Wilson, Lister, and Riemann (2012) showed that forest phenometrics can be derived from Fourier analysis of vegetation index and climatological time series. Their method was inspired by the FASIR NDVI time series smoothing technique developed in Sellers et. al. (1994).
Fourier analysis is based on the principle that any discrete or continuous function (e.g. a signal or time series) can be decomposed into a series of triogonometric functions with different frequencies which, when summed, will constructively and destructively interfere with one another to produce the original function. These trigonometric functions can then be described simply by their coefficients, which reflect the amplitudes of each frequency (one for the sine component and one for the cosine). The original function can then be represented as a series of the trigonometric coefficients, plus a single “offset” coefficient (corresponding to a frequency of zero).
Discrete Fourier transformation transforms time series data from the time domain to the frequency domain. It provides the amplitudes of the harmonics at each frequency relative to the total duration of the time series, which can be depicted as a frequency spectrum.
In temperate regions, vegetation index and climatological time
series exhibit obvious annual harmonics over multi-year periods (Figure
1). They also contain less obvious sub-annual (higher-frequency)
harmonics, which can be revealed by discrete Fourier transformation
(Figure 2). The amplitudes of these harmonics (corresponding to the
coefficients of the underlying trigonometric functions) can be
interpreted as phenometrics, since they reflect the significance of
seasonal cycles in the time series. For example, the time series of a
vegetation index of a primarily deciduous forest plot would be expected
to exhibit a more significant (greater amplitude) annual harmonic than
that of a primarily coniferous plot.
The time-domain trigonometric functions of the harmonics can be derived from the transformed time series by applying the inverse discrete Fourier transformation. The sum of the first few harmonic functions (plus the “offset”) can then be shown to approximate the original time series (Figure 3), demonstrating that the defining coefficients of the Fourier series can meaningfully represent the harmonic properties of the time series and, therefore, the underlying phenology.
In Figure 3 below, we start with the first EVI harmonic at the top, and successively add harmonics up to the fifth, at the bottom. In these graphs, the black lines represent summands, and the blue lines represent sums. In the bottom graph, the actual EVI values are plotted.
Note: These harmonics were derived from a Fourier analysis of a longer time series, from 2002-2019. The graphs in Figures 1 and 3 show a magnified segment of the time series for clarity.
Figure 3. Approximation of EVI Time Series by Fourier Series Summation
References
Neto, João. (2019). Fourier Transform: A R Tutorial. Fc.ul.pt. http://www.di.fc.ul.pt/~jpn/r/fourier/fourier.html
Sellers, P. J., Tucker, C. J., Collatz, G. J., Los, S. O., Justice, C. O., Dazlich, D. A., & Randall, D. A. (1994). A global 1 by 1 NDVI data set for climate studies. Part 2: The generation of global fields of terrestrial biophysical parameters from the NDVI. International Journal of remote sensing, 15(17), 3519-3545.
Wilson, B. T., Lister, A. J., & Riemann, R. I. (2012). A nearest-neighbor imputation approach to mapping tree species over large areas using forest inventory plots and moderate resolution raster data. Forest ecology and management, 271, 182-198.