Harmonic Analysis of Environmental Time Series with Missing Data or Irregular Sample Spacing

The Lomb periodogram and discrete Fourier transform are described and applied to harmonic analysis of two typical data sets, one air quality time series and one water quality time series. The air quality data is a 13 year series of 24 hour average particulate elemental carbon data from the IMPROVE station in Washington, D.C. The water quality data are from the stormwater monitoring network in Milwaukee, WI and cover almost 2 years of precipitation events. These data have irregular sampling periods and missing data that preclude the straightforward application of the fast Fourier transform (FFT). In both cases, an anthropogenic periodicity is identified; a 7-day weekday/weekend effect in the Washington elemental carbon series and a 1 month cycle in several constituents of stormwater. Practical aspects of application of the Lomb periodogram are discussed, particularly quantifying the effects of random noise. The proper application of the FFT to data that are irregularly spaced with missing values is demonstrated on the air quality data. Recommendations are given when to use the Lomb periodogram and when to use the FFT.