Derivation of directional ocean-wave spectra by integral inversion of second-order radar echoes

The derivation of sea-state from a single decameter radio wave-scatter observation requires the inversion of a two-dimensional nonlinear integral equation which gives the second-order radar cross section in terms of the ocean-wave spectrum. The second-order structure is sensitive to the directional properties of waves moving slower than the wind and the total nondirectional energy spectrum. We show how these properties, together with the rms waveheight and the wind direction, may be obtained by inverting the integral equation. The inversion method depends on recognition of physical properties of the wave-wave interactions which allow linearization of the equation, followed by application of standard inversion methods. This method is tested by application to simulated Doppler spectra corresponding to known ocean wave spectra. For typical examples (radar frequency 30 MHz, Doppler resolution 0.01 Hz, with 20 range and 10 time cells averaged) the directional ocean-wave spectrum was reproduced to within 10%, the true wind direction obtained to within 5 deg, and the rms waveheight predicted to within 5%.