TIME SERIES ANALYSIS Time series – collection of observations in time: x( ti ) x( ti ) discrete time series with Δt Deterministic process: Can be predicted exactly for all the values of the independent variable ti Stochastic process: Basically unpredictable – most geophysical phenomena

FOURIER ANALYSIS OF DETERMINISTIC PROCESS Fourier Analysis is concerned with orthogonal functions: Any time series y(t) can be reproduced with a summation of cosines and sines: Fourier series Average Constants – Fourier Coefficients

Any time series y(t) can be reproduced with a summation of cosines and sines: Collection of Fourier coefficients An and Bn forms a periodogram Fourier series defines contribution from each oscillatory component n to the total ‘energy’ of the observed signal – power spectral density Both An and Bn need to be specified to build a power spectrum periodogram. Therefore, there are 2 dof per spectral estimate for the ‘raw’ periodogram. n ( )

Construct y(t) through infinite Fourier series To obtain coefficients, multiply equation above times cos and sin and integrate over all frequencies: An and Bn provide a measure of the relative importance of each frequency to the overall signal variability. e.g. if there is much more spectral energy at frequency 1 than at 2

Fourier series can also be expressed in compact form: Phase between each Fourier coefficient Bn and An, or Phase displacement at the beginning of the time series for each Cn

(j)

SUMMARY Divide by 2 for “mathematical convenience”

To obtain coefficients: Multiplying data times sin and cos functions picks out frequency components specific to their trigonometric arguments Orthogonality requires that arguments be integer multiples of total record length T = Nt, otherwise original series cannot be replicated correctly Arguments2nj/N, are based on hierarchy of equally spaced frequencies n=2n/Nt and time increment j

Steps for computing Fourier coefficients: 1) Calculate arguments nj = 2nj/N, for each integer j and n = 1. 2) For each j = 1, 2, … , N evaluate the corresponding cos nj and sin nj ; effect sums of yj cos nj and yj sin nj 3) Increase n and repeat steps 1 and 2. Requires ~N2 operations (multiplication & addition) Data from James River

Cn An Bn

m radians