Spectral analysis of air quality data

UnrestrictedTypical air quality time series consist of consecutive hourly averages of concentrations of gaseous pollutants or composition of airborne particles. More often, time series of concentrations of particulate species are 24-hour averages observed every third or sixth day. Time series may give us important information about sources of pollutants. For instance, 7-day and monthly cycles are mostly associated with anthropogenic sources and can be analyzed to study the anthropogenic cycles and their variation over time. Moreover, species with the same periodicity might be related to the same sources and can be characterized based on their periodic behavior.; The fast Fourier transform (FFT) is an efficient means to computationally calculate the discrete Fourier transform (DFT). However, the FFT requires equally spaced data, which means therecan be no missing observations. A few isolated missing points can be filled in by interpolation, but air quality data often has long runs of missing data that cannot be replaced by interpolatedvalues in any reasonable manner. Thus, the FFT cannot usually be applied to air quality data, and that perhaps explains why there have been very few reports in the literature of Fourier methods applied to air quality time series.; In statistics, an estimate of the power spectrum at a finite set of frequencies is known as a periodogram. The periodogram is commonly used to identify significant periodicities in the data.However, it is well known that air quality time series exhibit yearly, seasonal, weekly, and daily periodicities; therefore, identifying the periodicities is not the main goal of this study. The primary purpose of this research is to show how the Fourier transform and the periodogram can be used to quantitatively characterize and classify the periodicities in time series of air pollutants. Currently, plots of hourly and monthly averages are the only way to show daily and seasonal variations. This graphical means is semi-quantitative and subjective. In this case, to systematically characterize and quantify periodic behavior of air pollutants that have been measured in different sites, an objective and quantitative method is required.; In this research several methods of addressing the problem of missing or irregularly spaced data are developed and applied to analyze air quality data. Least square fitting of sines and cosines to the data is an alternative means to estimate the discrete Fourier transform (DFT) when missing data are presented. But, because of its computational inefficiency due to repeatedly calculating sines and cosines, it was not a popular means of harmonic analysis and it was replaced by the FFT, which is very fast. However, using current powerful computers, the least square approach can be applied to air quality time series. Schuster, in 1897, realized that the periodogram, which is calculated by the Fourier transform, allows one to "separate any true periodicity of a variable from among its irregular changes" [67]. Using the Schuster periodogram allows the exhibition of all the periodicities in one graph, and it avoids plotting hundreds of time-averaged concentration graphs. In 1975 , Lomb developed a version of the least squares fitting equations to directly estimate the power spectrum in the presence of missing or irregularly spaced data. This research applied the Lomb periodogram to estimate the fraction of the variance explained by the yearly, seasonal, weekly, and daily periodicities. In many cases, the classic or Schuster periodogram can be used instead of the more esoteric Lomb periodogram and can be normalized to give the fraction of variance explained. Another major goal of this project is to characterize the periodic variation in air quality data. For this purpose, a method is developed to automatically classify and group species based on the configuration of the harmonics, and is demonstrated in this report.; Air quality time series are non-stationary and often have a trend; therefore, periodic behavior of long time series will vary over time. To specifically study time to time variation of power, sliding window Fourier transform is applied to air quality data. This method enables us to analyze the Fourier transform of a specific window of time, i.e. one year, which slides through the whole sampling time. Since weekly and monthly periodicities are mostly associated with anthropogenic sources, these frequencies are analyzed to study the changes in anthropogenic cycles.; A computer code is also implemented to estimate the inverse Fourier transform by the Lomb method. This method can be applied to reproduce the initial time series while its missing points are fully treated. Moreover, when frequency or period of the variabilities are known, the least square fitting can be used to evaluate the daily, weekly, monthly, seasonal, or yearly patterns of concentration of different species.; In conclusion, the Lomb periodogram is an appropriate tool when there are many missing values and long gaps in the data, and when the sampling time is irregular or random; otherwise, the Schuster periodogram mean-gapping would be a reasonable method to apply, and would provide the same result as the Lomb method. It must be emphasized that detecting peaks in spectral analysis of unequally spaced data requires cautious concern, and for this reason several procedures are discussed and a modified computer code is developed, which could be applied for our specific needs in air quality analysis