Fig. 2 shows the frequency response (amplitude response) of a seismic transducer, i.e. the amplitude ao of the "output" (xi(t) - x(t) in Fig. 1) divided by the amplitude ai of the "input" (xi(t)) whether this is displacement, velocity or acceleration, of a frequency in the range around fr, which has been oscillating long enough that the output amplitude (and phase) has stabilized. Therefore, Fig. 2 cannot be used to predict the response to a single pulse input like an FWD deflection.

This is clearly illustrated in the example in Fig. 3, which shows a (calculated) response of a seismic transducer (with fr = 4.5 Hz and D = 0.7) to a 18.5 Hz sinusoidal oscillation starting at t = 0. Fig. 2 would predict an amplitude response close to 1 (100 %) for f/fr = (18.5 / 4.5) = 4.1, which Fig. 3 also indicates after some full cycles (phase shift should be disregarded), but it is also clearly seen that during the first half cycle, which could very well resemble an FWD deflection pulse, the amplitude (i.e. peak) response is only 0.72, i.e. 28 % lower than the input amplitude.

Fig. 4 shows the same example as in Fig. 3; however, the input is zero after the first half cycle. The negative peak on the output is equal to the accumulated displacement of the seismic mass during the half cycle (half-sine) input, and certainly not close to zero as intended (and as one might assume if Fig. 2 was believed to apply for this case). When the input has finished its half-sine part and stays at zero, the seismic mass will then settle (with a damped (decaying) oscillation of frequency f_r), creating a "false" output signal.

Fig. 5 shows a (calculated) chart that may be used to find the error epr (on the peak response only) of a seismic transducer’s response to a half-sine input.

f_r = 4.5 Hz and t_p = 27 msec (= 0.027 sec, as in Fig.’s 3 and 4) gives 1/(2 f_rt_p) = 4.1, which for D = 0.7 gives e_pr = -28% as seen in Fig. 3.

Fig. 5 indicates that in general a small peak error e_pr can be achieved by using a seismic transducer with a low damping and a low f_r, but this will introduce some negative "side effects" as explained in the following

Example 2:

t_p = 27 msec., f_r = 3.3 Hz and D = 0 gives 1/(2 f_rt_p) = 5.6 and e_pr = -1.8 %, which may be acceptable, but as shown in Fig. 6, the time history will be seriously distorted, in particular for t > 27 msec., where the input is zero. Another disadvantage of a low damping (and small f_r) is that the seismic mass will need significant time to settle (i.e. stop oscillating) after any disturbance, like e.g. upon lowering of the seismometer to the ground and upon release of the FWD drop weight, which will cause the road surface to move upwards slightly, initiating a seismic mass movement which for some combinations of drop height, mass of drop weight and type of pavement may cause a significant error, especially on the small deflections far from the loading center.

For a given loading time (t_p) and seismometer damping (D), the measured peak value may of course be corrected by a correction factor (which can be established by a calibration procedure), but firstly, such factor will not be usable to correct time history data, and secondly, if t_p or D subsequently changes, then this factor will no longer be valid even for peak value correction. For the example with a seismometer with f_r = 3.3 Hz and a low damping (0 < D < approx. 0.1), the (theoretical) peak value sensitivity to variations in t_p will be some 0.2 - 0.3 % per msec, and sensitivity to damping will be some 1.5 - 2% for a change in D of 0.05. E.g., D = 0.1 gives e_pr = approx. -5% in Example 2.

For the first example (see also Fig.’s 3 and 4) which applies for a typical geophone, the sensitivities to t_p and D will be approx. 0.8% per msec and approx. 1% per 0.05, respectively, if a fixed correction factor (based only on the peak value) was used. The sensitivity to D is not really a problem in case of a geophone, as D is easy to control and keep constant (by means of a shunt resistor), but the sensitivity to t_p is severe, and please keep in mind that Fig. 5 only deals with peak value errors - the post-peak responses (time histories) will be much more erroneous. Therefore, a more comprehensive method of compensation will be necessary to obtain acceptable FWD deflection responses.

In selecting a seismic transducer, seismometers with low f_r and D should probably be ruled out because they are in general relatively heavy, bulky and fragile.
Furthermore, damping is hard to control (i.e., keep at a constant, low level), and the settling time of the seismic mass is long compared to the falling time of the FWD drop weight.
It may seem an advantage that the sensing element of a seismometer, typically an LVDT, can be calibrated statically;

However, the error sources mentioned earlier are dependent on the mechanical properties of a seismometer only, so a dynamic calibration procedure is a must to evaluate these properties and will then in addition account for any LVDT calibration error.