show up in a standard barometer because of ``stiction''-due to the coefficient of static friction being greater than that of kinetic friction, and the conventional sensor gets stuck in a localized potential well of metastability. The author's experiments with a host of different pendula has demonstrated that the same phenomenon is important to seismic instruments. The conversion to force-balance instruments probably exacerbates the problem. Because the process is not visible in the macro regime, there is no evidence to the observer of its presence. Consequently, a calibration procedure, based on linear system assumptions, is employed. Such techniques are not valid for predicting system performance where the stiction is important, since it is a nonlinear phenomenon.]

Typically, the operating voltage for a sensor is in the one to ten volt range, and it is difficult to sense (over long times) changes less than about 1.0×10-6 V. In other words, electronics limitations borne of things like 1/f noise place a lower limit of detectability that is about 1 ppm (part per million) in the voltage output of the sensor. Now k depends on many factors, but let's look at a specific case. The symmetric differential capacitive (SDC) sensor array that I am using on a WWSSN (Sprengnether) vertical seismometer has a sensitivity approaching 2000 V/m, over a range (requiring gain switches to realize) of about 0.5 cm. The minimum detectable DV  =  1 mV gives rise to the following requirements on the vertical position of the masses of our seismometer:
D >  2 ×10-9m,
Dynamic range   =  2.5 ×106 ® 130 dB
[Note: The Leaf-Spring seismometer by E. Wielandt and G. Streckeisen (W/L) has a stated Dymin  =  10-10m with an operating range of 0.5 mm, using an LVDT. We see that the dynamic range is about the same as my SDC outfitted vertical seismometer.]

1.4  Tradeoff between Sensitivity and Dynamic Range

There are two properties of any measuring system that work in conflict with each other-dynamic range and sensitivity. It is very difficult to measure a length strain of one part per million (ppm) anywhere it occurs in a distance of many centimeters. The numbers above are probably fairly typical of sensors used in seismometers (parts per million with a range near 100 dB). We will shortly see that great sensitivity implies weakness of spring, but the greater the weakness the less the mechanical integrity. In particular, a sensitive system is prone to creep, which is strongly coupled to stress/strain hysteresis in the form of anelasticity (internal friction).

1.5  Importance of the ``Spring''

It will now be shown that the spring becomes the ``heart and soul'' of a practical seismometer. It is the simplest case of a ``strain-gage'' to consider, because ideally it obeys Hooke's law. A displacement of the test mass vertically through a distance y gives rise to a restoring force from the spring given by F  =  -k y, where the spring constant k will be shown to be a very important factor in the sensitivity of our instrument.

The differential equation for the test mass (in the absence of external forces and friction) is given by

m d2 y/d t2 + k y  =  0  =  d2 y/d t2 + w2 y

When there is an external acceleration of the case, the test mass equation of motion takes the form (including friction to damp the system)

d2 y/d t2 + damping + w2 y  =  aext

If aext were a Heaviside step function, then the spring length steady state change y in response to this function is given by y  =  aext/w2. Since w  =  2p/T, it is seen that the spring changes by an amount proportional to the square of the period of the mass/spring system. Even if aext is a complicated function of the time, and steady state is not realized; it is still true that the sensitivity of the instrument is determined (mechanical contribution) by the period of the instrument-being proportional to T2.

1.6  Considerations for isolating from the environment

The ``obvious'' (though erroneous) way to build a sensitive vertical seismometer is to thus use a very long spring, so that k is very small. The problem with this (other than its impracticality), is that a large system is susceptible to so many of the noninertial properties of the environment-ones that are manmade and have nothing to do with earth accelerations that we're interested in.

Other fields of physics have faced this challenge, in particular the development of the scanning tunneling microscope. Binnig and Rohrer deserved the Nobel prize, in part, for the herculean effort to isolate their first instrument, which was large. Once miniaturization had occurred, STM's became easy to use in a noisy environment. The eigenmodes of the instrument had been moved to a sufficiently high frequency (by size reduction)-that coupling to the typically lower frequencies of the environment were no longer so important.

2  Mass size optimization

The simple minded (empirical) description of Coulombic friction falls way ``short of the mark''. The force of kinetic friction fk acting on an object of mass, M, resting on a surface- is simply a coefficient mk times the normal force, n; i.e.,

fk  =  mk n

And if the mass is at rest on the surface, then the static force maximum value is given by the same thing replaced by a static coefficient ms which is always greater than mk. Because the normal force, for a horizontal surface, is given by n  =  M g, it is seen that M should have no bearing on the design of a seismometer, if Coulombic friction were both significant and adequately described by the equations indicated. It is known, however, that test mass size is important-even when the instrument is evacuated. For viscosity of air to be unimportant, the pressure must be dropped to a value below which the air molecules become ballistic; i.e., the mean free path between collisions is bigger than chamber dimensions. Even though instruments typically satisfy this condition, test mass size is still known to be important. Typically, instruments use a test mass in the order of a kilogram, and the choice is based on considerations of Brownian motion. The author believes that spring anelasticity could be equally if not more important to mass selection. Part of the logic involved in this thinking proceeds as follows. Larger mass implies larger spring, implies more material. More material implies more defects, and thus greater averaging over all the effects of anelasticity; i.e., a grading of the washboard road of the potential well, as it were.

3  Work of LaCoste under Romberg's direction

As a student of physics under Prof. Romberg at the University of Texas at Austin, LeCoste addressed the problem of how to build a compact, low k effective spring in the 1930's. He inclined the spring at an angle with respect to vertical and provided twist in the coiling operation during fabrication. In another webplace, quantitative reasons are provided for why this zero-length design by LeCoste is a superior one for vertical seismometers. His patent has stood the test of time in terms of improved mechanical stability.

3.1  Modern instruments-Force balance technique

Many forms of modern seismometers do not use passive sensing as described above. Instead, a transducer is involved in a force-balance feedback network. In the absence of feedback, acceleration of the case would cause a change in y as indicated above. With the transducer and network, the feedback signal is adjusted to whatever value is necessary for Dy  =  0. In turn, the error signal (proportional to the amount of feedback) is a measure of how y would otherwise change. Such a system is always equivalent to a harmonic oscillator in its simplest form, and thus to a mass/spring system. Instead of relying totally on the mechanical spring to determine the dynamics of the instrument, the phase and amplitude of the feedback signal is adjusted to yield a seismometer with an effective spring constant keff, whose magnitude can be varied. [Note: W/L describe the force-balance seismometer in the following way: "The idea behind a conventional LP seismometer is to measure the motion of the ground against an inertial reference, i.e., against an elastically suspended mass that is supposed to stay at rest when the ground moves. A force-balance seismometer is a negative-feedback (i.e., electronic servo) system that causes the mass to follow the motion of the ground. The force required for this purpose is a measure of the ground acceleration.''] It should be noted that keff of the force-balance instrument must be small if the sensitivity is to be large, just as with the conventional instrument. To think that modern instruments are sensitive even though they have a short period (when operating without feedback) is to miss the point altogether.

3.2  Limitations of the Force-balance approach

An analysis of limitations inherent to the force-balance method requires some careful thought. Ostensibly, the method appears to solve many problems encountered in seismometer design. There are, however, serious issues of anelasticity that have never apparently been considered in relationship to the use of these instruments.

3.3  Gedanken to illustrate that limitations exist

Lest one believe that force-balance feedback is infinitely superior to conventional seismometry, consider the following logic. Why even bother with the leaf-spring that is commonly used to support the test mass in these instruments? Why not just add a feedback network to a solid state mass balance instrument that works with resistive strain gauges? Place a big test mass on the pan of the modified mass balance (mmb), add a magnetic transducer of some type to provide a significant lifting force on the mass, and ``voila''-with proper feedback adjustment we suddenly can see earthquakes with the simplest of instruments. Hopefully everyone will quickly recognize the folly of this reasoning and know that such a modified mass measuring instrument is not capable of functioning as a bonafide seismometer. But why? The answer to this question lies in the following observation. System adaptability is no better than the integrity of the ``spring'' used in generation of the error signal. As noted earlier, any error signal requires the measurement of strain. In the case of the hypothetical modified mmb, the ``spring'', in the absence of feedback, has an exceedingly large k. In the case of the W/S leaf-spring seismometer, the leaf has a considerably larger k than that of the conventional seismometer. Can electronics soften even the hardest springs? The answer is obviously no! What are the limitations to softening? I submit to the reader that there are a host of unanswered questions in the matter. It is easy to see that electronics limitations (addressed earlier) pose an ultimate upper limit on the size of k. But anelasticity of the support is probably even more important than the electronics-and the problems borne of it are mostly unstudied. This is true in spite of the fact that practictioners understand that an instrument must be allowed to settle for some time after initial loading, before it becomes dependable. This settling is necessary to minimize the effects of anelasticity, through a type of work-hardening.

3.4  Commonly held erroneous thinking

A decade of careful research on the author's part has shown that mesoanelastic complexity disallows the creation of an ideal spring, no matter whether LeCoste zero-length, W/S leaf, or any other type. Practically speaking, the mesodynamic range where we want to study earth motions is one in which the assumed parabolic (harmonic) potential well is not valid. There is fine structure (FS) superposed on the harmonic potential (whether conventional or electronically ``softened''), and this FS is not even static (God help us!). Diffusion processes influenced by temperature cause the ``corrugations'' of the potential well to undergo continuous change. Bottom line-the system is not linear in the regime where we want to conduct experiments. To believe that the impulse response of the seismometer at larger levels gives a true measure of the low level sensitivity of the instrument is sheer folly. Every properly trained engineer knows that Green's function techniques do not apply to nonlinear systems.

3.5  Dithering and Stochastic resonance

It is possible that the conventional instrument may even be superior to the force-balance instrument at low levels and long periods. Why? Because of features of the new science of stochastic resonance. Engineers have long known how to deal with friction hindered systems. They use dithering, in which the instrument is shaken so as to avoid metastabilities that derive from static friction being greater than kinetic friction. Everyone knows this to be true of Coulombic friction (surfaces sliding relative to one another). It is also true, apparently, of internal friction borne of granularity (due to grain boundaries). Thus it appears that there is an advantage to operating away from critical damping, which tends to insure ``latching'' to a metastable state. Force-balancing might even be worse in forcing the system into such a state from which it does not exit.

It should be noted that latching, as  the word is used here, does not mean the instrument has "seized-up" and has no low level response at any frequency.  Rather, it means that the system refuses to oscillate at its natural frequency,  f, and evidently at other frequencies in an unknown region below f (and perhaps also above).  The system still responds to ultra-low frequencies, as evidenced by the seismometer's apparent response to tidal forcing with a 12-hr period.   

4  Proposal

The assumption of system linearity results in a specific functional form for the free-period decay of an instrument. Specifically, linear theory requires that the logarithmic decrement be proportional to the period of oscillation raised to the first power. All long-period mechanical oscillators studied by the author deviate radically from this assumption. In particular, the log-decrement is nearly proportional to the square of the period-because of internal friction borne of anelasticity. Thus it is suggested that users of the modern instruments do experiments in which the damping is measured as a function of adjustable period.

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On 9 Aug 2000, 12:24.