Estimation of Directional Wave Spectra from Multicomponent Observations

T. H. C. Herbers Center for Coastal Studies, Scripps Institution of Oceanography, La Jolla, California

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R. T. Guza Center for Coastal Studies, Scripps Institution of Oceanography, La Jolla, California

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Abstract

An improved method for estimating the directional spectrum of linear surface gravity waves from in Situ observations is presented. The technique, a refinement and extension of the inverse method of Long and Hasselmann, is applicable to multicomponent wave measurements at fixed locations in constant or slowly varying depth water. On a frequency band by frequency band basis, an estimate of the directional distribution of wave energy S(θ) is obtained by minimizing a roughness measure of the form ∫dθ[d2S(θ)/dθ2]2 subject to the constraints: (i) S(θ) is nonnegative with unit integral, (ii) S(θ) fits the data within a chosen statistical confidence level, and (iii) S(θ) is zero on any directional sectors where energy levels are always relatively low because of the influence of geographic surroundings. The solution to this inverse problem is derived through a variational formulation with Lagrange multipliers.

A series of simulations using the new estimator show the fundamental limitations of sparse array data and the importance of using all available data-independent information [i.e., constraints (i) and (iii)] for achieving optimal estimates. The advantages of smoothness optimization are illustrated in a comparison of the present and Long and Hasselmann methods. The present method yields smooth estimates where Long and Hasselmann obtained rough estimates with multiple spurious peaks. A smooth solution to the inverse problem that has only truly resolved features is both easier to interpret and more readily evaluated numerically than wildly spurious solutions. The examples also demonstrate the subjectivity of intercomparing estimation techniques.

A few illustrative examples are presented of S(θ) estimates obtained from a two-dimensional array (aperture 120 m × 96 m) of 14 pressure transducer in 6 m water depth. Estimates using the full array and no geographic constraints are smooth and exhibit the expected refractive columnation of shoreward propagating energy towards normal incidence. Additionally, reflection from the mildly sloping beach 310 m shoreward of the center of this array is very weak at wind wave and swell frequencies. Estimates of S(θ) made using only the sensors on a longshore line, and a constraint of no reflected energy, are very similar to S(θ) obtained with the full array and no constraint.

Abstract

An improved method for estimating the directional spectrum of linear surface gravity waves from in Situ observations is presented. The technique, a refinement and extension of the inverse method of Long and Hasselmann, is applicable to multicomponent wave measurements at fixed locations in constant or slowly varying depth water. On a frequency band by frequency band basis, an estimate of the directional distribution of wave energy S(θ) is obtained by minimizing a roughness measure of the form ∫dθ[d2S(θ)/dθ2]2 subject to the constraints: (i) S(θ) is nonnegative with unit integral, (ii) S(θ) fits the data within a chosen statistical confidence level, and (iii) S(θ) is zero on any directional sectors where energy levels are always relatively low because of the influence of geographic surroundings. The solution to this inverse problem is derived through a variational formulation with Lagrange multipliers.

A series of simulations using the new estimator show the fundamental limitations of sparse array data and the importance of using all available data-independent information [i.e., constraints (i) and (iii)] for achieving optimal estimates. The advantages of smoothness optimization are illustrated in a comparison of the present and Long and Hasselmann methods. The present method yields smooth estimates where Long and Hasselmann obtained rough estimates with multiple spurious peaks. A smooth solution to the inverse problem that has only truly resolved features is both easier to interpret and more readily evaluated numerically than wildly spurious solutions. The examples also demonstrate the subjectivity of intercomparing estimation techniques.

A few illustrative examples are presented of S(θ) estimates obtained from a two-dimensional array (aperture 120 m × 96 m) of 14 pressure transducer in 6 m water depth. Estimates using the full array and no geographic constraints are smooth and exhibit the expected refractive columnation of shoreward propagating energy towards normal incidence. Additionally, reflection from the mildly sloping beach 310 m shoreward of the center of this array is very weak at wind wave and swell frequencies. Estimates of S(θ) made using only the sensors on a longshore line, and a constraint of no reflected energy, are very similar to S(θ) obtained with the full array and no constraint.

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