• # 4 Calibration

## 4.1 The nature of the problem

### 4.1.1 The basic objective

The basic objective of calibration is to determine the nature of the transfer function which relates the original motion of the seismometer support to the motions of the indicator on the final record. This relationship can he expressed either in terms of the relative amplitudes and phases of the harmonic components of the earth motion, or as the response of the indicator to an impulse in the ground. As an arbitrary ground motion can, in principle, be expressed either as the sum of a set of infinite sinusoidal wave trains or as a succession of impulses, the two representations are analytically equivalent.

In practice, conversion from one form of calibration function to the other is possible only over the range of periods for which the harmonic components are sufficiently distinguishable from noise generated in the ground or in the recording system. If conversion is made by applying the ordinary methods of Fourier transformation to an experimentally recorded impulse, the result will typically provide a good match to an independently determined harmonic response for frequencies close to that of maximum sensitivity of the seismograph. At frequencies remote from the favourable region, the noise content of the record overwhelms the comparatively small harmonic components in the instrumental response to the input signal, and the computed harmonic components rise uncontrollably above the proper values.

This problem can be avoided (Mitchell and Landisman, 1969) by recognizing that the calibration curve is defined by a limited number of constants and that these can be determined from a calibration pulse by the application of the method of least squares.

In modern seismometry, the formation of an intermediate record (e.g. analogue or digital on magnetic tape) is becoming an increasingly common step in advance of final presentation and interpretation. Considerable filtering, or other types of data processing, may be included in the preparation of the final output, and it is therefore becoming increasingly important to provide a stored representation of simulated earth motion in the intermediate record which can ultimately appear as the calibration signal on the final output. An impulse response is particularly convenient in this respect.

### 4.1.2 The uses of calibration data

Calibration data are now being used in a variety of ways in which the advantages and disadvantages of different methods differ in relative importance. These are as follows:

1. For the classification of instruments. In conducting research on a suite of seismograms, or in interpreting the significance of bulletin readings, it is necessary to have a preliminary classification of instrumental type. Until fairly recently, the simple division into long-period and short-period classes sufficed for this purpose, but the great diversity of modern instrumentation has made it necessary to adopt the much more detailed system of Section INST 1, and the calibration curves on which the classification depends must therefore be universally determined.

2. For matching the elements of arrays. In this application, matching must be sufficiently accurate to permit the recognition of common features in the wave-form on neighboring channels and to permit significant features of the output to add in phase in the velocity-filtering process. In this application, the absolute levels of sensitivity of the whole array is not relevant to the summation process, but relative phase shifts must be small over the region of maximum sensitivity. Moderate variations of sensitivity within the array can be tolerated in the summation process, but the mean level should be known for amplitude interpretation.

3. For matching the elements of 3-component sets. The chief requirement is to analyse the nature of the particle motions to be recognized in an advancing wave front, and the most sensitive procedure is to determine the direction of an axis along which the component of interest exhibits a null value. This process inherently involves the subtraction of wave-forms, leaving a minimum output composed of noise and of mismatch between components of the signal. Experience has shown that this method can determine the orientation of a null axis to within better than 1°, which requires that the signal components at all frequencies should match to within about 1% of the amplitude of the strongest component. The process, however, involves matching of relative levels between the three components and is independent of the mean level of the set.

4. For the determination of absolute signal level. This application most frequently occurs in magnitude studies, in which the relevant observation determines the parameter A/T for an incoming phase. The most desirable characteristic for the seismograph is for the response to earth velocity to be fairly flat in the vicinity of the dominant period of the seismic wave. If this condition is satisfied, the shape of the calibration curve at other periods will be comparatively unimportant, but the level at the period of interest should be known to within about 5%.

### 4.1.3 The nature of the solution

All methods of calibration can be divided between those in which the overall response is calculated from the constants of the individual elements of the system, and those in which the system is regarded as a 'black box' in which an input simulating earth motion produces output on the final record. The first class contains inherent redundancy, in so far as a given system response can correspond to any one of an infinite number of combinations of individual parameters (Coulomb and Grenet, 1935). In comparing the two classes of calibration methods we note that step-by-step calibration concentrates attention on each parameter in turn, and is therefore particularly useful in setting up matched systems in which all corresponding elements are interchangeable. In contrast, whole-system calibrations avoid redundancy and are much faster to apply, but they cannot directly indicate the source of any departure from a desired characteristic, and offer no guarantee that systems which match in overall response will continue to match if elements are interchanged.

The redundancy of instrumental constants in the equations of motion leads to a dilemma in the presentation of calibration results. The seismologist interpreting ground motion is concerned only with the overall response. The technician assembling a set of instruments will usually wish to set up each element to a pre-determined specification. The common practice of putting the technician's data (periods and damping constants) in the user's Bulletin is now seen to involve the user in unnecessary computation if anything more than a rough classification of data is needed. The modern requirement is to provide access to the full calibration in graphical or tabular form.

## 4.2 Direct-coupled seismometers (classical method)

### 4.2.1 Static magnification, for horizontal seismometers with calibrated leveling screw

Some seismometers are designed in such a way that a small measured rotation of a leveling screw will introduce a known tilt of psi radians about a horizontal axis perpendicular to the direction of free oscillation. This process is equivalent to the application of a horizontal acceleration

where g is the acceleration due to gravity. Apply the tilt, observe the resulting trace deflection a, and substitute for d2chi/dt2 in equation [4] as follows:

We shall determine the natural period Ts in the next section, so we can write Omega0 = 2pi/Ts and hence find the static magnification l'/l.

### 4.2.2 Static magnification, for other seismometers

When the tilt test is not applicable, the seismometer must be dismantled, and a calibration support arranged so that the boom can swing in a vertical plane. Set the pendulum into small oscillation and determine the duration of 50 swings. Hence measure the period tau, and determine the reduced pendulum length from the formula

While the seismometer is dismantled, it may be convenient to weigh the boom, and to determine the position of the centre of gravity by balancing on a knife edge, or suspending in a loop of fine thread. In vertical seismometers, weigh the spring, and count one third of its mass as part of the mass of the boom. If this is done, an independent measure of static magnification may be obtained after reassembly by placing a small test weight in on the boom at the centre of gravity, and lifting it off. This is equivalent to an earth acceleration of mg/M, which is entered in Equation [4] to yield the static magnification, as in the tilt test. In horizontal seismometers, a small auxiliary platform can be supported on threads inclined at 45° to the vertical, so that a horizontal force of mg is applied.

If the seismometer uses an optical lever, remember that the angular deflection of the light beam is twice the deflection of the mirror, and hence put l' equal to twice the physical length of the light beam. If the system contains auxiliary levers, multiply the length of the final indicating lever by the total ratio of the lever train which connects it to the main hinge of the boom.

### 4.2.3 Period

Remove the damping system, set the boom into oscillation, and measure the apparent period T's by visual timing with a stopwatch, or by recording and measuring the trace. Also measure the amplitudes alpha1, alpha2, ... alphan of successive swings on opposite sides of the zero line, and determine the mean value of the logarithmic decrement

Now determine the true period Ts from the formula

In these equations we have used the logarithm to base 10, instead of the more fundamental natural logarithm, for ease of entry to common tables.

### 4.2.4 Damping

When the period has been determined replace the damping system and adjust to a trial value. To obtain critical damping, for which beta =1, increase the damping until the indicator first fails to cross the zero line after an initial deflection.

If the desired value of damping is less than critical, several swings of the boom can be observed. Measure the logarithmic decrement lambda as above, and use the equation

to determine beta.

For values of beta greater than 1, apply an impulse, and record deflection as a function of time. Measure the time interval t between the peak deflection alpha1 and the instant at which the deflection has declined to a deflection alpha2. Determine a normalized time t' = Omega0 t, and hence determine beta by entering the family of curves shown on Fig. 4.2.4. The figure is adapted from a paper by Julio Morencos (Morencos, 1966).

## 4.3 Electromagnetic seismographs (classical method)

### 4.3.1 Galvanometer constants

Disconnect the galvanometer from the rest of the circuit, apply a short electrical impulse to set it swinging and record the resulting oscillations. Determine the logarithmic decrement, the natural period and the open circuit damping constant alpha0 by the methods which have been described for the mechanical seismometer. Hence determine the constant d/k from the formula alpha0 = d/2komega0.

Now connect a known shunt Si across the galvanometer terminals, and repeat the damping experiment to determine a new damping constant alphai. This will satisfy the equation

Thus we can determine g2/k if we know the galvanometer resistance rg. Usually, manufacturers quote the total resistance of the coil and suspension of a galvanometer. If this figure is not available, it may be determined by setting up the galvanometer as the unknown resistance in a Wheatstone bridge circuit, and connecting a key across the terminals normally used for the null-detecting galvanometer. A small current is passed through the bridge from a battery with a high resistance in series. The balance resistors are then adjusted until the galvanometer deflection is unaltered by opening and closing the key, which shows that the balance condition has been attained.

The values of damping which are normally used in seismology cause oscillations to die away too rapidly to yield a good estimate of logarithmic decrement (and hence for a) in the working condition. We therefore determine alphai for a number of different values of the shunt resistance Si. A plot of alphai against (rg + Si)-t yields a straight line which can be extrapolated, either graphically or by the method of least squares, to reveal the value of the shunt resistance S corresponding to the desired value of alpha.

### 4.3.2 Seismometer constants

Determine the constants, Omega0, D/K and Delta/K by making experiments analogous to those performed on the galvanometer, and determine I by any of the methods which were described for the mechanical seismometer. In these experiments it will be necessary to observe, and preferably to record, small oscillations of the boom. This may be done by observing a mark by means of a microscope with a scale in the eyepiece, or by attaching a mirror to the boom, and recording the deflections photographically. Alternatively, an overdamped galvanometer, coupled to the transducer through a high resistance, may be used.

The disadvantages of the above method are that the minimum available damping may be high enough to limit the number of visible oscillations to an undesirably low number, and that the smallest amplitude which can conveniently be observed may be much larger than the amplitude of motion in normal service. Some faults in the seismometer, such as dirt in the gap or slight buckling in a strip hinge, can have serious effects on small-motion performance without showing up on a large-motion calibration experiment, so that the detecting system used should have the highest possible magnification.

### 4.3.3 Constants of the coupling circuit

Connect the galvanometer to the seismometer through a high series resistance R1 and set up a pair of stops on either side of the seismometer boom. These stops may conveniently be the jaws of a micrometer, set up in such a way as to clamp the seismometer boom, and then opened by a measured amount.

Move the seismometer boom quickly from one stop to the other, and record the free oscillations of the galvanometer. Note the amplitude alpha1, alpha2 . . . of successive half-swings, and thereby calculate the amplitude alpha0 of undamped oscillation from the formula

Calculate the initial angular velocity of the galvanometer coil from the equation

Then calculate the constant gG/k from the equation

We can now determine sigma2 from the equation

which tells us all the quantities which enter into equation [7].

If the contribution of sigma2 can be neglected, the graphical method of plotting the overall response (Section 3.2) may be substituted for the more tedious procedure of substituting numerical values for the parameters in equation [7].

### 4.3.4 Units

Because of the complicated interaction of electrical and mechanical units in the equations, it is essential to use a consistent series of units throughout. The Sl system, in which distances are measured in meters and masses in kilograms is strongly recommended. The system is consistent with the practical electrical units (volts, amperes and ohms). The constant g may be taken as 9.81 m/s2.

## 4.4 Harmonic drive methods

The difficulties which arise in the conventional methods of calibration can be avoided if means can be found of injecting a continuous sinusoidal signal into the system, and of displaying the response. The methods may be applicable either to complete systems, or to individual elements. The common feature of these methods is the steady state of the output, which greatly improves the precision of reading amplitudes and periods in the presence of noise, permits a wide range of amplitude to be used as a check on linearity of response, and enables phase relations between input and output to be measured directly.

### 4.4.1 Shaking tables

The most perfect simulation of earth motion is, of course, to move the platform on which the seismometer has been set up. In spite of this apparent simplicity the method involves numerous difficulties, of which the following are the most important:

1. The table must in practice be considerably larger than the seismometers under test. If the seismometers themselves are full-sized observatory instruments, the table becomes a piece of heavy engineering beyond the reach of most observatories.

2. If the table is to be designed for horizontal motion the effects of tilt, especially at long periods, are likely to be comparable to the desired horizontal accelerations.

3. The seismometer has to be removed from its normal position of operation for calibration, and there is no guarantee that this can be done, and the instrument subsequently returned to normal service, without disturbing the calibration. This point is particularly serious for horizontal seismometers, for which the natural period depends critically on leveling.

For these reasons, large shaking tables tend to be a designer's tool in the investigation of problems such as the existence of parasitic modes of oscillation. Their principal use for calibration is in the exploration industry, where quite small, portable tables can be used to check the response of several geophones at the one time.

### 4.4.2 Seismometers with auxiliary coil

If an auxiliary moving-coil transducer is attached to the suspended mass of the seismometer, sinusoidal or other forms of input can be provided in the form of an electric current, the effect of which will be directly proportional to acceleration of the ground. The constant of proportionality can, if necessary, be determined by comparing the effect of switching a known current on or off, with the effect of a tilt or weight-lift experiment (Section 4.2.1 or 4.2.2).

If the auxiliary coil and the main transducer are in close proximity to each other (and especially if they take the form of two coils wound on the same bobbin) it is important to ensure that electrical coupling through mutual inductance or capacity is at a much lower level than mechanical coupling through the mass. This can be checked by comparing output with the mass clamped and unclamped. Calibration should be restricted to the range of periods for which output in the clamped condition is insignificant.

Output can be displayed either in the form of an oscillogram showing output as a function of time, or as a Lissajous figure in which input and output are shown as X and Y components of a closed loop on the face of an oscilloscope or on an X-Y plotter. A useful review of these methods, with special reference to precise phase calibration, has been given by Mitronovas and Wielandt (1975).

#### 4.4.2.1 Oscillogram display

Output may be presented either on the normal recording system of the seismograph, or on a separate oscillogram or oscilloscope. In the first case the ratio of equivalent input acceleration to recorded output is a complete calibration for the operational seismograph. In the second case, normal practice is to use a display device with a very short response time (e.g. cathode ray tube or short-period galvanometer) so that any difference of response between the indicator used in the calibration experiment and the indicator used in normal service will require to be determined separately.

An important special case is the one in which the system under test responds like a simple pendulum. In that case the relative velocity between the mass and its supporting framework will be a symmetrical function of input acceleration (which follows by differentiation of equation [2] of Section 3.1) and this will be proportional to the output e.m.f. if the main transducer is of the moving-coil type. The ratio of output voltage to input current will, therefore, fit one of the family of curves in Fig. 3.1a. The natural period of the pendulum will be on the axis of symmetry of the response curve. The damping constant beta can be derived from the fact that the peak of the response curve falls above or below the point of intersection of the asymptotes in the ratio 2beta:1.

The time lag (TL) between input and output can be measured by displaying the two wave-forms on identical channels of an oscillograph and scaling between them, or by displaying input and output separately and timing the interval between zero crossings with a stop-watch or electronic timer. If a cathode-ray tube display is used, it is highly desirable to use only DC coupled channels, as AC amplifiers can distort amplitude and phase relationships at the low-frequency end of their pass-band, even when set up to yield identical response at high frequencies. Similarly, mechanical galvanometers will modify the transfer function near the high-frequency end of their pass-band, the phase and amplitude response being the same as those for mechanical seismographs with the input current of the galvanometer simulating earth acceleration as a seismometer input (see Section 3.1).

Time lags determined in this way are converted into values of the angular phase shift i, by using the conversion formula xi = 2pi TL/T .

#### 4.4.2.2 Display by Lissajous figure

In this group of methods, the input voltage which is used to drive the auxiliary coil is supplied also to the X-deflection input of an oscilloscope or X-Y plotter, and the output of the system is used to generate the Y-deflection of the display.

If the input is a pure sine wave and the seismic system has linear response, the image will be one of a family of ellipses, which degenerate into straight lines whenever the phase angle takes the form xi = npi/2, n integer (see Fig. 4.4.2.2).

The most important parameters which can be determined from observations on the Lissajous figure are as follows:

1. Phase lag Time the passage of the indicator from the crossing of the Y-axis (X = 0) to the grossing of the X-axis as a measure of the time-lag TL. Alternatively, measure the quantities OA, OB or OC, OD from Fig. 4.4.2.2 and use one of the following formulae:

All methods should give the same result, although relative sensitivity will vary with phase angle.

2. Natural period for simple pendulum When the seismometer is a simple pendulum (or has an electronic system which simulates this response) the Lissajous figure will close on resonance. This method can be extremely sensitive at low values of damping, and departures from resonance as low as ±0.1% can often be recognized. In these circumstances, however, initial transients take a long time to die out, and it may be desirable to introduce some damping to reduce the settling time.

3. Damping constant for a simple pendulum Measure the phase angle xi as in Method 1 above for a suitable range of periods, and use the formula

Note that substantial changes in xi are concentrated into a small range of changes in the driving frequency omega if beta is small. If low damping is a property of the normal operating condition, most faults will increase it, and the test provides a very sensitive check on malfunctioning of the equipment.

4. Linearity Look for any departure from the ellipsoidal form of the output, especially for any tendency for the figure to close into figure-of-eight form instead of to a straight line, or for the resonant frequency of a simple pendulum to vary with amplitude. Remember, however, that any harmonic content in the oscillator output will also distort the wave-form. If this is suspected, energize the Y-input with an R-C coupling and check the figure for ellipticity.

### 4.4.3 Bridge methods

When an auxiliary coil is not available, the main coil can be used for this purpose, if one remembers that the output which appears at the terminals is the superposition of the e.m.f. which arises entirely from the circulation of electric current in the network (i.e. that which is observable when the seismometer is clamped), and the additional e.m.f. arising from deflections of the mass.

#### 4.4.3.1 Bridge for Lissajous figure

Set up the circuit as in Fig. 4.4.3 .1. This circuit is adequate for checking resonance and linearity but, because of the superposition of two e.m.f.'s in the Y deflection input, is not very suitable for the determination of phase shift.

#### 4.4.3.2 The Maxwell impedance bridge

In this system (Grenet, 1946, Willmore, 1959) the seismometer is connected in one arm of the bridge circuit (Fig. 4.4.3.2) and the electrical coupling is eliminated by balancing the bridge with the seismometer clamped. The e.m.f. which appears on unclamping then arises entirely from motions of the seismometer coil. The method is convenient for use when the output is detected by a galvanometer. If electronic displays are used, it is necessary to ensure that the oscillator and detector have independently balanced output and input, as any coupling (as through common resistive or capacitative coupling to ground) will destroy the balance condition of the bridge.

The steps in the calibration are as follows:

1. Balance the bridge with the seismometer clamped and the drive connected to the 'main output' of Fig. 4.4.3.2. Because of the balance condition the galvanometer is undeflected, but a current ic is flowing through the coil, where

and a force of moment Gic is being exerted on the clamp. The mechanical system is therefore in the same condition as if the seismometer support were being subjected to an acceleration of -Gic/LM. In a translatory system, the expression Gic corresponds to a force causing translation, and the corresponding acceleration is Gic/M.

2. Unclamp the mass, which now begins to oscillate as though the seismometer were on a shaking table with acceleration proportional to the driving current. Record the galvanometer deflection theta as a function of oscillator frequency, thereby obtaining the form of the acceleration response.

3. To obtain the constant of proportionality switch the drive to the 'substitution input' without changing the driving voltage frequency, and observe the new deflection thetas. The constant G may now be determined from the equation

where |ZM| is the mechanical impedance of the seismometer mass, excluding the loading effect of the galvanometer circuit.

The equation for |ZM| is

If we plot thetas/theta as a function of frequency, we obtain a curve with a minimum where omega = Omega0. Knowing Omega0, plot (thetas/theta)2 against (omega - Omega02/omega)2, which yields a straight line, which intersects the line (thetas/theta)2 = 0 at the point -D2/K2. Hence we find G, and can derive the absolute calibration if we know the mass and moment of inertia of the boom.

## 4.5 Whole-system impulse methods for electromagnetic seismographs

The injection of a single impulse into the seismometer is the quickest method of calibration, as all the information needed to define the transfer function is contained in the output transient.

Input can take the form of a short impulse (e.g. by discharge of a condenser) or step function and can be injected into the seismometer through an auxiliary coil, through a balanced impedance bridge (Section 4.4.3), or as current voltage input into the main coil. When the injection is into the main coil, the desired electromechanical response will be superposed on to the direct response of the indicator to the circulation of current in the electrical net work so that, in the most general case, we have to perform the experiment once with the seismometer clamped and once with it unclamped. This enables the electromechanical response to be separated by subtraction.

### 4.5.1 Impulse response for wide-band indicator

An important special case is that in which the indicator can take up a new position in a time which is short compared with the seismometer period and can sustain constant deflection for constant input signal for a time much longer than the seismometer period. The major classes of indicator which fulfill this condition are the short-period galvanometer and the oscillograph driven by a D.C.-coupled amplifier. In such cases, we can determine the indicator response by observing the static or quasi-static response to a step function, and assume that the single value of indicator sensitivity derived in that way applies to the whole of the significant energy spectrum of the seismometer transient.

The following method, out of many possible minor variations, is recommended for this purpose:

1. Set up the seismometer in the circuit shown in Fig. 4.5.1, choosing R1 and R2 to be about 50 times the seismometer coil resistance. Rd should be adjustable through the value required to give the desired degree of damping. For values near critical, and periods near 1 second, Rd will typically be a few times greater than the coil resistance Rc.

The voltage source must be capable of generating adequate deflections of the indicator, when it acts through the attenuation of the network R1Rc.

This will be about 50 times the voltage needed to give full-scale deflection when connected directly to the indicator input. The operation of the circuit, as drawn with solid lines, requires that the indicator input impedance is much higher than R2 + Rd. If this is not the case, the alternative connection (shown as a dotted line in Fig. 4.5.1) should be made instead of the solid connection A-B, and V will then require a much wider range of adjustment.

The indicator unit must have a response time much shorter than the quarter-period of the seismometer. Subject to this condition, either an oscillograph or an oscilloscope may be used, but the former, with its permanent, measurable record, is of course more accurate and convenient.

2. Close S1 and turn up the response of the indicator as far as possible or until background noise is just visible.

3. Clamp the seismometer, and actuate S2. Observe the wave-form (Fig. 4.5.1a), watching for any transient which could seriously confuse subsequent interpretation. If the wave-form is sufficiently simple, unclamp the seismometer and continue to subsequent steps.

4. Actuate S2 repeatedly, allowing the transient to die away between each movement. The indicator wave form will show steps corresponding to the indicator response, with a superimposed signal representing the mechanical response of the seismometer mass (Fig. 4.5.1b, c or d).

5. Adjust the voltage source until the indicator deflection has a convenient value.

6. Adjust the resistance Rd to provide the required degree of damping for the seismometer. Normally, damping near critical is chosen, in which case the oscillatory character of the response will be almost completely suppressed.

The actual value of damping constant beta can, if desired, now be determined by the method of Section 4.2.4.

7. Open S1 which puts the resistor R2 in circuit and should restore the under-damped wave-form when S2 is actuated. Adjust the source voltage until this waveform is well displayed (Fig. 4.5.1e).

8. Measure the deflections alpha1, alpha2, alpha3 of the indicator from its final rest position, and the deflection dc which represents the response to the input current when the oscillations have died away (Fig. 4.5.1e). Also measure the apparent free period Ts of the seismometer mass, and the value Vc of the source voltage used for this part of the experiment.

9. Estimate the amplitude alpha0 which would have occurred in the complete absence of damping. With some types of indicator unit the initial swing alpha1 will be distorted by an indicator transient, so it is better to measure a series of later swings alphar, alphar+l ... and then to determine alpha0 from the formula

where alphan and alphan+1 are the amplitudes of any other pair of successive swings. The effects of background noise and errors of measurement can be reduced by taking the mean value of several determinations, using several pairs of values of r and n from a given wave train, or by taking measurements from several wave trains. Then determine the logarithmic decrement and the true period Ts as in Section 4.2.3.

10. Restore S1 to the 'normal-damping' position. The equipment will now be in its operating mode.

It may now be shown that the electromagnetic flux-linkage G for the seismometer is given by the formula

and will be measured in volt-seconds per meter if M is in kilograms, Ts in seconds and Rc in ohms.

The indicator deflection dg produced by unit relative velocity between the coil and the magnet will be

As the indicator has been assumed to have a flat response over the entire range of frequencies which can be generated by the seismometer, the overall response to unit particle velocity in the ground will be described by one of the family of curves on Fig. 3.1a, transformation from displacement to velocity response having arisen in the transducer. The value of dg derived above sets the levels of the horizontal asymptote of the response curve. The derived values of beta and Ts select the appropriate member of the family and determine the position along the time axis.

### 4.5.2 Matching by simple superposition

When it is desired to set up a multichannel system with identical response in each channel, it becomes unnecessary to follow through the algebra in detail. The alternative, which is very easy to apply, is to generate the impulse response, and to adjust parameters by trial and error until the output wave-form of each channel can be superposed on a standard wave-form (typically drawn or photographed on a transparent base) within predetermined limits. Fig. 4.5.2 shows tracings from an actual series of such wave- forms superposed to show the effects of variations of ±10% variation of gain, damping resistance and seismometer period in relation to a standard curve. With care, final precision of the order of 1% in each of these parameters can be obtained.

### 4.5.3 Total inversion of calibration impulse

The method of Mitchell and Landisman (1969) uses the inverse Fourier transformation to predict the form of an impulse response for a given set of periods and damping constants, and then adjusts these constants to provide a least-squares fit to a digitized representation of the impulse response of the instrument under test. This process enables the harmonic response to be determined in a way which is much less sensitive to noise contamination than the more obvious method of direct Fourier transformation.

The method yields estimates of the periods and damping constants on the assumption that the coupling constant sigma2 is zero, and different sets of constants would be obtained if different values of sigma2 were to be used.

The authors make the partially erroneous comment that the solution is, therefore, only an approximation. The proper view, as we have seen above, is that the instrumental constants which would provide a given 'black box' response with sigma2 = 0 are perfectly valid combinations of the actual constants of the seismometer and galvanometer with the actual value of sigma2

### 4.5.4 Application to standard galvanometer systems

Impulse calibration checks are routine procedure on the WWSSN and some other systems, and standard impulse response curves for many combinations of seismometer and galvanometer have been published (Espinosa, Sutton and Miller, 1965).

## 4.6 Summary of recommendations

In view of the numerous possibilities which have been raised, recommendations for choosing the method most appropriate for each case are summarised as follows:

1. Piecemeal and whole-system calibrations have clearly defined areas of superiority, for setting up and for record interpretation respectively.

2. Impulse and harmonic drive methods are applicable in both types of calibration, impulses being much easier and quicker to apply, continuous drive being easier to interpret in terms of A/T values. Accuracy is comparable when applied to linear systems at levels well above noise but, as sinusoidal drive can be used over a wider range of signal levels than impulse methods, it provides superior protection against the possibility of instrumental malfunction.

3. The whole-system impulse response, stored on the intermediate record and ultimately appearing on processed output, is of growing value in the context of modern tape recording and electronic processing, and should be applied universally.

4. The least-squares method of fitting a harmonic response to an impulse provides a considerable improvement in noise immunity in comparison with direct Fourier transformation.

5. The outstanding problem lies not in the availability or accuracy of calibration methods, but in gaining worldwide acceptance of the requirement.

• ```Date created: 1/7/97