**Weigen Huang, Bin Fu, Changbao Zhou, Jingsong Yang, Aiqin Shi and Donglin Li**

Second Institute of Oceanography

State Oceanic Administration

Hangzhou, P. R. China

**Abstract**

A simulation model for the radar backscattering cross section of the sea surface has been developed based on the radar imaging mechanism of underwater bottom topography. The model consists of the Navier-Stokes equation, the action balance equation and the radar backscatter model. The simulation model is solved numerically using the method of characteristics. The results of the simulation model have been used to study the optimal SAR parameters (frequency, polarization and incidence angle) for mapping underwater bottom topography. It is shown that long wavelengths are required. P band is the optimal band for mapping underwater bottom topography, followed by L, C and X bands. Although mapping underwater bottom topography is independent of the polarization, the V V polarization is the best choice because of its large signal-to-noise ratio. The incidence angle between 20°to 40°are needed to map underwater bottom topography by spaceborne SAR.

**Introduction**

It is well known now that under certain environmental conditions underwater bottom topography can be mapped by imaging radars operating at different frequency, polarization and incidence angles. The phenomenon was first imaged by a Ka band side looking airborne radar (SLAR) off the Dutch coast in 1969(de Loor, 1981). Later, bottom topography was found on images acquired by spaceborne synthetic aperture radar (SAR) systems such as Seasat HH polarized L band SAR, ERS-1/2 VV polarized C band SAR and Radarsat HH polarized C band SAR. The optimal SAR parameters (frequency and polarization) for mapping bathymetric features based on the Spaceborne Imaging Radar-C, X-Band Synthetic Aperture Radar (SIR-C/X-SAR) observations have been reviewed by Schmullius and Evans (1997). The aim of this study is to simulate the radar backscattering cross section of the sea surface and present the optimal SAR frequency, polarization and incidence angles for mapping underwater bottom topography.

**Model Description**

The understanding of the radar imaging mechanism of underwater bottom topography has been improved since the first explanation by Alpers and Hennings in 1984. It is generally accepted that the imaging mechanism of mapping underwater bottom topography by imaging radar consists of three stages: the interaction between tidal flow and bottom topography which results in modulations in the surface flow velocity, the interaction between the variable surface flow and the short surface water waves, and the interaction between the short surface water waves and radar signal. The present simulation model for radar signatures of underwater bottom topography includes the Navier-Stokes equation, the action balance equation and the radar backscatter model, which describe above three stages.

**Navier-Stokes equation**

The Navier-Stokes equation describes the interaction between tidal flow and underwater bottom topography and is given by

where (u,v) is depth averaged flow vector in (x,y) direction ,x is water elevation relative to reference surface, h is distance between bottom and reference surface, g is acceleration due to gravity, r is water density, C is chezy coefficient modeling bottom roughness, and (tx ,ty ) is wind stress in (x,y) direction.

** Action balance equation**

The action balance equation describes the evolution of the energy of a wave packet that travels through a slowly varying surface current field and reads

where A is the action spectral density of the wave packet, t is the time,W is the apparent frequency in the moving medium, (kx,ky ) is the wavenumber vector of the wave packet, and S is a source function. The action spectrum related to the energy spectrum E by E =wA. The intrinsic frequency is w=(gk+Tk3)1/2 , T being the ratio of surface tension to water density.

The apparent frequency W is related to the intrinsic wave frequency by

whereis the surface current vector. For the source function, the linear equation suggested by Alpers and Hennings(1984) is used because it allows an explicit solution of the action balance equation. S is given by

S=-m(A-A0) (6)

where m is a relaxation parameter and A0 the equilibrium spectrum. The relationship between the action spectrum A and the waveheight spectrum y is given by

A=yrw/k (7)

** Radar backscatter model**

The radar backscatter is based on the Bragg mechanism. The radar backscatter model has the form

s 0pol=16p k04 cos4(q)|gp (q |2y(kb) (8)

where k0 is the radar wavenumber and q the incidence angle of the radar. The magnitude of the Bragg wavenumber vector kb is given by

kb=2k0sinq (9)

The complex scattering coefficient gp can be approximated for horizontal (HH) polarization by

and for vertical (V V) polarization by

where e is the relative dielectric constant of sea water.

** Method of simulation**

** Description of the parameters**

The radar, environmental and underwater bottom topography parameters for simulation are listed in table 1. Satellite is assumed to fly from south towards north. Current and wind flow along the x direction.

The underwater bottom topography is shown in Fig.1. ** Table 1 The radar, environmental and underwater bottom topography parameters for simulation **

Radar band | P,L,C and X |

Polarization | V V and HH |

Incidence angle | 20°,23°,30°,40°, 50°,60° and 70° |

Current speed (u0) | 0.5m/s |

Wind speed (u10) | 5m/s |

Height of sandwave (d) | 1,4,7 and 10m |

Water depth above sandwave (h1) | 1,5,10,15,20,25 and 30m |

** Fig.1 Schematic of the underwater bottom Topography. **

**Method of solution**

The calculation of surface currents from equations (1)-(3) has been reduced to a quasi-one-dimensional problem since the underwater bottom topography in fig.1 is perpendicular to the current direction. It is also assumed that the current flow is laminar, free of vertical current shear and quasi-stationary. Thus, the current (x) is derived from the simple continuity equation. The action balance equation (4) is solved numerically following the method of characteristics employed by Hughes (1978). The normalized radar backscattering cross section (s 0) is finally calculated from equations (7) and (8).

** Results**

** Optimal radar frequency**

Fig.2 is an example of the simulation for optimal radar frequency showing the variations of s 0 with underwater bottom topography for q=23° , d=10m , h1=10m , V V polarized P, L, C and X bands. D s 0=½ s 0 -s 00½ , s 00 being the normalized radar backscattering cross section at x=0 . It can be seen from fig.2(a) that s 0 for all bands decreases slowly with increase of the height of the sandwave and reaches their minimum near the crest at x=573m. The s 0 then increases rapidly with decrease of the height of the sandwave. s 0p(s 0 for P band) has a largest variation while s 0x (s 0 for X band) has a smallest variation across the sandwave. This means that long wavelength radars can see the pattern of the sandwave more clearly than the short wavelength radars. It can be found from fig.2(b) that for each location of the sandwave ½ D s 0P½> ½ D s 0 L½> ½ D s 0C½ >½ D s 0x½. This indicates that underwater bottom topography can be detected more easily by the longer wavelength radars. Simulation for other cases shows similar results. It can be concluded from our simulation that long wavelengths are required. P band is the optimal band for mapping underwater bottom topography, followed by L, C and X bands.

** Optimal polarization**

Our simulation results show that s 0 for P, L, C and X bands do not depend on the polarization. However, s 0 from the sea surface for V V polarization is highest, which yield the best signal-to-noise ratio. Thus, V V polarization is to be preferred for mapping underwater bottom topography.

** Fig.2 Variations of s 0 (a) and D s 0 (b) with underwater bottom topography(c).**

** **

** Optimal incidence angle**

Fig.3 depicts the relationship between the incidence angles and the s 0 for d=7m, h1=10m, V V polarized P, L, C and X bands. It can be seen that the s 0 decreases with incidence angles. The signal from the sea surface is too small to be detected by spaceborne radar when the incidence angle is very large. Compared with the Radarsat SAR noise equivalent sigma naught of –18.5 dB (minimum detectable signal) (parashar, et al., 1993), the optimal range of the incidence angle for mapping underwater bottom topography is between 20° to 40°.

** Conclusions**

In this work the radar backscattering cross section of the sea surface has been simulated and analyzed. Form the results of the simulation the following conclusions are drawn. For mapping underwater bottom topography, large wavelengths (P and L bands), V V polarization and small incidence angles (20° to 40°) are prefered.

** Fig.3 Variations of s 0 at P((a)),L((b)),C((c)) and X((d)) bands with the radar incidence angles **

**Acknowledgments**

This work was supported by the China 863 Program under the Projects 818-06-02 and 2-7-4-15.

**References**

- Alpers, W. and Hennings, I., 1984, A theory of the imaging mechanism of underwater bottom topography by real and synthetic aperture radar. Journal of Geophysical Research, 89C, 10529-10546.
- de Loor, G. P., 1981, The observation of tidal patterns, currents and bathymetry with SLAR imagery of the sea. I.E.E.E Journal of Oceanic Engineering, 6, 124-129.
- Hughes, B. A., 1978, The effect of internal waves on surface waves. 2. Theoretical analysis, Journal of Geophysical Research, 83C, 455-465.
- Parashar, S., et al., 1993, Radarsat mission requirements and concept. Canadian Journal of Remote Sensing, 19, 280-288.
- Schmullius, C. C. and Evans, D. L., 1997, Synthetic aperture radar (SAR) frequency and polarization requirements for applications in ecology, geology, hydrology and oceanography: a tabular status quo after SIR-C/X-SAR. Int. J. Remote Sensing, 18, 2713-2722.