harmonic analysis
DESCRIPTION
Harmonic Analysis. The observed flow u’ may be represented as the sum of M harmonics: u’ = u 0 + Σ j M =1 A j sin ( j t + j ). For M = 1 harmonic (e.g. a diurnal or semidiurnal constituent): u’ = u 0 + A 1 sin ( 1 t + 1 ). With the trigonometric identity: - PowerPoint PPT PresentationTRANSCRIPT
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