Environmental Data Analysis with MatLab Lecture 10: Complex Fourier Series.
1.Our Solar System: What does it tell us? 2. Fourier Analysis i. Finding periods in your data ii....
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1. Our Solar System: What does it tell us?
2. Fourier Analysis
i. Finding periods in your data
ii. Fitting your data
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Earth
Distance: 1.0 AU (1.5 ×1013 cm)
Period: 1 year
Radius: 1 RE (6378 km)
Mass: 1 ME (5.97 ×1027 gm)
Density 5.50 gm/cm3 (densest)
Satellites: Moon (Sodium atmosphere)
Structure: Iron/Nickel Core (~5000 km), rocky mantle
Temperature: -85 to 58 C (mild Greenhouse effect)
Magnetic Field: Modest
Atmosphere: 77% Nitrogen, 21 % Oxygen , CO2, water
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Internal Structure of the Earth
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Venus
Distance: 0.72 AU
Period: 0.61 years
Radius: 0.94 RE
Mass: 0.82 ME
Density 5.4 gm/cm3
Structure: Similar to Earth Iron Core (~3000 km), rocky mantle
Magnetic Field: None (due to slow rotation)
Atmosphere: Mostly Carbon Dioxide
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Crust:
1. Silicate Mantle
Nickel-Iron Core
Venus is believed to have an internal structure similar to the Earth
Internal Structure of Venus
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Mars
Distance: 1.5 AU
Period: 1.87 years
Radius: 0.53 RE
Mass: 0.11 ME
Density: 4.0 gm/cm3
Satellites: Phobos and Deimos
Structure: Dense Core (~1700 km), rocky mantle, thin crust
Temperature: -87 to -5 C
Magnetic Field: Weak and variable (some parts strong)
Atmosphere: 95% CO2, 3% Nitrogen, argon, traces of oxygen
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Internal Structure of Mars
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Mercury
Distance: 0.38 AU
Period: 0.23 years
Radius: 0.38 RE
Mass: 0.055 ME
Density 5.43 gm/cm3 (second densest)
Structure: Iron Core (~1900 km), silicate mantle (~500 km)
Temperature: 90K – 700 K
Magnetic Field: 1% Earth
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Internal Structure of Mercury
1. Crust: 100 km
2. Silicate Mantle (25%)
3. Nickel-Iron Core (75%)
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Moon
Radius: 0.27 RE
Mass: 0.011 ME
Density: 3.34 gm/cm3
Structure: Dense Core (~1700 km), rocky mantle, thin crust
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The moon has a very small core, but a large mantle (≈70%)
Internal Structure of the Moon
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Comparison of Terrestrial Planets
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http://astronomy.nju.edu.cn/~lixd/GA/AT4/AT411/HTML/AT41105.htm
R = 0.28 REarth
M = 0.015 MEarth
= 3.55 gm cm–3
R = 0.25 REarth
M = 0.083MEarth
= 3.01 gm cm–3
R = 0.41 REarth
M = 0.025MEarth
= 1.94 gm cm–3
R = 0.38 REarth
M = 0.018 MEarth
= 1.86 gm cm–3
Note: The mean density increases with increasing distance from Jupiter
Satellites
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Internal Structure of Titan
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Mercury
MarsVenus
Earth
Moon
1
2
3
4
5
7
10
0.2 0.4
Radius (REarth)
(g
m/c
m3)
0.6 0.8 1 1.2 1.4 1.6 1.8 2
No iron
Earth-likeIron enriched
From Diana Valencia
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Jupiter
Distance: 5.2 AU
Period: 11.9 years
Diameter: 11.2 RE (equatorial)
Mass: 318 ME
Density 1.24 gm/cm3
Satellites: > 20
Structure: Rocky Core of 10-13 ME, surrounded by liquid metallic hydrogen
Temperature: -148 C
Magnetic Field: Huge
Atmosphere: 90% Hydrogen, 10% Helium
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From Brian Woodahl
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Saturn
Distance: 9.54 AU
Period: 29.47 years
Radius: 9.45 RE (equatorial) = 0.84 RJ
Mass: 95 ME (0.3 MJ)
Density 0.62 gm/cm3 (least dense)
Satellites: > 20
Structure: Similar to Jupiter
Temperature: -178 C
Magnetic Field: Large
Atmosphere: 75% Hydrogen, 25% Helium
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Uranus
Distance: 19.2 AU
Period: 84 years
Radius: 4.0 RE (equatorial) = 0.36 RJ
Mass: 14.5 ME (0.05 MJ)
Density: 1.25 gm/cm3
Satellites: > 20
Structure: Rocky Core, Similar to Jupiter but without metallic hydrogen
Temperature: -216 C
Magnetic Field: Large and decentered
Atmosphere: 85% Hydrogen, 13% Helium, 2% Methane
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Neptune
Distance: 30.06 AU
Period: 164 years
Radius: 3.88 RE (equatorial) = 0.35 RJ
Mass: 17 ME (0.05 MJ)
Density: 1.6 gm/cm3 (second densest of giant planets)
Satellites: 7
Structure: Rocky Core, no metallic Hydrogen (like Uranus)
Temperature: -214 C
Magnetic Field: Large
Atmosphere: Hydrogen and Helium
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NeptuneUranus
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http://www.freewebs.com/mdreyes3/chaptersix.htm
Comparison of the Giant Planets
1.24 0.62 1.25 1.6
Mean density (gm/cm3)
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Jupiter
Saturn
Neptune
Uranus
Log
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CoRoT 7b
CoRoT 9b
Jupiter
Saturn
Uranus
Neptune
Earth
Venus
Pure H/He
50% H/He
10% H/He
Pure Ice
Pure Rock
Pure Iron
H/He dominated planets
Ice dominated planets
Rock/Iron dominated planets
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Reminder of what a transit curve looks like
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II. Fourier Analysis: Searching for Periods in Your Data
Discrete Fourier Transform: Any function can be fit as a sum of sine and cosines (basis or orthogonal functions)
FT() = Xj (t) e–it
N0
j=1
A DFT gives you as a function of frequency the amplitude (power = amplitude2) of each sine function that is in the data
Power: Px() = | FTX()|2
1
N0
Px() =
1
N0
N0 = number of points
[( Xj cos tj + Xj sin tj ) ( ) ]2 2
Recall eit = cos t + i sint
X(t) is the time series
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A pure sine wave is a delta function in Fourier space
t
P
Ao
FT
Ao
1/P
Every function can be represented by a sum of sine (cosine) functions. The FT gives you the amplitude of these sine (cosine) functions.
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Fourier Transforms
Two important features of Fourier transforms:
1) The “spatial or time coordinate” x maps into a “frequency” coordinate 1/x (= s or )
Thus small changes in x map into large changes in s. A function that is narrow in x is wide in s
The second feature comes later….
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A Pictoral Catalog of Fourier Transforms
Time/Space Domain Fourier/Frequency Domain
Comb of Shah function (sampling function)
x 1/x
Time Frequency (1/time)
Period = 1/frequency
0
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Time/Space Domain Fourier/Frequency Domain
Cosine is an even function: cos(–x) = cos(x)
Positive frequencies
Negative frequencies
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Time/Space Domain Fourier/Frequency Domain
Sine is an odd function: sin(–x) = –sin(x)
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Time/Space Domain Fourier/Frequency Domain
The Fourier Transform of a Gausssian is another Gaussian. If the Gaussian is wide (narrow) in the temporal/spatial domain, it is narrow(wide) in the Fourier/frequency domain. In the limit of an infinitely narrow Gaussian (-function) the Fourier transform is infinitely wide (constant)
w 1/w
e–x2 e–s2
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Time/Space Domain Fourier/Frequency Domain
Note: these are the diffraction patterns of a slit, triangular and circular apertures
All functions are interchangeable. If it is a sinc function in time, it is a slit function in frequency space
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Convolution
Fourier Transforms : Convolution
f(u)(x–u)du = f *
f(x):
(x):
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Fourier Transforms: Convolution
(x-u)
a1
a2
g(x)a3
a2
a3
a1
Convolution is a smoothing function
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2) In Fourier space the convolution is just the product of the two transforms:
Normal Space Fourier Space f*g F G
Fourier Transforms
The second important features of Fourier transforms:
f g F * G
sinc sinc2
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Alias periods:
Undersampled periods appearing as another period
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Nyquist Frequency:
The shortest detectable frequency in your data. If you sample your data at a rate of t, the shortest frequency you can detect with no aliases is 1/(2t)
Example: if you collect photometric data at the rate of once per night (sampling rate 1 day) you will only be able to detect frequencies up to 0.5 c/d
In ground based data from one site one always sees alias frequencies at + 1
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What does a transit light curve look like in Fourier space?
In time domain
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A Fourier transform uses sine function. Can it find a periodic signal consisting of a transit shape (slit function)?
This is a sync function caused by the length of the data window
P = 3.85 d= 0.26 c/d
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A longer time string of the same sine
A short time string of a sine
Wide sinc function
Narrow sinc function
Sine times step function of length of your data window
-fnc * step
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The peak of the combs is modulated with a shape of another sinc function. Why?
What happens when you carry out the Fourier transform of our Transit light curve to higher frequencies?
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= * XTransit shape Comb
spacing of P
Length of data string
In time „space“
In frequency „space“
X
* = convolution
=
Sinc of data window
Sinc function of transit shape
Comb spacing of 1/P
*
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But wait, the observed light curve is not a continuous function. One should multiply by a comb function of your sampling rate. Thus this observed transform should be convolved with another comb.
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When you go to higher frequencies you see this. In this case the sampling rate is 0.005 d, thus the the pattern is repeated on a comb every 200 c/d. Frequencies at the Nyquist frequency of 100 d.
One generally does not compute the FT for frequencies beyond the Nyquist frequencies since these repeat and are aliases.
Nyquist
Frequencies repeat
This pattern gets repeated in intervals of 200 c/d for this sampling. Frequencies on either side of the peak are – and +
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t = 0.125 d
1/t
The duration of the transit is related to the location of the first zero in the sinc function that modulates the entire Fourier transform
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In principle one can use the Fourier transform of your light curve to get the transit period and transit duration. What limits you from doing this is the sampling window and noise.
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The effects of noise in your data
Little noise
More noise
A lot of noise
Noise level
Signal level
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Frequency (c/d)
Transit period of 3.85 d (frequency = 0.26 c/d)
20 d?
Time (d)
20 d
Sampling creates aliases and spectral leakage which produces „false peaks“ that make it difficult to chose the correct period that is in the data.
This is the previous transit light curve with more realistic sampling typical of what you can achieve from the ground.
The Effects of Sampling
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A very nice sine fit to data….
That was generated with pure random noise and no signal
P = 3.16 d
After you have found a periodic signal in your data you must ask yourself „What is the probability that noise would also produce this signal? This is commonly called the False Alarm Probability (FAP)
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1. Is there a periodic signal in my data?
2. Is it due to Noise?
3. What is its Nature?
yesStop
no
4. Is this interesting?
noStopyes
yesFind another starno
5. Publish results
A Flow Diagram for making exciting discoveries
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Period Analysis with Lomb-Scargle Periodograms
LS Periodograms are useful for assessing the statistical signficance of a signal
In a normal Fourier Transform the Amplitude (or Power) of a frequency is just the amplitude of that sine wave that is present in the data.
In a Scargle Periodogram the power is a measure of the statistical significance of that frequency (i.e. is the signal real?)
1
2Px() =
[ Xj sin tj–]2
j
Xj sin2 tj–
[ Xj cos tj–]2
j
Xj cos2 tj–j
+1
2
tan(2) = sin 2tj)/cos 2tj)j j
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Fourier Transform Scargle Periodogram
Am
plit
ude
(m/s
)
Note: Square this for a direct comparison to Scargle: power to power
FT and Scargle have different „Power“ units
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Period Analysis with Lomb-Scargle Periodograms
False alarm probability ≈ 1 – (1–e–P)N ≈ Ne–P
N = number of indepedent frequencies ≈ number of data points
If P is the „Scargle Power“ of a peak in the Scargle periodogram we have two cases to consider:
1. You are looking for an unknown period. In this case you must ask „What is the FAP that random noise will produce a peak higher than the peak in your data periodogram over a certain frequency interval 1 < < 2. This is given by:
Horne & Baliunas (1986), Astrophysical Journal, 302, 757 found an empirical relationship between the number of independent frequencies,
Ni, and the number of data points, N0 :
Ni = –6.362 + 1.193 N0 + 0.00098 N02
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Example: Suppose you have 40 measurements of a star that has periodic variations and you find a peak in the periodogram. The Scargle power, P, would have to have a value of ≈ 8.3 for the FAP to be 0.01 ( a 1% chance that it is noise).
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2. There is a known period (frequency) in your data. This is often the case in transit work where you have a known photometric period, but you are looking for the same period in your radial velocity data. You are now asking „What is the probability that noise will produce a peak exactly at this frequency that has more power than the peak found in the data?“ In this case the number of independent frequencies is just one: N = 1. The FAP now becomes:
False alarm probability = e–P
Example: Regardless of how many measurements you have the Scargle power should be greater than about 4.6 to have a FAP of 0.01 for a known period (frequency)
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In a normal Fourier transform the Amplitude of a peak stays the same, but the noise level drops
Noisy data
Less Noisy data
Fourier Amplitude
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In a Scargle periodogram the noise level drops, but the power in the peak increases to reflect the higher significance of the detection.
Two ways to increase the significance: 1) Take better data (less noise) or 2) Take more observations (more data). In this figure the red curve is the Scargle periodogram of transit data with the same noise level as the blue curve, but with more data measurements.
versus Lomb-Scargle Amplitude
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Assessing the False Alarm Probability: Random Data
The best way to assess the FAP is through Monte Carlo simulations:
Method 1: Create random noise with the same standard deviation, , as your data. Sample it in the same way as the data. Calculate the periodogram and see if there is a peak with power higher than in your data over a speficied frequency range. If you are fitting sine wave see if you have a lower 2 for the best fitting sine wave. Do this a large number of times (1000-100000). The number of periodograms with power larger than in your data, or 2 for sine fitting that is
lower gives you the FAP.
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Assessing the False Alarm Probability: Bootstrap Method
Method 2: Method 1 assumes that your noise distribution is Gaussian. What if it is not? Then randomly shuffle your actual data values keeping the times fixed. Calculate the periodogram and see if there is a peak with power higher than in your data over a specified frequency range. If you are fitting sine wave see if you have a lower 2 for the best fitting sine function. Shuffle your data a large number of times (1000-100000). The number of periodograms in your shuffled data with power larger than in your data, or 2 for sine fitting that
are lower gives you the FAP.
This is my preferred method as it preserves the noise characteristics in your data. It is also a conservative estimate because if you have a true signal your shuffling is also including signal rather than noise (i.e. your noise is lower)
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Least Squares Sine Fitting
Fit a sine wave of the form:y(t) = A·sin(t + ) + ConstantWhere = 2/P, = phase shiftBest fit minimizes the 2:
2 = di –gi)2/N
di = data, gi = fit
Most algorithms (fortran and c language) can be found in Numerical Recipes
Period04: multi-sine fitting with Fourier analysis. Tutorials available plus versions in Mac OS, Windows, and Linux
http://www.univie.ac.at/tops/Period04/
Sine fitting is more appropriate if you have few data points. Scargle estimates the noise from the rms scatter of the data regardless if a signal is present in your data. The peak in the periodogram will thus have a lower significance even if there is really a signal in the data. But beware, one can find lots of good sine fits to noise!
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The first Tautenburg Planet: HD 13189
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Least squares sine fitting: The best fit period (frequency) has the lowest 2
Discrete Fourier Transform: Gives the power of each frequency that is present in the data. Power is in (m/s)2 or (m/s) for amplitude
Lomb-Scargle Periodogram: Gives the power of each frequency that is present in the data. Power is a measure of statistical signficance
Am
plit
ude
(m/s
)
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Fourier Analysis: Removing unwanted signals
Sines and Cosines form a basis. This means that every function can be modeled as a infinite series of sines and cosines. This is useful for fitting time series data and removing unwanted signals.
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Example. For a function y = x over the interval x = 0,L you can calculate the Fourier coefficients and get that the amplitudes of the sine waves are
Bn = (–1) n+1 (2kL/n)
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Fitting a step functions with sines
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See file corot2b.dat for light curve
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Prot = 23 d
See file corot7b.dat and corot7b.p04
0.035%
PTransit = 0.85 d = 1.176 d