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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 Presentation

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Harmonic AnalysisThe observed flow u may be represented as the sum of M harmonics:

u = u0 + jM=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 1b1 = A1 sin 1

so u is the harmonic representationThe squared errors between the observed current u and the harmonic representation may be expressed as 2 :2 = N [u - u ]2 = u 2 - 2uu + u 2 Then:

2 = N {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., 2/ u0 , 2/ a1 , 2/ b1 :2/ u0 = N { -2u +2u0 + 2a1 sin (1t ) + 2b1 cos (1t ) } = 02/ a1 = N { -2u sin (1t ) +2u0 sin (1t ) + 2b1 sin (1t ) cos (1t ) + 2a1 sin2(1t ) } = 02/ b1 = N {-2u cos (1t ) +2u0 cos (1t ) + 2a1 sin (1t ) cos (1t ) + 2b1 cos2(1t ) } = 0N { -2u +2u0 + 2a1 sin (1t ) + 2b1 cos (1t ) } = 0N {-2u sin (1t ) +2u0 sin (1t ) + 2b1 sin (1t ) cos (1t ) + 2a1 sin2(1t ) } = 0N { -2u cos (1t ) +2u0 cos (1t ) + 2a1 sin (1t ) cos (1t ) + 2b1 cos2(1t ) } = 0Rearranging:

N { u = u0 + a1 sin (1t ) + b1 cos (1t ) } N { u sin (1t ) = u0 sin (1t ) + b1 sin (1t ) cos (1t ) + a1 sin2(1t ) }N { u cos (1t ) = u0 cos (1t ) + a1 sin (1t ) cos (1t ) + b1 cos2(1t ) }And in matrix form:N u cos (1t ) N cos (1t ) N sin (1t ) cos (1t ) N cos2(1t ) b1 N u N N sin (1t ) N cos (1t ) u0N u sin (1t ) = N sin (1t ) N sin2(1t ) N 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 + a12 )Pay attention to the arc tangent function used. For example, in IDL you should use atan (b1,a1) and in MATLAB, you should use atan2For M = 2 harmonics (e.g. diurnal and semidiurnal constituents): u = u0 + A1 sin (1t + 1) + A2 sin (2t + 2)

N cos (1t ) N sin (1t ) cos (1t ) N cos2(1t ) N cos (1t ) sin (2t ) N cos (1t ) cos (2t ) N N sin (1t ) N cos (1t ) N sin (2t ) N cos (2t ) N sin (1t ) N sin2(1t ) N sin (1t ) cos (1t ) N sin (1t ) sin (2t ) N sin (1t ) cos (2t ) Matrix A is then:N sin (2t ) N sin (1t ) sin (2t ) N cos (1t ) sin (2t ) N sin2(2t ) N sin (2t ) cos (2t ) N cos (2t ) N sin (1t ) cos (2t ) N cos (1t ) cos (2t ) N sin (2t ) cos (2t ) N cos2 (2t ) Remember that: X = A-1 B

and B =N u cos (1t ) N u sin (2t )N u cos (2t ) N u N u sin (1t )u0

a1

b1

a2

b2X =Goodness of Fit:

[< uobs > - upred] 2-------------------------------------

[ - uobs] 2Root mean square error:

[1/N (uobs - upred) 2] Fit with M2 only

Fit with M2, K1

Fit with M2, S2, K1

Rayleigh Criterion: record frequency 1 2Fit with M2, S2, K1,M4, M6

M2S2K1Tidal Ellipse ParametersMajor axis: Mminor axis: mellipticity = m / MPhase Orientation

Tidal Ellipse Parameters

ua, va, up, vp are the amplitudes and phases of the east-west and north-south components of velocityamplitude of the clockwise rotary component

amplitude of the counter-clockwise rotary component

phase of the clockwise rotary component

phase of the counter-clockwise rotary componentThe characteristics of the tidal ellipses are:Major axis = M = Qcc + Qcminor axis = m = Qcc - Qcellipticity = m / MPhase = -0.5 (thetacc - thetac)Orientation = 0.5 (thetacc + thetac)Ellipse Coordinates:

M2S2K1

Two Years of Tide Data at Trident Pier, Florida (Cape Caaveral)Use U-tide routine

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 DemodulationTime series X(t) taken as nearly periodic plus non-periodic Z(t), still varying in time. Amplitude A and phase of the nearly periodic signal are allowed to be time-dependent but vary slowly compared to the frequency .X(t) = A(t) cos(t +(t))+ Z(t)

Demodulate by multiplying times

Varies slowly, independent of Varies at frequency 2Varies at frequency Low-pass filter to remove frequencies at or above

Varies slowly, independent of (low-pass filter smooths this term denoted by )Separate A and

Sea level at Cape Caaveral, Floridam2 years of dataX(t) = A(t) cos(t +(t))+ Z(t)