What are the dominant frequencies? Fourier transforms decompose a data sequence into a set of discrete spectral estimates – separate the variance of a

What are the dominant frequencies?

Fourier transforms decompose a data sequence into a set of discrete spectral estimates – separate the variance of a time series as a function of frequency.

A common use of Fourier transforms is to find the frequency components of a signal buried in a time domain signal.

1

0

2)(1 N

t

NtnietfN

FFT

FAST FOURIER TRANSFORM (FFT)

In practice, if the time series f(t) is not a power of 2, it should be padded with zeros

What is the statistical significance of the peaks?

Each spectral estimate has a confidence limit defined by a chi-squared distribution 2

Spectral Analysis Approach

1. Remove mean and trend of time series

2. Pad series with zeroes to a power of 2

3. To reduce end effect (Gibbs’ phenomenon) use a window (Hanning, Hamming, Kaiser) to taper the series

4. Compute the Fourier transform of the series, multiplied times the window

5. Rescale Fourier transform by multiplying times 8/3 for the Hanning Window

6. Compute band-averages or block-segmented averages

7. Incorporate confidence intervals to spectral estimates

Sea level at Mayport, FL

July 1, 2007 (day “0” in the abscissa) to September 1, 2007

mm

Raw data and Low-pass filtered data

High-pass filtered data

1. Remove mean and trend of time series (N = 1512)

2. Pad series with zeroes to a power of 2 (N = 2048)

Cycles per day

m2/c

pd

m2/c

pd

Spectrum of raw data

Spectrum of high-pass filtered data

Day from July 1, 2007

Val

ue

of

the

Win

do

w

Hanning Window

Hamming Window

102121 NnNnw ...,)/cos(

102460540 NnNnw ...),/cos(..



Val

ue

of

the

Win

do

w

Hanning WindowHamming WindowKaiser-Bessel, α = 2Kaiser-Bessel, α = 3

2

00

212

0

0

2

21

20

k

k

kx

xI

Nn

NnII

w

!

)(

/...,)()(


mm

Raw series x Hanning Window(one to one)

Raw series x Hamming Window(one to one)


To reduce side-lobe effects


mm

High-pass series x Hanning Window(one to one)

High pass series x Hamming Window(one to one)


To reduce side-lobe effects


High pass series x Kaiser-Bessel Windowα=3 (one to one)

m

Day from July 1, 20074. Compute the Fourier transform of the series, multiplied times the window

Cycles per day

m2/c

pd

m2/c

pd

Original from Raw Data

with Hanning window

with Hamming window

Windows reduce noise produced by side-lobe effects

Noise reduction is effected at different frequencies

Cycles per day

m2/c

pd

m2/c

pd

with Hanning window

with Hamming and Kaiser-Bessel (α=3) windows

5. Rescale Fourier transform by multiplying:

times 8/3 for the Hanning Window

times 2.5164 for the Hamming Window

times ~8/3 for the Kaiser-Bessel (Depending on alpha)

6. Compute band-averages or block-segmented averages

7. Incorporate confidence intervals to spectral estimates

Upper limit:

2

,21

2,2

Lower limit:

1-alpha is the confidence (or probability)nu are the degrees of freedomgamma is the ordinate reference value

0.995 0.990 0.975 0.950 0.900 0.750 0.500 0.250 0.100 0.050 0.025 0.010 0.005 7.88 6.63 5.02 3.84 2.71 1.32 0.45 0.10 0.02 0.00 0.00 0.00 0.00 10.60 9.21 7.38 5.99 4.61 2.77 1.39 0.58 0.21 0.10 0.05 0.02 0.01 12.84 11.34 9.35 7.81 6.25 4.11 2.37 1.21 0.58 0.35 0.22 0.11 0.07 14.86 13.28 11.14 9.49 7.78 5.39 3.36 1.92 1.06 0.71 0.48 0.30 0.21 16.75 15.09 12.83 11.07 9.24 6.63 4.35 2.67 1.61 1.15 0.83 0.55 0.41 18.55 16.81 14.45 12.59 10.64 7.84 5.35 3.45 2.20 1.64 1.24 0.87 0.68 20.28 18.48 16.01 14.07 12.02 9.04 6.35 4.25 2.83 2.17 1.69 1.24 0.99 21.95 20.09 17.53 15.51 13.36 10.22 7.34 5.07 3.49 2.73 2.18 1.65 1.34 23.59 21.67 19.02 16.92 14.68 11.39 8.34 5.90 4.17 3.33 2.70 2.09 1.73 25.19 23.21 20.48 18.31 15.99 12.55 9.34 6.74 4.87 3.94 3.25 2.56 2.16 26.76 24.72 21.92 19.68 17.28 13.70 10.34 7.58 5.58 4.57 3.82 3.05 2.60 28.30 26.22 23.34 21.03 18.55 14.85 11.34 8.44 6.30 5.23 4.40 3.57 3.07 29.82 27.69 24.74 22.36 19.81 15.98 12.34 9.30 7.04 5.89 5.01 4.11 3.57 31.32 29.14 26.12 23.68 21.06 17.12 13.34 10.17 7.79 6.57 5.63 4.66 4.07 32.80 30.58 27.49 25.00 22.31 18.25 14.34 11.04 8.55 7.26 6.26 5.23 4.60 34.27 32.00 28.85 26.30 23.54 19.37 15.34 11.91 9.31 7.96 6.91 5.81 5.14 35.72 33.41 30.19 27.59 24.77 20.49 16.34 12.79 10.09 8.67 7.56 6.41 5.70 37.16 34.81 31.53 28.87 25.99 21.60 17.34 13.68 10.86 9.39 8.23 7.01 6.26 38.58 36.19 32.85 30.14 27.20 22.72 18.34 14.56 11.65 10.12 8.91 7.63 6.84 40.00 37.57 34.17 31.41 28.41 23.83 19.34 15.45 12.44 10.85 9.59 8.26 7.43 41.40 38.93 35.48 32.67 29.62 24.93 20.34 16.34 13.24 11.59 10.28 8.90 8.03 42.80 40.29 36.78 33.92 30.81 26.04 21.34 17.24 14.04 12.34 10.98 9.54 8.64 44.18 41.64 38.08 35.17 32.01 27.14 22.34 18.14 14.85 13.09 11.69 10.20 9.26 45.56 42.98 39.36 36.42 33.20 28.24 23.34 19.04 15.66 13.85 12.40 10.86 9.89 46.93 44.31 40.65 37.65 34.38 29.34 24.34 19.94 16.47 14.61 13.12 11.52 10.52 48.29 45.64 41.92 38.89 35.56 30.43 25.34 20.84 17.29 15.38 13.84 12.20 11.16 49.64 46.96 43.19 40.11 36.74 31.53 26.34 21.75 18.11 16.15 14.57 12.88 11.81 50.99 48.28 44.46 41.34 37.92 32.62 27.34 22.66 18.94 16.93 15.31 13.56 12.46 52.34 49.59 45.72 42.56 39.09 33.71 28.34 23.57 19.77 17.71 16.05 14.26 13.12 53.67 50.89 46.98 43.77 40.26 34.80 29.34 24.48 20.60 18.49 16.79 14.95 13.79 55.00 52.19 48.23 44.99 41.42 35.89 30.34 25.39 21.43 19.28 17.54 15.66 14.46 56.33 53.49 49.48 46.19 42.58 36.97 31.34 26.30 22.27 20.07 18.29 16.36 15.13 57.65 54.78 50.73 47.40 43.75 38.06 32.34 27.22 23.11 20.87 19.05 17.07 15.82 58.96 56.06 51.97 48.60 44.90 39.14 33.34 28.14 23.95 21.66 19.81 17.79 16.50 60.27 57.34 53.20 49.80 46.06 40.22 34.34 29.05 24.80 22.47 20.57 18.51 17.19

Probability1234567891011121314151617181920212223242526272829303132333435

Deg

rees

of

fre

edo

m

Includes low frequency

N=1512

Excludes low frequency

N=1512

N=1512

N=1512

Least Squares Fit to Main Harmonics

The observed flow u’ may be represented as the sum of M harmonics:

u’ = u0 + ΣjM

=1 Aj sin (j t + j)

For M = 1 harmonic (e.g. a diurnal or semidiurnal constituent):

u’ = u0 + A1 sin (1t + 1)

With the trigonometric identity: sin (A + B) = cosBsinA + cosAsinB u’ = u0 + a1 sin (1t ) + b1 cos (1t )

taking:a1 = A1 cos 1

b1 = A1 sin 1

The squared errors between the observed current u and the harmonic representation may be expressed as 2 :

2 = ΣN [u - u’ ]2 = u 2 - 2uu’ + u’ 2

Then:

2 = ΣN {u 2 - 2uu0 - 2ua1 sin (1t ) - 2ub1 cos (1t ) + u02 + 2u0a1 sin (1t ) +

2u0b1 cos (1t ) + 2a1 b1 sin (1t ) cos (1t ) + a12 sin2 (1t ) +

b12 cos2 (1t ) }

Using u’ = u0 + a1 sin (1t ) + b1 cos (1t )

Then, to find the minimum distance between observed and theoretical values we need to minimize

2 with respect to u0 a1 and b1, i.e., δ 2/ δu0 , δ 2/ δa1 , δ 2/ δb1 :

δ2/ δu0 = ΣN { -2u +2u0 + 2a1 sin (1t ) + 2b1 cos (1t ) } = 0

δ2/ δa1 = ΣN { -2u sin (1t ) +2u0 sin (1t ) + 2b1 sin (1t ) cos (1t ) + 2a1 sin2(1t ) } = 0

δ2/ δb1 = ΣN {-2u cos (1t ) +2u0 cos (1t ) + 2a1 sin (1t ) cos (1t ) + 2b1 cos2(1t ) } = 0

ΣN { -2u +2u0 + 2a1 sin (1t ) + 2b1 cos (1t ) } = 0

ΣN {-2u sin (1t ) +2u0 sin (1t ) + 2b1 sin (1t ) cos (1t ) + 2a1 sin2(1t ) } = 0

ΣN { -2u cos (1t ) +2u0 cos (1t ) + 2a1 sin (1t ) cos (1t ) + 2b1 cos2(1t ) } = 0

Rearranging:

ΣN { u = u0 + a1 sin (1t ) + b1 cos (1t ) }

ΣN { u sin (1t ) = u0 sin (1t ) + b1 sin (1t ) cos (1t ) + a1 sin2(1t ) }

ΣN { u cos (1t ) = u0 cos (1t ) + a1 sin (1t ) cos (1t ) + b1 cos2(1t ) }

And in matrix form:

ΣN u cos (1t ) ΣN cos (1t ) ΣN sin (1t ) cos (1t ) ΣN cos2(1t ) b1

ΣN u N ΣN sin (1t ) Σ N cos (1t ) u0

ΣN u sin (1t ) = ΣN sin (1t ) ΣN sin2(1t ) ΣN sin (1t ) cos (1t ) a1

B = A X X = A-1 B

Finally...

The residual or mean is u0

The phase of constituent 1 is: 1 = atan ( b1 / a1 )

The amplitude of constituent 1 is: A1 = ( b12 + a1

2 )½

Pay attention to the arc tangent function used. For example, in IDL you should use atan (b1,a1) and in MATLAB, you should use atan2

For M = 2 harmonics (e.g. diurnal and semidiurnal constituents):

u’ = u0 + A1 sin (1t + 1) + A2 sin (2t + 2)

ΣN cos (1t ) ΣN sin (1t ) cos (1t ) ΣN cos2(1t ) ΣN cos (1t ) sin (2t ) ΣN cos (1t ) cos (2t )

N ΣN sin (1t ) Σ N cos (1t ) ΣN sin (2t ) Σ N cos (2t )

ΣN sin (1t ) ΣN sin2(1t ) ΣN sin (1t ) cos (1t ) ΣN sin (1t ) sin (2t ) ΣN sin (1t ) cos (2t )

Matrix A is then:

ΣN sin (2t ) ΣN sin (1t ) sin (2t ) ΣN cos (1t ) sin (2t ) ΣN sin2(2t ) ΣN sin (2t ) cos (2t )

ΣN cos (2t ) ΣN sin (1t ) cos (2t ) ΣN cos (1t ) cos (2t ) ΣN sin (2t ) cos (2t ) ΣN cos2 (2t )

Remember that: X = A-1 B

and B =ΣN u cos (1t )

ΣN u sin (2t )

ΣN u cos (2t )

ΣN u

ΣN u sin (1t )

u0

a1

b1

a2

b2

X =

Goodness of Fit:

Σ [< uobs > - upred] 2

-------------------------------------

Σ [<uobs > - uobs] 2

Root mean square error:

[1/N Σ (uobs - upred) 2] ½

Fit with M2 only

Fit with M2, K1

Fit with M2, S2, K1

Fit with M2, S2, K1,M4, M6

Tidal Ellipse Parameters

21

)sin(221 22

ppaaaac uvvuvuQ

ua, va, up, vp are the amplitudes and phases of the east-west and north-south components of velocity

amplitude of the clockwise rotary component

21

)sin(221 22

ppaaaacc uvvuvuQ amplitude of the counter-clockwise rotary component

papa

papac vvuu

vvuu

sincos

cossintan 1 phase of the clockwise rotary component

papa

papacc vvuu

vvuu

sincos

cossintan 1 phase of the counter-clockwise rotary component

The characteristics of the tidal ellipses are: Major axis = M = Qcc + Qc

minor axis = m = Qac - Qc

ellipticity = m / MPhase = -0.5 (thetacc - thetac)Orientation = 0.5 (thetacc + thetac)

Ellipse Coordinates:tmtMy

tmtMx

sincoscossin

sinsincoscos

M2

S2

K1

Fit with M2 only (dotted) and M2 + M4 (continuous)

Fit with M2 only (dotted) and M2 + M6 (continuous)

Documents

What are the dominant frequencies? Fourier transforms decompose a data sequence into a set of discrete spectral estimates – separate the variance of a