Inversion and assessment of swell waveheights from HF radar spectra in the Iroise Sea

Abstract

As an extension of previous work in Wang et al. (Ocean Dyn 64:1447–1456, 2014), this article presents significant waveheights of swell inverted from a 13 month dataset of two high-frequency (HF) phased array radars. As an important intermediate variable in the calculation of significant waveheights, relative swell directions obtained by two different methods from a single radar station are also presented. The impact of the inaccuracy of relative swell direction on the calculation of waveheight is investigated and an alternative way of using constant swell direction is proposed. Radar-inverted swell significant waveheights using different ranges of relative swell directions are investigated. Results are assessed by WAVEWATCH III model hind casts. Analysis of the complete database shows that radar-inverted swell significant waveheights agree reasonably well with model estimates with large scatter. Standard deviation of the difference between the two estimations increases with waveheight, whereas the relative standard deviation, normalized by waveheight, keeps nearly constant. The constant direction scheme of waveheight inversion gives good estimations except for energetic swell exceeding the small perturbation assumption. Statistical analysis suggests that radar measurement uncertainty explains a considerable part of the difference between radar and model estimates. Swell estimates from both radar stations are consistent. This enables combined use of both radar spectra at common radar cells. Use of double spectra solves the ambiguity of relative swell direction, i.e., absolute swell direction is obtained, and effectively improves the accuracy of swell direction by the least-squares method.

Appendix A: The coupling coefficient

The coupling coefficient Γ in the expression of the second-order sea echo (Eq. 2) is obtained from a perturbational solution of the nonlinear boundary conditions at the ocean surface, and is the sum of electromagnetic and hydrodynamic components, Γ _H and Γ _EM , respectively (E.g. Barrick 1977; Lipa and Barrick 1986). The expression of Γ is given by

$$ \varGamma ={\varGamma}_{EM}-i{\varGamma}_H $$

(A1)

with i is the imaginary number $\sqrt{-1}$.

In deep-water condition, the two components are defined by

$$ {\varGamma}_{EM}=\frac{1}{2}\frac{\left({\overrightarrow{K}}_1\cdot {\overrightarrow{k}}_0\right)\left({\overrightarrow{K}}_2\cdot {\overrightarrow{k}}_0\right)/{k}_0^2-2{\overrightarrow{K}}_1\cdot {\overrightarrow{K}}_2}{\sqrt{{\overrightarrow{K}}_1\cdot {\overrightarrow{K}}_2}-{k}_0\varDelta } $$

(A2)

$$ {\varGamma}_H=\frac{1}{2}\Big[{K}_1+{K}_2-\frac{\left({K}_1{K}_2-{\overrightarrow{K}}_1\cdot {\overrightarrow{K}}_2\right)}{m_1{m}_2\sqrt{K_1{K}_2}}\frac{\omega^2+{\omega}_B^2}{\omega^2-{\omega}_B^2} $$

(A3)

with ω the angular Doppler frequency; Δ = 0.011 − 0.012i the impedance.

In shallow water condition, the electromagnetic component remains the same form, while the hydrodynamic component is related with water depth, given by

$$ {\varGamma}_H=\frac{1}{2}\left\{\begin{array}{l}{K}_1^{\prime }+{K}_2^{\prime }-\frac{\left({K}_1^{\prime }{K}_2^{\prime }-{\overrightarrow{K}}_1\cdot {\overrightarrow{K}}_2\right)}{m_1{m}_2\sqrt{K_1^{\prime }{K}_2^{\prime }}}\frac{\omega^2+{\omega}_B^2}{\omega^2-{\omega}_B^2}\\ {}\kern2em +\frac{\omega \left[{\left({m}_1\sqrt{g{K}_1^{\prime }}\right)}^3{\operatorname{csch}}^2\left({K}_1d\right)+{\left({m}_2\sqrt{g{K}_2^{\prime }}\right)}^3{\operatorname{csch}}^2\left({K}_2d\right)\right]}{g\left({\omega}^2-{\omega}^2\right)}\end{array}\right\} $$

(A4)

with K ^′₁  = K ₁ tanh(K ₁ d), K ^′₂  = K ₂ tanh(K ₂ d).