Removal of periodicities by differencing and monthly mean subtraction

Abstract A monthly hydrologic time series, rescaled to obtain a constant variance, has a spectrum which, in turn, has a discrete and a continuous part representing the contributions of the circularly stationary (periodic) and stationary random components, respectively. A spectral representation can be constructed for this mixed spectrum which is observed in monthly runoff and rainfall time series. Seasonal and nonseasonal differencing and monthly mean subtraction methods for the removal of the circularly stationary (periodic) component were analyzed making use of this spectral representation for the case of monthly series. It is shown both analytically and by an example that seasonal differencing removes and nonseasonal differencing effectively reduces the periodicity in the covariance function. Both kinds of differencing greatly distort the original spectrum, wiping out the original spectral contribution at the origin and greatly reducing it near the origin. Fitting an ARMA model with few parameters to the distorted spectrum is almost impossible and the fitted models are impractical for hydrologic simulation since they either yield an infinite variance or are undefined at the spectral origin. Subtraction of monthly means completely removes the periodicity in the covariance but introduces a nonstationarity which is negligible for practical purposes.