Many environmental time series have seasonal structures. This article presents an approach to identifying the rare events of such series based on time-series residuals. The methods justify the application of classical peaks over threshold methods to estimated versions of the one-step-ahead prediction errors of the series. Such methods enable the seasonal means, variances and autocorrelations of the series to be taken into account. Even in stationary settings, the proposed strategy is useful as it bypasses the need for blocking runs of extremes. The mathematics are justified via a limit theorem for a periodic autoregressive moving-average time series. A detailed application to a daily temperature series from Griffin, Georgia, is pursued.