B.3. INTEGRATION Integration of acceleration-time data involves summation of the trapezoidal areas defined by pairs of consecutive data points and the axis of zero acceleration. Magnitudes of elemental areas are derived by multiplying the mean acceleration amplitudes of a pair of adjacent data by the time interval between them. These areas represent incremental velocities, and their cumulative sum represents approximately the instantaneous velocity at the time represented by the last datum included in the summation. Evidently, if adjacent points are sufficiently close, the stepwise summation approximates closely the true area between the acceleration-time curve and zero axis. It is also evident that both stability of the zero balance of the record trace and its proper evaluation within the time span of the integration exercise strong control on the relative error in the result. B.4. INFLUENCE OF ERRORS The acceleration-time signal is some undefined function of time to which an error, either constant or a linear or sinusoidal function of time, has been added in recording or reduction. Since the error is added, the effect of integration upon it may be considered independently of the signal. The relative effect of the error on the integrated data can then be estimated roughly. The integral of a sine function of the form E sin wt is —(E/w) cos wt, where E is the am- plitude and w ts the frequency in radians per second. Consequently the amplitude of the integral (disregarding sign) will be less than E for all values of w greater than 1. Or, since w = anf, where f is frequency in cps, then E/w will be less than E for all frequencies greater than Yn, or approximately 4 cps. This suggests that any sinusoidal error of frequency greater than Y cps will be diminished in amplitude by integration. But the importance of an error is its relative magnitude with respect to the signal, and relative error will remain unchanged or be diminished by integration only for those components of the signal which have frequencies equal to or greater than the error frequency regardless of its relation to the -cps limit. It is apparent that high-frequency error such as noise will become negligible with respect to signals of normal ground-motion range, less than 40 cps; but even the higher frequency ground-motion signals will be diminished in processing. It is also evident that integrations over short intervals, of the order of a few seconds, will be affected principally by constant or linear errors or by portions of periodic errors which may be approximated by linear or parabolic error functions. Integrations extending over very long intervals will show the influence of sinusoidal errors as such. A constant error a becomes upon integration at + b, and double integration makes it hat? + bt +c. Similarly a linear error, at + b, in primary data becomes Yat? + bt + c in the first inte- gral and ¥,at® + Abt? + ct +d in the double integral. The significance of these errors becomes evident if it is assumed that the primary data are represented by sine function of amplitude A and frequency f and that there is a constant error a. The relative error is expressed as the ratio of the error at any time t to the maximum signal amplitude. The maximum signal amplitude for the primary data is A; for the once-integrated data, velocity V is A/f; and for the doubly integrated data, displacement D is A/f*. The relative errors are then for the initial constant error a Ea = Ey v = (at + b) £ A f? Ep = (fat? + bt + ez It is evident that, as t increases, the relative errors Ey and Ey become rapidly greater than 51