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EGS XXVI General Assembly, Nice 25-29 March 2001

TRIPLET MEASUREMENTS OF DIRECTIONAL WAVE SPECTRA Harald E. Krogstad NTNU , Trondheim, NORWAY. PDF version available: http://www.math.ntnu.no/~hek/. EGS XXVI General Assembly, Nice 25-29 March 2001. THE WORK-HORSES OF DIRECTIONAL WAVE INSTRUMENTATION. Heave/pitch/roll buoys

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EGS XXVI General Assembly, Nice 25-29 March 2001

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  1. TRIPLET MEASUREMENTS OF DIRECTIONAL WAVE SPECTRA Harald E. Krogstad NTNU, Trondheim, NORWAY PDF version available: http://www.math.ntnu.no/~hek/ EGS XXVI General Assembly, Nice 25-29 March 2001

  2. THE WORK-HORSES OF DIRECTIONAL WAVE INSTRUMENTATION • Heave/pitch/roll buoys • Small array elevation/slope gauges • Displacement buoys • Velocity tracking buoys • p/u/v-systems • u/v/w-systems

  3. Triplets are compact • Triplets are ”complete” w.r.t. first order quantities • Triplets utilize data analysis techniques which • always produce • consistent and correct results for ideal instruments • feasible results under wrong calibration, non-ideal • behaviour of the instrument, and improper surface models (!)

  4. The ocean surface: The full spectrum: Linear wave theory: The directional spectrum: Triplets provide 3 time series:

  5. Fourier coefficient estimates (illustrated for a heave/slope instrument): The dispersion (”check”) ratio:

  6. GENERAL DATA ANALYSIS APPROACH: INVERSE PROBLEM FORMULATION Find the optimal distribution in the feasible domain that is, : The fit to the data : ”(Non)suitability” measure

  7. Tikhonov Regularization (Long/Hasselmann) Shannon Entropy (Hashimoto et al.) Burg Entropy Relative Entropy (Cross Entropy) Weighted Eucledian norm: Mahanobis distance: Bayesian techniques, (Hashimoto)

  8. ....WHAT IF THERE ARE... • 1. Improper calibration functions or noise in the data • 2. Significant steady currents • 3. Horizontal excursions and mooring effects (for buoys) • 4. Non-linearities in the wave field

  9. 1. TRANSFER FUNCTION ERRORS AND NOISE Some filter Noise • Generally easy to analyse, e.g. for a rotational symmetric • heave/pitch roll buoy: • The heave spectrum is only dependent on the amplitude • of the heave transfer function • The Fourier coefficients a1 and b1 are only dependent on the phase. • a2 and b2 are totally independent of transfer function errors • noise only affects the results for low signal levels

  10. THE DATA CONSISTENCY CHECKS • 1. The phases of the cross spectra should be either • purely imaginary or real. May be used to check • electronic transfer functions • hydromechanical transfer functions • 2. The check ratio is equal to 1 when LWT is valid: • Should apply around and above the main spectral peak • Questionable for high and low frequencies • (discussed below)

  11. 2. WAVE MEASUREMENTS IN CURRENTS Stationary system Advected system • The wavenumber spectrum transforms easily • A well-defined directional spectrum exists only for ”small” currents • Dispersion relation in the stationary system is not independent • of the direction: • Some transfer functions are very sensitve to the wavenumber • (e.g.) • For triplets: No direct estimation of Fourier coefficients anymore!

  12. THE TECHNIQUE FOR SMALL CURRENTS: i) Carry out the analysis in a stationary frame and estimate the apparent directional spectrum, : ii) Transform from apparent to actual directional spectrum, : Measurements in large currents (e.g., measuring from a moving platform) must be based on the wavenumber spectrum

  13. 3. THE EFFECT OF HORIZONTAL EXCURSIONS Consider a measurement from a horizontally moving buoy: Assume the motion is linearly connected to the surface motion: Expand to second order:

  14. By inserting the expansions above and using the zero (>2) cumulant property for Gaussian variables: • The horizontal motion of a H/P/R buoy is less damaging than • expected since the leading order perturbation due to the • horizontal motion vanishes. • A similar conclusions is found if we assume that the • horizontal excursions are independent of the surface motion. • (Of course, it is hard to say what happens when the mooring creates • something in between!)

  15. 4. IMPACT OF HIGHER ORDER SPECTRA First order spectrum supported on the dispersion surface . Second order spectra (4th in wave slope):

  16. Hm0=8m, Tp=12s, Sigma1 = 20 degs 3 2 log10(Spectra) 1 1 and 2 0 0 0.1 0.2 0.3 0.4 Freq. 80 60 Sigma1 40 Dir. spread 20 Sigma2 0 0 0.1 0.2 0.3 0.4 Freq. 5 4 3 Disp. ratio 2 1 0 0 0.1 0.2 0.3 0.4 Freq.

  17. PRELIMINARY CONCLUSIONS FOR HIGHER ORDER SPECTRA: • The second order spectrum is well below the first order spectrum • around the peak in the spectrum • The strong increase in the directional spread below the • spectral peak that is always observed in real spectra, may • partly be due to the impact of the second order spectrum • The strong deviation from 1 in the dispersion ratio between • the spectral peak may also be due to the second order spectrum

  18. CONCLUSIONS • The standard estimates are robust, also when the data are erroneous/biased: • Never believe the sign-conventionsand filters stated by themanufacturer! • Apply the data consistency checks before any serious • analysis • Inspect the raw time series visually for obvious errors hard to • find from the spectral analysis • Be aware of currents contaminating the measurements • Interpretation of directional coefficients may be influenced by • higher order spectra

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