Radar Back-Scatter from the Sea Surface 



2 2 



For a Pierson-Moskowitz spectrum, ( c*)|) ~ U and 



(ciO|n.) ~ U. The slope moments (njn:) diverge logarithmically at 



high' wave numbers. To obtain finite ^njnj) , the "carrier -wave" 



spectrum was cut off at an upper wavenumber kg/lO. The exact 



position of the cut-off is not critical for the evaluation of (njnj) , 



and the slope moments themselves enter only rather weakly in the 



second-order term q| of the expansion (11). However, the 



existence of a divergence as such points to a conceptual difficulty of 



the wave-facet interaction model. It appears that for an asymptotic 



k"* spectrum the carrier-wave region of the spectrum cannot be 



rigorously separated from the Bragg-scattering region. 



Figure 5 shows the computed half-power bandwidths for the 

 lowest-order Gaussian spectrum as a function of wave height. The 

 values compare well with measurements by Valenzuela and Laing 

 [20]. 



Deviations from the Gaussian form due to the higher-order 

 corrections qf and q| are represented in Figs. 6-9 in terms of 

 the mean frequency (ci^/cog and the frequency bandwidth 

 ((w- (co)) )/(cof) , normalised by their appropriate values for the 

 zero'th order Gaussian spectrum. 



The strongest correction is found for the mean frequency, 

 particularly for horizontal polarisation. The dependence on depres- 

 sion-angle and polarisation, shown in Fig. 6 for U = 20 m/s , is found 

 to be very similar at all wind speeds. The absolute values of the 

 frequency shifts increase approximately linearly with wind speed. 

 Fig. 7. Qualitatively, the polarisation and wind-speed dependence of 

 the mean Doppler frequency are in agreement with measurements 

 made by Hicks et al. [ 13] at low depression angles of about 5°. How- 

 ever, the theory is not strictly applicable in this case on account of 

 shadowing effects. 



The bandwidth corrections (Figs. 8 and 9) remain rather 

 small for depression angles less than 45° and limited azimuth angles 

 ijj relative to the wind. Larger deviations in the cross-wind direc- 

 tions depend strongly on the spreading factors, which are rather un- 

 certain for these angles. The experimental dependence of the Doppler 

 bandwidth on radar frequency and polarisation [ 20] tends to be some- 

 what larger and have a different trend than the corrections shown in 

 Figs. 8 and 9. Valenzuela and Laing [ 20] suggest that these effects 

 may be due partly to spray. To a fair approximation, the observed 

 bandwidths can be represented for small and intermediate angles i|j 

 and by the zero'th order Gaussian bandwidth. 



Both the bandwidth and mean frequency vary significantly with 

 wave height and can therefore be used for estimates of sea state. 

 For the one-parametrical family of spectra considered in the present 

 example, the two estimates are not independent. However, in general 

 the mean square bandwidth ( (co - ( co) ) ) ~ ( cof) (Eq. 14) and the nxean 



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