March 1999: LATEST EDITION: A Basic Theory of an Electromagnetic Seismograph
List of figures and illustrations: (last change April 2000 )
1. Drawing of the Hinge Arrangement of the Long Period Horizontal Seismometer that is used in the WWNSS (worldwide network of standard seismograph stations). This is the old W.F.Sprengnether model 5000 of about 1955.
2. A micropower preamplifier suitable for most seismometers and geophones: The schematic drawing of the seismic preamplifier used in the telemetry network This has a response from 20hz to 60 seconds, gain settings from 24 to 90 db, (x16 to x31623) and operates from +,- 3 to 4.5 volts at 80 microamps, suitable for 2 years of borehole operation from photo- lithium batteries. Changing the 4 filter resistors to 10x values lowers the corner to 2 hz, and replacing the large 330ufarad capacitors with jumpers makes it suitable for a long period amplifier. Other amplifiers can be used if low power is not a consideration. In April 1999 the old 1980 schematic was brought up to date and re-drawn, and I just got around to scanning it. I can send you a copy of it and a transparancy copy of the printed circuit board if you send three stamps and an address.
3. An IMAGE of the micropower preamplifier suitable for seismometers and geophones: A flatbed scan (photo) of the seismic preamplifier used in the telemetry network This also shows the foil side of the printed circuit board, as well as the same amplifier assembled on a perforated board.
"A Basic and Direct Formulation of the Seismometer Response"
(by Sean-Thomas Morrissey; original from April 1992)
To measure motion of the earth we need to detect it with respect to something that is not moving, at least momentarily. This requires an inertial mass m that is either hanging from a spring that is attached to a frame resting on the earth, or a mass suspended to swing horizontally with only a fraction of the force of gravity keeping it centered.
First we will consider a vertical movement of the mass; later the same formulas will be seen to apply to the horizontal configuration. The mass is supported by a spring attached to a frame; the spring has a constant k and produces a force proportional to its extension. We establish a reference system at the center of the mass, with x being the instantaneous vertical position of the mass with respect to the frame: ie the length of the spring. If we move the frame down toward the mass, the force of the spring on the mass is k*-x, the "-" meaning that the spring exerts less force on the mass if the frame is moved down because of the inertia of the mass.
If the mass hangs freely on the spring, it will oscillate up and down. The force equation F = m*a can be written -k*x = m*x", where x" is the acceleration of the mass. A solution of this shows that the mass will oscillate at an angular frequency w that is equal to the square root of k/m, or w = sqrt(k/m), where w is omega measured in radians per second, and equals 2*pi*f, f being the frequency of the oscillation, or w = 2*pi/T, where T is the period of oscillation in seconds.
If we want to stop the oscillations, viscous damping is introduced, such as shunting the velocity sensing coil with an appropriate resistance. By viscous we mean a non-frictional force that is proportional to velocity. The force produced by such damping is D*-x', where -x' represents the velocity of the mass, and D is the damping coefficient with units of force per unit of velocity (like dynes/cm/second).
Now the equation of motion of the spring-mass system is:
m*x" = -k*x -D*x'.
or, in standard form, x" + (D/m)*x' + (k/m)*x = 0
A solution of this is: (it looks messy when written here in ascii)
x(t) = e^(-D*t/2*m)*|Ac*e^(R*t) + Bc*e^(R*t)|,
where R = sqrt((D/2*m)^2 -k/m). (Ac and Bc are constants)
For a solution where R = 0, which is the critical damping value for x(t),
Dc = 2*sqrt(k*m). (critical damping)
But we generally consider various amounts of damping D with respect to critical, so a relative damping term is defined as B, which is the ratio of the actual damping to the value needed to critically damp the mass movement, (critical means stopping the mass in a time about equal to 1/w, for which B = 1).
So B = D/Dc = D/(2*sqrt(k*m)) = D/(2*m*w), where k/m = w^2 from above or D = 2*B*w*m.
So now the solution of the spring-mass oscillator is:
x(t) = e^(-B*w*t) *|Ac*e^(R*t) + Bc*e^(R*t)|, where R = sqrt((B^2-1)*w). (if the natural log constant e = 2.718 bothers you, just go on; we manage to avoid it later)
Or for the critically damped case: x(t) = e^(-w*t)
For a seismometer, we are interested in the relative position of the mass with respect to the frame (eg. the seismometer case) after the frame is moved by the earth. (We get an output when a coil attached to the mass moves inside a magnet attached to the case, or some such detection method.) The instantaneous position of the frame is u; the instantaneous position of the mass relative to the frame is x; the instantaneous position of the mass with respect to the earth is um. A change in um gives the change in position of the earth with respect to the inertial frame. That is um = x + u.
So now to describe the equation of motion of the seismometer, we still use the ever popular F = m*a, where the forces on the mass are -k*x and -D*x'.
So we can write -k*x -D*x' = m*(x" + u")
This can be rearranged to read: x" + D/m*x' + k/m*x = -u".
Substituting for D = 2*B*w*m from above, and k/m = w^2 this reads:
x" + 2*B*w*x' + w^2*x = -u" , a familiar differential equation.
Taking Fourier transforms where X(w) = the transform of x(t) and U"(w) is the transform of u"(t), the transfer function between the mass displacement and the acceleration of the inertial frame is:
X(w) = -U"(w)/(-we^2 - i(2*B*we*wo) + wo^2)
where w or we is the angular frequency of the ground motion and wo is the angular frequency of the seismometer suspension.
Note that this expression applies to both vertical and horizontal seismometers, since neither the spring constant nor the mass is considered.
But modern seismometers produce an output that is a function of velocity, since this avoids the problems of the mass wandering around with earth tides, local tilt noise, and the inevitable thermal problems of the seismometer suspension.
The output is generally from a coil moving in a magnet that has a "generator constant" called G, with units of volts/meter/second. But this can apply to any velocity detection system, including feedback.
When we differentiate the above equation to get the velocity of the mass X'(w) = i*we*X(w), and multiply by G, the output voltage of the seismometer is:
Eo = G*X'(w) = -i*G*we*U"(w)/(-we^2 - i(2*B*we*wo) + wo^2)
THis voltage is the signal that is generally recorded by a visible (paper on a rotating drum) seismograph or digitized for digital data systems.
Notice that the seismometer mass does not appear directly in the equations. The ratio B of the actual damping to critical damping essentially determines the shape of the response. If B is small, it is peaked or resonant or ringy, if B is large, like up to 10, the response is very broad, which can be used with a low noise amplifier to simulate a somewhat broadband response. Obviously the mass and the spring constant determine the natural period wo, but these are generally fixed parameters of the mechanical spring-mass system, and are generally determined by practical considerations of the design.
However the mass does determine the noise of the system. We usually consider the value of the M*T*Q product as an indicator of the inverse of the noise of the seismometer, where a value less than 1 is considered poor. Q is usually calculated as the inverse of the ratio of the intrinsic or mechanical damping to critical damping (0.7071). SO a simple seis like an L4-C, with a Bo (intrinsic damping) of 0.236, a mass of 1 kg, a period of 1 second, has an MTQ of 3. A quartz torsion fiber seis can have a Bo of 0.001, so a very high Q, and a workable MTQ product even with a small mass. Obviously, a vanishing mass makes the seismometer totally insensitive and raises the Brownian noise, which has a PSD of about 10^-19 for the L4-C, but about 6 x 10^-16 for a 4.5 hz phone with a 29 gram mass. But with either a small or a large mass, the response is determined by the total damping, even if most of the ambient mass movement is Brownian noise or thermal convection.
|
