Restoration

Whenever a time-varying quantity has to be measured, there is an inevitable blurring due to the nonzero time interval necessary to make a single measurement. Consequently, the measurement relating to a given instant always lumps together values that occurred during the measurement interval. the time-varying quantity is varying slowly, or if the time resolution is short, the measurement may be very good; but some degree of smoothing is in principle always present. That means that a measured waveform never faithfully represents the original quantity. Therefore, one may ask what correction has to be applied to the measurement. In the development given below it is supposed that convolution is involved, but it is understood that convolution is merely an important case of the more general linear functional. Measurements may also involve a little nonlinearity and a little hysteresis. Even so, as will be seen, there is enough complication in the simple presentation below to suffice for a first study. A further limitation may be mentioned. In some subjects, including astronomy, meteorology, and geophysics, the measurements, or observations, may be all we know about the time-varying quantity; in fact the purpose of the observations may be to find out what is there. In such a case the “time-varying quantity” is just a concept without any reality of its own; only the measurements exist and are available to work with. In experimental subjects, as distinct from the observational, the underlying quantity may be verifiable by alternative methods, but in the final analysis measurement is always conducted to a finite resolution. Procedures for combating the smoothing effect of instrumental intervention are known as restoration.