Environmental Data Analysis with MatLab Lecture 12: Power Spectral Density.

Environmental Data Analysis with MatLab

Lecture 12:

Power Spectral Density

Lecture 01 Using MatLabLecture 02 Looking At DataLecture 03 Probability and Measurement Error Lecture 04 Multivariate DistributionsLecture 05 Linear ModelsLecture 06 The Principle of Least SquaresLecture 07 Prior InformationLecture 08 Solving Generalized Least Squares ProblemsLecture 09 Fourier SeriesLecture 10 Complex Fourier SeriesLecture 11 Lessons Learned from the Fourier TransformLecture 12 Power Spectral DensityLecture 13 Filter Theory Lecture 14 Applications of Filters Lecture 15 Factor Analysis Lecture 16 Orthogonal functions Lecture 17 Covariance and AutocorrelationLecture 18 Cross-correlationLecture 19 Smoothing, Correlation and SpectraLecture 20 Coherence; Tapering and Spectral Analysis Lecture 21 InterpolationLecture 22 Hypothesis testing Lecture 23 Hypothesis Testing continued; F-TestsLecture 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

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, dduration, Tlength, N

one quantity that might be stationary is …

“Power”

0T

0T

Power

mean-squared amplitude of time series

How is power related topower spectral density ?

write Fourier Series asd = Gmwere m are the Fourier coefficients

now use

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

unitsif 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. u2-s or equivalently u2/Hz

we will assume that thepower 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 samebecause of statistical fluctuation

finally

we will normally subtract out the mean of the time series

so that power spectral densityrepresents fluctuations about the

mean value

Example 1Ground vibration at Palisades NY

0 200 400 600 800 1000 1200 1400 1600

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

velo

city,

mic

rons/s

time, seconds

enlargement

0 5 10 15 20 25 30 35 40 45-0.4

-0.2

0

0.2

0.4

velo

city,

mic

rons/s

time, seconds

enlargement

0 5 10 15 20 25 30 35 40 45-0.4

-0.2

0

0.2

0.4

velo

city,

mic

rons/s

time, seconds

periods of a few seconds

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

p.s

.d,

um2 /s

2 per

Hz

frequency, Hz

power spectral density

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

p.s

.d,

um2 /s

2 per

Hz

frequency, Hz

power spectral density

frequencies of a few tenths of a Hzperiods of a few seconds

cumulative power

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

pow

er

frequency, Hz

power in time series

0 500 1000 1500 2000 2500 3000 3500 40000

1

2

x 104

time, days

disc

harg

e, c

fs

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

2

4

6

8

x 109

frequency, cycles per dayPS

D,

(cfs

)2 p

er c

ycle

/day

Example 2Neuse River Stream Flow

0 500 1000 1500 2000 2500 3000 3500 40000

1

2

x 104

time, days

disc

harg

e, c

fs

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

2

4

6

8

x 109

frequency, cycles per dayPS

D,

(cfs

)2 p

er c

ycle

/day

Example 2Neuse River Stream Flow

period of 1 year

0 500 1000 1500 2000 2500 3000 3500 40000

1

2

x 104

time, days

disc

harg

e, c

fs

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

2

4

6

8

x 109

frequency, cycles per dayPS

D,

(cfs

)2 p

er c

ycle

/day

power spectral density, s2(f)

frequency f, cycles/day

pow

er s

pect

ra d

ensi

tys2 (

f), (

cfs)

2 per

cyc

le/d

ay

0 500 1000 1500 2000 2500 3000 3500 40000

1

2

x 104

time, days

disc

harg

e, c

fs

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

2

4

6

8

x 109

frequency, cycles per dayPS

D,

(cfs

)2 p

er c

ycle

/day

power spectral density, s2(f)

frequency f, cycles/day

pow

er s

pect

ra d

ensi

tys2 (

f), (

cfs)

2 per

cyc

le/d

ay

period of 1 year

Example 3Atmospheric CO2

(after removing anthropogenic trend)

0 5 10 15 20 25 30 35 40 45 50

-4

-2

0

2

4

time, years

CO

2, p

pm

0 1 2 3 4 50

1

2

3

frequency, cycles per year

log1

0 ps

d of

CO

2

0 0.5 1 1.5 2 2.5 3-3

-2

-1

0

1

2

3

4

time, years

CO

2, p

pmenlargement

0 0.5 1 1.5 2 2.5 3-3

-2

-1

0

1

2

3

4

time, years

CO

2, p

pmenlargement

period of 1 year

0 5 10 15 20 25 30 35 40 45 50

-4

-2

0

2

4

time, years

CO

2, p

pm

0 1 2 3 4 50

1

2

3

frequency, cycles per year

log1

0 ps

d of

CO

2

power spectral density

frequency, cycles per year

0 5 10 15 20 25 30 35 40 45 50

-4

-2

0

2

4

time, years

CO

2, p

pm

0 1 2 3 4 50

1

2

3

frequency, cycles per year

log1

0 ps

d of

CO

2

power spectral density

frequency, cycles per year

1 year period ½ year

period

0 0.5 1 1.5 2 2.5 3-3

-2

-1

0

1

2

3

4

time, years

CO

2, p

pm

0 0.5 1 1.5 2 2.5 3-3

-2

-1

0

1

2

3

4

time, years

CO

2, p

pm

shallow side: 1 year and year½ out of phase steep side: 1 year and ½year in phase

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

frequency, cycles per year

pow

er

cumulative power

power in time series

Example 3:Tides

0 20 40 60 80 100 120-3

-2

-1

0

1

2

3

4

5

Ele

vation,

ft

time, days

90 days of data

enlargement

0 1 2 3 4 5 6 7-2

-1

0

1

2

3

4

Ele

vation,

ft

time, days

7 days of data

enlargement

0 1 2 3 4 5 6 7-2

-1

0

1

2

3

4

Ele

vation,

ft

time, days

7 days of data

period of day½

0 0.5 1 1.5 2 2.5 3-1

0

1

2

3

frequency, cycles per day

log1

0 ps

d

0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

frequency, cycles per day

log1

0 ps

d

power spectral density

cumulative power

power in time series

0 0.5 1 1.5 2 2.5 3-1

0

1

2

3

frequency, cycles per day

log1

0 ps

d

0 0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

frequency, cycles per day

log1

0 ps

d

power spectral density

cumulative power

power in time series

about ½ day period

about1 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

total power

should be the same!