Environmental Data Analysis with MatLab
2nd Edition
Lecture 12:
Power Spectral Density
SYLLABUS
Lecture 01
Lecture 02
Lecture 03
Lecture 04
Lecture 05
Lecture 06
Lecture 07
Lecture 08
Lecture 09
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Lecture 23
Lecture 24
Using MatLab
Looking At Data
Probability and Measurement Error
Multivariate Distributions
Linear Models
The Principle of Least Squares
Prior Information
Solving Generalized Least Squares Problems
Fourier Series
Complex Fourier Series
Lessons Learned from the Fourier Transform
Power Spectral Density
Filter Theory
Applications of Filters
Factor Analysis
Orthogonal functions
Covariance and Autocorrelation
Cross-correlation
Smoothing, Correlation and Spectra
Coherence; Tapering and Spectral Analysis
Interpolation
Hypothesis testing
Hypothesis Testing continued; F-Tests
Confidence Limits of Spectra, Bootstraps
Goals of the lecture
compute and understand
Power Spectral Density
of indefinitely-long time series
ground vibrations at the Palisades NY seismographic station
Nov 27, 2000
time, minutes
Jan 4, 2011
time, minutes
similar appearance of measurements separated by 10+ years apart
stationary time series
indefinitely long
but
statistical properties don’t vary with time
assume that we are dealing with a fragment
of an indefinitely long time series
time, minutes
time series, d
duration, T
length, N
one quantity that might be stationary is …
“Power”
T
0
Power
T
0
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
now use
coefficients of complex
exponentials
equals 2/T
coefficients of
sines and cosines
Fourier
Transform
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 u2×time2/time
e.g.
or equivalently
u2-s
u2/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
0.8
0.6
velocity, microns/s
0.4
0.2
0
-0.2
-0.4
-0.6
0
200
400
600
800
1000
time, seconds
1200
1400
1600
enlargement
velocity, microns/s
0.4
0.2
0
-0.2
-0.4
0
5
10
15
20
25
time, seconds
30
35
40
45
enlargement
periods of a few seconds
velocity, microns/s
0.4
0.2
0
-0.2
-0.4
0
5
10
15
20
25
time, seconds
30
35
40
45
power spectral density
1
p.s.d, um2/s 2 per Hz
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
frequency, Hz
0.7
0.8
0.9
1
power spectral density
1
p.s.d, um2/s 2 per Hz
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
frequency, Hz
frequencies of a few tenths of a Hz
periods of a few seconds
0.7
0.8
0.9
1
cumulative power
0.025
power
0.02
0.015
0.01
power
in time
series
0.005
0
0
0.1
0.2
0.3
0.4
0.5
0.6
frequency, Hz
0.7
0.8
0.9
1
Example 2
Neuse River Stream Flow
4
discharge, cfs
x 10
2
1
ycle/day
0
0
500
9
x 10
8
1000
1500
2000 2500
time, days
3000
3500
4000
Example 2
Neuse River Stream Flow
4
discharge, cfs
x 10
1
0
ycle/day
period of 1 year
2
0
500
9
x 10
8
1000
1500
2000 2500
time, days
3000
3500
4000
discharge, cfs
2
1
PSD, (cfs) per cycle/day
power spectra density
s2(f), (cfs)22 per cycle/day
0
0
500
1000
1500
2000 2500
time, days
3000
3500
4000
9
x 10
power spectral density, s2(f)
8
6
4
2
0
0
0.005
0.01
0.015 0.02 0.025 0.03 0.035
frequency, cycles per day
frequency f, cycles/day
0.04 0.045
0.05
discharge, cfs
2
1
PSD, (cfs) per cycle/day
power spectra density
s2(f), (cfs)22 per cycle/day
0
0
500
1000
1500
2000 2500
time, days
3000
3500
4000
9
x 10
power spectral density, s2(f)
8
6
4
2
0
0
0.005
0.01
0.015 0.02 0.025 0.03 0.035
frequency, cycles per day
frequency f, cycles/day
period of 1 year
0.04 0.045
0.05
Example 3
Atmospheric CO2
(after removing anthropogenic trend)
CO2, ppm
4
2
0
-2
-4
0
log10 psd of CO2
3
2
1
5
10
15
20
25
30
time, years
35
40
45
50
4
enlargement
3
CO2, ppm
2
1
0
-1
-2
-3
0
0.5
1
1.5
time, years
2
2.5
3
4
enlargement
3
CO2, ppm
2
1
0
-1
-2
-3
0
period of 1 year
0.5
1
1.5
time, years
2
2.5
3
CO2, ppm
4
2
power spectral density
0
-2
-4
0
5
10
15
20
25
30
time, years
35
40
45
log10 psd of CO2
3
2
1
0
0
1
2
3
frequency, cycles per year
4
frequency, cycles per year
5
50
CO2, ppm
4
2
power spectral density
0
-2
-4
0
5
10
log10 psd of CO2
3
15
20
1 year
period
2
25
30
time, years
35
40
45
½ year
period
1
0
0
1
2
3
frequency, cycles per year
4
frequency, cycles per year
5
50
4
3
CO2, ppm
2
1
0
-1
-2
-3
0
0.5
1
1.5
time, years
2
2.5
3
shallow side: 1 year and
½ year out of phase
4
3
steep side: 1
year and ½
year in phase
CO2, ppm
2
1
0
-1
-2
-3
0
0.5
1
1.5
time, years
2
2.5
3
cumulative power
5
4.5
4
3.5
power
3
power
in time
series
2.5
2
1.5
1
0.5
0
0
1
2
3
4
frequency, cycles per year
5
6
Example 3:
Tides
5
4
Elevation, ft
3
2
1
0
-1
-2
-3
0
20
40
60
time, days
90 days of data
80
100
120
enlargement
4
3
Elevation, ft
2
1
0
-1
-2
0
1
2
3
4
time, days
7 days of data
5
6
7
enlargement
period of
day½
4
3
Elevation, ft
2
1
0
-1
-2
0
1
2
3
4
time, days
7 days of data
5
6
7
power spectral density
3
log10 psd
2
1
0
-1
0
1
0.5
1
1.5
frequency, cycles per day
2
2.5
3
cumulative power
log10 psd
0.5
0
power
in time
series
-0.5
-1
0
0.5
1
1.5
frequency, cycles per day
2
2.5
3
power spectral density
3
log10 psd
2
0
-1
about
1 day
period
fortnighly
(2 wk)
tide
1
0
1
0.5
about
½ day
period
1
1.5
frequency, cycles per day
2
2.5
3
cumulative power
log10 psd
0.5
0
power
in time
series
-0.5
-1
0
0.5
1
1.5
frequency, cycles per day
2
2.5
3
MatLab
dtilde= Dt*fft(d-mean(d)); Fourier Transform
dtilde = dtilde(1:Nf); delete negative frequencies
psd = (2/T)*abs(dtilde).^2; 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
total power
should be
the same!