harmonic analysis

Harmonic Analysis

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

so u’ is the ‘harmonic representation’

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

Rayleigh Criterion: record frequency ≤ ω1 – ω2

M2

K1

Tidal Ellipse Parameters

Major axis: Mminor axis: mellipticity = m / MPhase Orientation

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 = Qcc - Qc

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

Ellipse Coordinates:

time frequency; harmonic

norientatio

sincoscossin

sinsincoscos

t

tmtMy

tmtMx

M2

K1

Two Years of Tide Data at Trident Pier, Florida (Cape Canaveral)

Use “U-tide” routine

“utide” scripts

SA = Solar annualSSA = Solar SemiannualMSM = Lunar synodic monthly (29.53 d)MM = Lunar Monthly (27.55 d)MSF = Lunisolar synodic fortnightly (14.76 d)MF = Lunisolar fortnightly (13.66 d)

SA = Solar annualSSA = Solar SemiannualMSM = Lunar synodic monthly (29.53 d)MM = Lunar Monthly (27.55 d)MSF = Lunisolar synodic fortnightly (14.76 d)MF = Lunisolar fortnightly (13.66 d)

Complex Demodulation

Time series X(t) taken as nearly periodic plus non-periodic Z(t), still varying in time.

Amplitude A and phase ϕ of the nearly periodic signal are allowed to be time-dependent but vary slowly compared to the frequency ω.

X(t) = A(t) cos(ωt +ϕ(t))+ Z(t)

tZee)t(A ttitti

2

Demodulate by multiplying times tie tietXtY

tittiti etZetA

etA

tY 2

22

Varies slowly, independent of

Varies at frequency 2Varies at frequency

Low-pass filter to remove frequencies at or above

tietA

tY '

2

''

'2' YtA

2122 'Im'Re2)(' YYtA

'Re

'Imtan'

Y

Yat

Varies slowly, independent of (low-pass filter smooths this term – denoted by ’)

Separate (or extract) A’ and ’

tA

tYe ti

'

'2'

Sea level at Cape Cañaveral, Florida

m

2 years of data ( variables t and s)

X(t) = A(t) cos(ωt +ϕ(t))+ Z(t)

tietXtY

Ensenada de la Paz

Ensenada de La Paz, Mexico '2' YtA

tietXtY

Amplitude of complex demodulated series at semidiurnal and diurnal frequencies

YAVAROS BAY, MEXICO

Dworak, J. A., and J. Gomez-Valdes (2005), J. Geophys. Res., 110, C01007, doi:10.1029/2003JC001865.

Dworak, J. A., and J. Gomez-Valdes (2005), J. Geophys. Res., 110, C01007, doi:10.1029/2003JC001865.

Station M

Puerto Morelos Coral Reef Lagoon

31Sabrina Parra

Pacific Ocean

Atlantic Ocean

Gulf of Mexico

Caribbean Sea

North America

Mexico

Yucatan Peninsul

a

Pargos Spring

Northern Inlet

Central Inlet

Southern Inlet

Puerto Morelos Lagoon

Coronado et al. 2007

WINDOWED FOURIER TRANSFORM