A. If it is assumed that a = 0.001A; that b and c are successively an order of magnitude
greater than a, i.e., b= 0.01A and c = 0.1A; and that the sigaal frequency { is 10 cps, then the
relative errors at the end of integration periods of 5an¢ 10 sec are
~
For 5 sec
For 10 sec
Eq = 0.1%
Ey = 15% Ep = 1625%
Eq = 0.1%
Ey= 20%
Ep = 2500%
Similarly a linear error at + b in the primary data woulc give errors of the order of 100 per
cent in the velocities and 10,000 per cent in the displacements. Of course, if all of the error
coefficients do not have the same sign, those of opposing sigr will counteract each other and
reduce the relative error in the integrals, although not usually to extinction.
B.5
CORRECTION PROCEDURE
Evidently, innocuous acceleration errors can become monstrous in their effect on the desired displacement information, and to obtain useful results correction must be made. Direct
correction in the primary data is often inadequate; it is difficult, for example, to recognize a
zero shift of 0.1 per cent of the peak signal amplitude. Correction can be made, however, in
the velocity data by fitting a continuous curve which when added or subtracted will cause the
velocity to satisfy both the initial and final conditions that it be zero before and after passage
of the transient signal. Sometimes it is necessary to approximate the second condition because
of the multiplicity of ground-motion signals which arrive at a station over various refraction
and reflection paths, but generally some reasonable form of correction can be applied.
Similar correction would appear to be applicable to displacement data, but, in the absence
of independent posttransient measurements of residual displacement, no independent criterion
for a thermal condition exists other than the intuitive one that residual displacement should be
less than its maximum value and that adjustment of the residual value to nearly zero should
not seriously affect amplitudes of early peaks.
B.6
EXAMPLE
The ground-transmitted acceleration data observed at Station 650.06 on Parry for Mike
shot provide an extreme example of multiple correction. Signal strength «-as very low, with
consequent poor signal-noise conditions, and duration was very long. Direct integration of the
acceleration data between about 7 and 35 sec yields the curve V, in Fig. B.1. The noise has
been reduced nearly to extinction in the integral, and the curve suggests a strong parabolic increase in velocity and a roughly sinusoidal variation with a period of about 30 sec. Considera-
tion of arrival time, frequency, and magnitudes suggests that the 1.5-cps component of the signal between about 10 and 16 sec may be a reflected pulse from the deep basalt and is probably
the strongest part of the true ground motion. This 1.5-cps signal is distinguishable but is
minor compared to the parabolic and 30-sec periodic components. A parabolic-error curve
(the dashed line in Fig. B.1) was fitted to the velocity curve at three points, and the primary
velocity error equation Ey, = 0.001875t? + 0.03375t — 0.325 was derived from {t. The derivative
of this equation, converted from feet per second to g units, gives the linear correction function
for the acceleration
Ca, = -E,, =—0.00011646t — 0.001048
Acceleration data within the interval between 7 and 24 sec were then corrected and integrated
to give the curve V, in Fig. B.2. This curve, which is plotted on a velocity scale 10 times that
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