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

2u0b1 cos (1t ) + 2a1 b1 sin (1t ) cos (1t ) + a12 sin2 (1t ) + 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 : δ2/ δu0 = ΣN { -2u +2u0 + 2a1 sin (1t ) + 2b1 cos (1t ) } = 0 δ2/ δa1 = ΣN { -2u sin (1t ) +2u0 sin (1t ) + 2b1 sin (1t ) cos (1t ) + 2a1 sin2(1t ) } = 0 δ2/ 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 X = A-1 B B = A X

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 )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 =

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

Tidal Ellipse Parameters K1 Major axis: M minor axis: m ellipticity = m / M Phase Orientation

Tidal Ellipse Parameters 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 amplitude of the counter-clockwise rotary component phase of the clockwise rotary component 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 / M Phase = -0.5 (thetacc - thetac) Orientation = 0.5 (thetacc + thetac) Ellipse Coordinates:

M2 K1

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

“utide” scripts

SA = Solar annual SSA = Solar Semiannual MSM = 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 annual SSA = Solar Semiannual MSM = 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) Demodulate by multiplying times Low-pass filter to remove frequencies at or above  Varies at frequency 2 Varies at frequency  Varies slowly, independent of 

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

Sea level at Cape Cañaveral, Florida m 2 years of data X(t) = A(t) cos(ωt +ϕ(t))+ Z(t)

Ensenada de la Paz

Amplitude of complex demodulated series at semidiurnal and diurnal frequencies Ensenada de La Paz, Mexico

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

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

Puerto Morelos Coral Reef Lagoon Pargos Spring Northern Inlet Central Inlet Southern Inlet Puerto Morelos Lagoon North America Atlantic Ocean Gulf of Mexico Mexico Caribbean Sea Yucatan Peninsula Here we have North America. Puerto Morelos coral reef lagoon is located on the Yucatan Peninsula, just south of Cancun. As we zoom in, here we have the lagoon. Coronado showed that the circulation is mostly forced by waves entering the lagoon over the shallow coral reefs and strong outflow through the 3 main inlets: Northern, Central and Southern. The buoyant jet discharge we studied is called Pargos and is located here, inside the lagoon. To understand the effects of waves and tides in the lagoon we deployed these instruments. Pacific Ocean Coronado et al. 2007 Sabrina Parra

WINDOWED FOURIER TRANSFORM