Power Spectrum Analysis:
Analysis of the power spectrum of the streamfunction output from a numerical simulation of a 2layer ocean model.
Data Analysis Project
Patricia Sánchez
April 26,2012
Motivation
• We have study Q-G systems
– Many assumptions
– Leads to set of equations of motions
– How to solve them???
– Numerical Solutions
• Quantify the Instabilities
– How they vary in time…
• GFD2 Project !!
Background
• Dataset:
– Numerical Model: solve the Q-G PV equation.
• Two-layer with thickness H1 and H2
• Uniform background current of speed Uo in top layer
and 0 in the bottom layer
• Output: PSI ψ(x,y,t) of each layer.
Layer #1
Longitude
(256)
Latitude (512 gridpoints)
Height
Method
• Power Spectrum:
– An often more useful alternative is the power spectral density (PSD), which
describes how the power of a signal or time series is distributed with frequency.
Here power can be the actual physical power, or more often, for convenience
with abstract signals, can be defined as the squared value of the signal,
– Fourier Transform has units of u×time
– power spectral density has units of u2×time2/time
Time Series
Fourier
Transform
Power
Spectral
Density (freq)
Experiment
• What do we want?
• Previous work: The estimation of the frequency-wavenumber
power spectral density is of considerable importance in the analysis
of propagating waves by an array of sensors . (Capon,J, 1969)
• Analyze the power spectrum density in terms of wavenumber.
Time Series of PSI
map
Example:
http://www.weizmann.ac.il/es
erpages/kaspi/jets/jas07_1.ht
m
1. Select a
snapshot (time)
2. Select spatial
latitude
Interpretation
Power Density
Spectrum in terms
of wavenumber
Cumulative power
Data: Output
• PSI ψ(x,y,t) snapshots of each layer. “Raw Data”
– Snapshots: every 100 days
• Animation
Time:100
Time:2400
Time:17000
Steady State
Small Scale features
Large Scale features but with small scale
ψ(x,y,t) Stream function Animation
Power density Spectrum
Time:100
Time:9100
Time:15100
Key Features:
• Power Spectrum analysis at each snapshot
• Variation in amplitude and slope at different times.
• Almost all the power (variance) is at “lower”
wavenumber.
• But, how “low” are these dominant wavenumbers?
• Next….
Next step:
• Since we know that most of the power is concentrated
in lower wavenumbers…
• Analyze the wavenumber where the power spectrum is
maximum.
• How changes over time?
Time Series
Of a Signal
Power
Density
Spectrum
Analyze the
maximum
wavenumber
Analysis of “Kmax” results
• 2nd Step: Evaluate the “Kmax”
• Remember, what do we want…
• Not clear interpretation of the results .
• Too much variability
• We need to define a new way/method to analyze the data
How to Interpret the Result:
• From power spectrum, we can see:
1. It changes in amplitude
2. It changes where is the “kmax” and how steep is
the slope
Time:15100
Refine the results
• From 1:
– Calculate how the total (maximum) cumulative
power changes with time.
• From 2:
– Approach: Locate the wavenumber at which half
the power is in lower wavenumbers.
Cumulative Power
• In an analogy to the energy signals, let us define a
function that would give us some indication of the
relative power contributions at various frequencies,
as Sf (ω), or in this experiment, wavenumbers Sk(k).
Normalized
From 0-1
½ of all cumulative
power
Wavenumber 1-50
Cumulative Power
Time:5100
Time:15100
Time series of total of cumulative
Power
• Tendency: to decrease for
the first half of the time
series.
• When the flow is more
perturbed the maximum
cumulative power increases.
K1/2 Analysis
• The wavenumber at which half the power is in lower wavenumbers)
• We know that the most of the total variance is contributed by the
“lower” wavenumbers.
• We want to quantify how of these wavenumbers vary with time.
• With this result, we could guess what
is the scale of the flow.
• It includes the signal (that your eye
already told you so you believe it),
but clean and objective
Summary
• Power Spectrum Density:
– Transform the signal from space (x,y t) domain to frequency/wavenumber
domain.
– The power spectrum show a concentration of power at low frequencies.
– It varies in amplitude and slope.
– Hypothesis: “The small scale perturbations could have more power than the
larger scale features” (Kamenkovich)
• Cumulative Power Spectrum:
– In snapshot: shows how “low” wavenumber dominates
– In time series:
• We could determine the scale of the signal
• Future work:
– Apply same analysis for:
• a different latitude to compare how different are the results.
• Instead of latitude-cross, try with longitude-cross, i.e. meridional wavenumber l analysis.
• Another experiment (model output)
– We could determine the fastest growing mode.