Displacement at very low frequencies produces very low accelerations since:

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1 SEISMOLOGY The ability to do earthquake location and calculate magnitude immediately brings us into two basic requirement of instrumentation: Keeping accurate time and determining the frequency dependent relation between the measurement and the real ground motion. In order to get there, we need to know a bit more. So what are the main topics of instrumental seismology? It all starts with being able to measure the ground motion, and this is the most important topic. Then it follows recording and/or transmission to a central site. The range of amplitudes measured in seismology is very large. The natural background noise, highly frequency dependent, sets the limit for the smallest amplitudes we can measure, which is typically 1 nm displacement at 1 Hz (Chapter 3), while the largest displacement is in the order of 1 m. This is a dynamic range of The band of frequencies we are interested in also has a large range, from 10-5 to 1000 Hz (Table 1). The challenge is therefore to construct seismic instruments, both sensors and recorders, which cover at least part of this large frequency and dynamic range.

2 Earlier, analog instrument were usually made to record one type of ground motion like velocity. Traditionally, seismologists prefer recording weak motion displacement or velocity, for easy interpretation of seismic phases, while engineers use strong motion acceleration, whose peak values are directly related to seismic load on structures. Today it makes less of a difference, since due to advancement in sensor and recording systems, the weak motion instruments can measure rather strong motions and the strong motion sensors are almost as sensitive as the weak motion sensors. SENSOR Since the measurements are done in a moving reference frame (the earth s surface), almost all seismic sensors are based on the inertia of a suspended mass, which will tend to remain stationary in response to external motion. The relative motion between the suspended mass and the ground will then be a function of the ground s motion (Figure 1.3). The swinging system will have a resonance frequency f 0 where k is the spring constant and m the mass. If the ground displacement frequency is near the resonance frequency, we get a larger relative motion (depending on damping) and, as it turns out, below the resonance frequency, the relative displacement, due to ground displacement, decreases. The sensor is moving with the ground and there is not any fixed undisturbed reference available. So displacement or velocity cannot be measured directly. According to the inertia principle, we can only observe the motion if it has a non zero acceleration. So if we put a seismic sensor in a train, we can only measure when the train is accelerating or braking. As seismologist, we want to measure displacement, but this is not possible to do directly. We have to measure the ground acceleration and integrate twice.

3 Displacement at very low frequencies produces very low accelerations since: where u is the ground displacement and f the frequency. It is therefore understandable why it is so difficult to produce seismometers that are sensitive to low frequency motion. It was, however, not possible to make sensors with a stable resonance frequency much lower than 0.03 Hz. Today, purely mechanical sensors are only constructed to have resonance frequencies down to about 1.0 Hz (short period sensors), while sensors that can measure lower frequencies are based on the Force Balance Accelerometer (FBA) Principle of measuring acceleration directly.

4 The Force Balanced Accelerometer (FBA) has a feedback coil, which can exert a force equal and opposite to the inertia force due to the largest acceleration we want to measure. The displacement transducer sends a current to this force coil through a resistor R in a negative feedback loop. The polarity of the current is such that it opposes any motion of the mass, and it will try to prevent the mass from moving at all with respect to the frame. A small permanent acceleration on the mass will therefore result in a small permanent current and a large acceleration will need a large current. The current is in fact proportional to the ground acceleration, so the voltage over the resistor gives a direct measure of acceleration. The FBA principle is now the heart of nearly all modern strong motion and broadband sensors (sensors recording in a large frequency band like 0.01 to 50 Hz). Currently, the best broadband sensors have a limit of about Hz, much lower than it was ever possible with purely mechanical sensors. The FBA principle also has the advantage of making linear sensors with a high dynamic range, since the mass almost does not move. Currently, the best sensors have a dynamic range of So, in summary, mechanical sensors are the most used when the natural frequency is above 1 Hz, while the FBA type sensors are used for accelerometers and broadband sensors. This means that a lot of sensors now are black boxes of highly complex mechanical and electronic components of which the user can do very little. Seismic sensor A seismic sensor is an instrument to measure the ground motion when it is shaken by a perturbation. This motion is dynamic and the seismic sensor or seismometer also has to give a dynamic physical variable related to this motion. Our objective is to measure the ground motion at a point with respect to this same point undisturbed. Unfortunately it is not so easy to measure this motion and the seismic sensor is the single most critical element (and can be the most expensive) of a seismograph (seismometer and recording unit). The main difficulties are: 1: The measurement is done in a moving reference frame, in other words, the sensor is moving with the ground and there is not any fixed undisturbed reference available. So displacement cannot be measured directly. According to the inertia principle, we can only

5 observe the motion if it has an acceleration. Seismic waves cause transient motions and this implies that there must be acceleration. Velocity and displacement may be estimated, but inertial seismometers cannot detect any continuous component1 of them. 2: The amplitude and frequency range of seismic signals is very large. The smallest motion of interest is limited by the ground noise. From Figure 3.3, it is seen that the smallest motion might be as small as or smaller than 0.1 nm. What is the largest motion? Considering that a fault can have a displacement of 10 m during an earthquake, this value could be considered the largest motion. This represents a dynamic range of (10/10-10 ) = This is a very large range and it will probably never be possible to make one sensor covering it. Similarly, the frequency band starts as low as Hz (earth tides) and could go to 1000 Hz. These values are of course the extremes, but a good quality all round seismic station for local and global studies should at least cover the frequency band 0.01 to 100 Hz and ground motions from 1 nm to 10 mm. The standard inertia seismometer Since the measurements are done in a moving reference frame (the earth s surface), almost all seismic sensors are based on the inertia of a suspended mass, which will tend to remain stationary in response to external motion. The relative motion between the suspended mass and the ground will then be a function of the ground s motion. Figure 2.1 shows a simple seismometer that will detect vertical ground motion. It consists of a mass suspended from a spring. The motion of the mass is damped using a dash pot, so that the mass will not swing excessively near the resonance frequency of the system. A ruler is mounted on the side to measure the motion of the mass relative to the ground. If the system is at rest, what would happen if a sudden (high frequency) impulse like the first swing of a P-wave occurred? Intuitively one would expect that the mass initially remains stationary while the ground moves up. Thus the displacement of the ground can be obtained directly as the relative displacement between the mass and the earth as read on the ruler. It is also seen that the ground moves up impulsively, the mass moves down relative to the frame, represented by the ruler, so there is a phase shift of π in the measure of ground displacement.

6 Similarly if the ground moves with a very fast sinusoidal motion, one would expect the mass to remain stationary and thus the ground sinusoidal motion could be measured directly. The amplitude of the measurement would also be the ground s amplitude and the seismometer would have a gain of 1. The seismometer thus measures relative displacement directly at high frequencies and we can say the seismometer response function (motion of mass relative to ground motion) is flat at high frequencies. The phase shift is also π in this case. Seismometer frequency response Let u(t) be the ground s vertical motion and z(t) the displacement of the mass relative to the ground, both positive upwards. There are two real forces acting on the mass m: The force of the deformed spring and the damping. Spring force. -kz, negative since the spring opposes the mass displacement, k is the spring constant. The resonance angular frequency of the mass spring system is ω k / m, where ω 0 = 2π/T 0 and T 0 is the corresponding natural period. Damping force. d z&, where d is the friction constant. Thus the damping force is proportional to the mass times the velocity and is negative since it also opposes the motion. 0 = The acceleration of the mass relative to an inertial reference frame, will be the sum of the acceleration & z& with respect to the frame (or the ground) and the ground acceleration u& &. Since the sum of forces must be equal to the mass times the acceleration, we have: 2.1

7 For practical reasons, it is convenient to use ω 0 and the seismometer damping constant, d h= instead of k and d, since both parameters are directly related to measurable quantities. 2mω 0 (2.1) can then be written: 2.2 This equation shows that the acceleration of the ground can be obtained by measuring the relative displacement of the mass, z, and its time derivatives. Before solving (2.2), let us evaluate it. If the frequency is high, the acceleration will be high compared to the velocity and displacement and the term & z& will dominate. The equation can then be written approximately as: This shows that the motion of the mass is nearly the same as the motion of the ground with reversed sign or a phase shift of π. This is the same result as obtained qualitatively above. If the frequency is low, the z-term will dominate and the equation can be approximated as: Therefore, for small frequencies, the relative displacement of the mass is directly proportional to the negative ground acceleration, 2 z = & & u/ω 0 and the sensitivity of the sensor to low frequency ground acceleration is inversely proportional to the squared natural frequency of the sensor. We can simply say that the seismometer works as a spring scale. This is also the principle behind the gravity meter. In the qualitative evaluation above, it seemed that there was no phase shift, so how can the minus sign be explained? This is simply due to the phase shift between displacement and acceleration. If the displacement is written u(t) = cos(ωt), then by double differentiating, the acceleration is obtained as u& ( t) = ω 2cos( ωt) = ω 2u( t). The fact that the seismometer measures acceleration linearly at low frequencies is used for constructing accelerometers. If the damping is very high, (2.2) is:

8 and the seismometer velocity is proportional to the ground acceleration, or, integrating once, the seismometer displacement is proportional to ground velocity. This property is also used in special applications, see later, under active seismometers. In the general case, there is no simple relationship between the sensor motion and the ground motion, and (2.2) will have to be solved, so that the input and output signals can be related. Ideally, we would like to know what the output signal is for any arbitrary input signal, however that is not so easy to solve directly. Since an arbitrary signal can be described as a superposition of harmonics (Fourier series, see Chapter 6), the simplest way to solve (2.2) is to assume an input of a harmonic ground motion and solve for the solution in the frequency domain. Lets write the ground motion as 2.6 where U(ω) is the complex amplitude and ω is the angular frequency. Note that, unless otherwise specified, we mean angular frequency when saying frequency. (2.6) is written in complex form for simplicity of solving the equations and the real part represents the actual ground motion, see also Chapter 6. Since a seismometer is assumed to represent a linear system, the seismometer motion is also a harmonic motion with the same frequency and amplitude is Z(ω) We then have: Inserting in (2.2) and dividing by the common factor e iωt, we can calculate the relationship between the output and input as T(ω)=Z(ω)/U(ω), the so called displacement frequency response function:

9 From this expression, the amplitude displacement response A d (ω) and phase displacement response Φ d (ω) can be calculated as the modulus and phase of the complex amplitude response: and Td(ω) can be written in polar form as: From (2.10), we can again see what happens in the extreme cases. For high frequencies, we get: This is a constant gain of one and the sensor behaves as a pure displacement sensor. For low frequencies, which is proportional to acceleration. For a high damping, and we have a pure velocity sensor, however the gain is low since h is high.

10 Figure 2.4 The amplitude and phase response function for a seismometer with a natural frequencyof 1 Hz. Curves for various level of damping h are shown.