Modulation model of radar backscatters is an important topic in the remote sensing of oceanic internal wave by synthetic aperture radar (SAR). Previous studies related with the modulation models were analyzed mainly based on the hypothesis that ocean surface waves are Gaussian distributed. However, this is not always true for the complicated ocean environment. Research has showed that the measurements are usually larger than the values predicted by modulation models for the high frequency radars (X-band and above). In this paper, a new modulation model was proposed which takes the third-order statistics of the ocean surface into account. It takes the situation into consideration that the surface waves are Non-Gaussian distributed under some conditions. The model can explain the discrepancy between the measurements and the values calculated by the traditional models in theory. Furthermore, it can accurately predict the modulation for the higher frequency band. The model was verified by the experimental measurements recorded in a wind wave tank. Further discussion was made about applicability of this model that it performs better in the prediction of radar backscatter modulation compared with the traditional modulation model for the high frequency band radar or under lager wind speeds.

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 [

Many joint experiments, such as SAXON-FPN [

However, the comparison between the theoretical model and experimental results showed that the measured modulation in SAR images is underestimated [

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 [

This paper is organized as follows: the modulation model of radar backscatter by internal wave based on the third-order statistics was derived in

An ocean surface scattering model, which is related to the ocean surface roughness spectrum, aims to quantify the relationship between the radar backscatter intensity and ocean surface statistics. The surface roughness spectrum is defined as the Fourier transform of the correlation of ocean surface wave [

The Fourier transform of the bicorrelation function

It is a function of four variables. Two special cases were considered in the following calculation of the model. When

The bispectrum is the Fourier transform of

They present the symmetric and asymmetric property of the random ocean surface waves, respectively. Radar backscatters of the ocean surface can be further calculated (see

To simplify the following analysis, we name the modulation model that only considers the contribution of second-order statistics of ocean surface IEM2 model. The model with consideration of third-order statistics is called the IEM3 model.

Defining

Therefore, Equation (9) can be rewritten as

We can get the modulation transfer function of radar backscatter by internal wave, that is,

For the case that the ocean surface is a Gaussian distribution, there will be no third-order component existing,

We can see that Equations (12) and (14) are exactly the same when the ocean surface is Gaussian distributed. The contribution of third-order statistics can be ignored as long as

We used the data of wind-wave tank experiment to validate the model proposed in

X-/Ka-band radars and CCD were employed in experiments to record the modulation of reflected microwave signals by internal waves. Specifications of radar system are listed in

The CCD array has high spatial and temporal resolution. It was used to record the information of surface waves in the tank. As the optical system, it can obtain the wave slope by retrieving the intensity of reflected light from the water surface. Specifications of the CCD are listed in

Radar system and CCD array are shown in

Experiments were carried out under different experimental conditions.

According to Bragg scattering theory [

We can also see that an interference frequency about 7 Hz was shown in

The surface wave height can be obtained by integrating the surface slope recorded by CCD array.

Given the high spatial and temporal resolution of CCD array, we took CCD data as the input of modulation models to calculate the theoretic value. Radar systems used in the experiments were not calibrated. Therefore, we cannot measure the absolute value of radar backscatters. In the further data processing, we only calculate the change of the radar backscatter modulated by the internal wave.

Results of the IEM2 model, IEM3 model and contribution of the bispectrum were compared with the radar data as shown in

From

The IEM2 model describes the contribution of surface roughness spectrum of the ocean. On the contrast, modulation calculated by IEM3 model is a weighted sum of second-order statistics (surface roughness spectrum) and third-order statistics (bispectrum) according to Equation (10). Contribution of bispectrum was compared with the radar data in

In this section, we will further discuss the modulation of high frequency band radar by internal waves at different wind speeds. To quantify the modulation, we introduce a new parameter modulation depth

We can see that all of the modulation depth measured by X-/Ka-band radar shows a decrease with increasing wind speed. It is similar for the predicted values calculated by modulated models except for Ka-band at 4.1 m/s. This is reasonable for the increasing wind speed corresponding to small relaxation rates [

Modulation depth calculated by the IEM3 model is superior to the results of the IEM2 model compared with the experimental measurements. The difference between them is more obviously for Ka-band than X-band. It is reasonable that Ka-band corresponds to a large

The predicted values by IEM2 of X-band and Ka-band are very close at wind speed larger than 4.1 m/s. However, the modulation depth of X-band is smaller than the Ka-band predicted by IEM3. With the increasing wind speed, the contribution of bispectrum increases especially for the higher band radar.

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.

This work was supported by the National Natural Science Foundation of China (No. 61302166 and No. 41406206).

Pengzhen Chen, Lei Liu and Xiaoqing Wang conceived and performed the experiments; Xiaoqing Wang and Jinsong Chong supervised and designed the research and made contribution to the article’s organization; Xin Zhang Xiangzhen Yu and provided help for the manuscript revision. Pengzhen Chen and Lei Liu drafted the manuscript, which was revised by all authors. All authors have read and approved the final manuscript.

The authors declare no conflict of interest.

Radar backscatter coefficient is a function of average receiving power

The scattering field

Since the averages of the quantities referred in Equation (A6) can be placed by [

Defining

For the second item in Equation (A2),

We take the situation that only single scattering occurs during the radar radiation, that is,

The third item in Equation (A2) is

For the same condition assumed above, we can get

Defining

By substituting Equations (A2), (A8), (A10) (A13) into Equation (A1), we obtain

Since

By substituting Equation (A17) into Equation (A16), we can further get

Schematic side view of the experimental wind-wave tank.

Experimental devices: (

Doppler spectrum at the wind speed 4 m/s (

Surface wave recorded by CCD: (

Comparison between measurements of radar and values calculated by models in theory. 10 m wind speed 5.2 m/s: (

Comparison between measurements of radar and values calculated by models in theory, 10 m wind speed 6.9 m/s; (

Modulation depth of radar backscatters as a function of wind speed: (

Specifications of radar system.

Specifications | Values | |
---|---|---|

Band | X | Ka |

Frequency | 9.4 GHz | 35 GHz |

Beam Width | 9° × 9° | 6° × 6° |

Incidence Angle | 50° | 57° |

Specifications of CCD.

Specifications | Values |
---|---|

Swath Width | 36 cm |

Resolution (Geometrical) | 0.3 mm |

Frame Repetition | 300 Hz |

Analog-to-Digital Convert Frequency | 300 KHz |

Description of experiments.

_{w} |
_{10} (m/s) |
_{i} |
|||
---|---|---|---|---|---|

No. 1 | 3.2 | 4.1 | 0.8 | 5 | 0.3 |

No. 2 | 4 | 5.2 | 0.8 | 5 | 0.3 |

No. 3 | 5 | 6.9 | 0.8 | 5 | 0.3 |

No. 4 | 6 | 8.6 | 0.8 | 5 | 0.3 |