Internal waves usually result from the sharp density change occurring along the interface of the stratified density structure of the two fluids and travel with the interior of a fluid [1
]. In the process of SAR imaging of internal waves, the internal wave firstly induce the variable current. Then, the current will directly interact with the surface waves, which results in the modulation of the radar backscatters [2
]. Therefore, modulation model building is very crucial for the study of interaction between the radar backscatter and internal wave.
Many joint experiments, such as SAXON-FPN [3
] (the Synthetic Aperture Radar and X Band Ocean Nonlinearities-Forschungs-platform Nordsee), JOWIP [4
] (Joint Canada-U.S. Ocean Wave Investigation Project), SARSEX [5
] (SAR Internal Wave Signature Experiment), CoastWatch-95 [6
], and SCSE [7
] (South China Sea Experiment) were carried out and in situ measurements [8
] were made to investigate the modulation mechanism of radar backscatter. Weak hydrodynamic interaction theory [13
] was used to describe the distribution of Bragg wave spectrum modulated by internal waves [2
]. The imaging of internal waves is attributed to variations in the spectral energy density of Bragg waves induced by weak current variations associated with internal waves, similarly as the analysis of the imaging of bottom topography. A two-scale composite surface model derived from a modified Kirchhoff model is used to calculate the L-/X-band radar backscatter modulation [16
]. A full-spectrum model of the modulation of internal wave is established taking account of the wave spectral perturbations over the entire spectrum of waves [17
]. Existing models are mainly based on the assumption that fluctuation of heights on the water surface is a random Gaussian distribution.
However, the comparison between the theoretical model and experimental results showed that the measured modulation in SAR images is underestimated [17
], especially for high frequency band (higher than X-band) radar signals. Some investigators pointed out that the contribution of the backscatter from breaking waves should not be ignored, especially for higher-band radar. RIM (Radar Imaging Model) [18
] adds the energy source of breaking waves into the formation of a wave-current model. RIM simulates the wave modulation induced by convergent current taking account of breaking waves and finds that the spectral modulation of the shorter wave (between 10 and 1000 rad/m) is larger than the modulation calculated by the wave-current model without waves breaking. The radar signatures of internal wave are more visible for HH polarization than VV polarization because of the impact of breaking waves, as reported in [7
]. In substance, RIM adopts the improved hydrodynamic model, the composite surface model and Phillips’s semi-empirical model [19
] of breaking waves scattering to describe the scattering processing and explain the discrepancy of the modulation.
A modulation model of internal wave based on the third-order statistics of surface backscattering is proposed in this paper. It can effectively explain the discrepancy mentioned above by taking the non-Gaussian distribution of ocean surface slope into consideration. The IEM [20
] (Integral Equation Model) was introduced to calculate radar backscatter coefficients. Compared with traditional models, the modulation model proposed in this paper combined the small perturbation method (SPM) [21
] and the physics optical method (POM) [22
], and it does not need to divide the ocean surface into different scales. As a result, the modulation of radar backscatter by internal wave could be calculated more precisely. The model explains the contradiction between the radar backscatter and the values predicted by traditional models. Experimental measurements were analyzed to verify the model. Information recorded by a CCD (Charge-coupled Device), which has high spatial and temporal resolution, was used to calculate the theoretical modulation attributed to second-order and third-order statistics. Results were compared with the data obtained by X and Ka band radar showing good agreement with the measured data by considering the third-order statistics. Moreover, these theoretical analyses and experimental observations demonstrate that the contribution of ocean surface third-order statistics to the modulation is significant for high frequency band radar. In other words, for high frequency band radar, it is necessary to add the contribution of ocean surface third-order statistics to the modulation by a variable surface current.
This paper is organized as follows: the modulation model of radar backscatter by internal wave based on the third-order statistics was derived in Section 2
. In Section 3
, an experiment was briefly described, as well as the data processing. In Section 4
, results of experimental data were analyzed and discussed to validate the proposed model. Finally, main conclusions were given in Section 5
In this paper, the discrepancy between traditional modulation model and the measurements of high frequency band radars are addressed. Based on the third-order statistics of ocean surface, a modulation model of high frequency band radar backscatters by internal wave was proposed. It takes the non-Gaussian distribution of the ocean surface into consideration.
Data of experiments conducted in a wind-wave tank was employed to evaluate performance of the proposed model. Modulation depth of radar backscatter coefficients were calculated based on the IEM model and compared with the measured results by X-/Ka-band radar. The IEM3 model that considers the third-order statistics shows a better consistency with the radar data than the IEM2 model. Further processing and analysis to the model were made and showed that the third-order statistics of ocean surface are more important to the high frequency band radar. The relation between modulation depth and wind speed are also given. The larger radar frequency as well as the wind speed corresponds to a greater weight to third-order statistics in the radar backscatters modulated by internal waves. For the Ka-band radar, there are some other scattering mechanisms at low wind speed, which will be explored in future studies.
This proposed model can be applied to high frequency band SAR imaging of internal waves. It can enhance the image quality and show more information. Furthermore, it can be used in other SAR’s marine applications such as the imaging of sea bottom topography and eddies, since they consist of similar imaging mechanisms.