Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density

Lecture 01Using MatLab Lecture 02Looking At Data Lecture 03Probability and Measurement Error Lecture 04Multivariate Distributions Lecture 05Linear Models Lecture 06The Principle of Least Squares Lecture 07Prior Information Lecture 08Solving Generalized Least Squares Problems Lecture 09Fourier Series Lecture 10Complex Fourier Series Lecture 11Lessons Learned from the Fourier Transform Lecture 12Power Spectral Density Lecture 13Filter Theory Lecture 14Applications of Filters Lecture 15Factor Analysis Lecture 16Orthogonal functions Lecture 17Covariance and Autocorrelation Lecture 18Cross-correlation Lecture 19Smoothing, Correlation and Spectra Lecture 20Coherence; Tapering and Spectral Analysis Lecture 21Interpolation Lecture 22 Hypothesis testing Lecture 23 Hypothesis Testing continued; F-Tests Lecture 24 Confidence Limits of Spectra, Bootstraps SYLLABUS

purpose of the lecture compute and understand Power Spectral Density of indefinitely-long time series

Nov 27, 2000 Jan 4, 2011 ground vibrations at the Palisades NY seismographic station similar appearance of measurements separated by 10+ years apart time, minutes

stationary time series indefinitely long but statistical properties don’t vary with time

time, minutes assume that we are dealing with a fragment of an indefinitely long time series time series, d duration, T length, N

one quantity that might be stationary is …

“Power” 0 T

0 T Power mean-squared amplitude of time series

How is power related to power spectral density ?

write Fourier Series as d = Gm were m are the Fourier coefficients

now use

coefficients of sines and cosines coefficients of complex exponentials Fourier Transform equals 2/T

so, if we define the power spectral density of a stationary time series as the integral of the p.s.d. is the power in the time series

units if time series d has units of u coefficients C also have units of u Fourier Transform has units of u × time power spectral density has units of u 2 × time 2 /time e.g.u 2 -s or equivalently u 2 /Hz

we will assume that the power spectral density is a stationary quantity

when we measure the power spectral density of a finite-length time series, we are making an estimate of the power spectral density of the indefinitely long time series the two are not the same because of statistical fluctuation

finally we will normally subtract out the mean of the time series so that power spectral density represents fluctuations about the mean value

Example 1 Ground vibration at Palisades NY

enlargement

periods of a few seconds

power spectral density

frequencies of a few tenths of a Hz periods of a few seconds

cumulative power power in time series

Example 2 Neuse River Stream Flow

period of 1 year

power spectral density, s 2 (f) frequency f, cycles/day power spectra density s 2 (f), (cfs) 2 per cycle/day

power spectral density, s 2 (f) frequency f, cycles/day power spectra density s 2 (f), (cfs) 2 per cycle/day period of 1 year

Example 3 Atmospheric CO 2 (after removing anthropogenic trend)

enlargement

period of 1 year

power spectral density frequency, cycles per year

power spectral density frequency, cycles per year 1 year period ½ year period

shallow side: 1 year and ½ year out of phase steep side: 1 year and ½ year in phase

cumulative power power in time series

Example 3: Tides 90 days of data

enlargement 7 days of data

enlargement 7 days of data period of day ½

power spectral density cumulative power power in time series

power spectral density cumulative power power in time series about ½ day period about 1 day period fortnighly (2 wk) tide

MatLab dtilde= Dt*fft(d-mean(d)); dtilde = dtilde(1:Nf); psd = (2/T)*abs(dtilde).^2; Fourier Transform delete negative frequencies power spectral density

MatLab pwr=df*cumsum(psd); Pf=df*sum(psd); Pt=sum(d.^2)/N; power as a function of frequency total power should be the same!