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

DIRECTIONAL WAVE INSTRUMENTATION 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

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 (!)

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

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

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

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

....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

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

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)

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!

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

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:

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!)

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

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

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

CONCLUSIONS The standard estimates are robust, also when the data are erroneous/biased: Never believe the sign-conventions and filters stated by the manufacturer! 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