# Bathymetry Determination via X-Band Radar Data: A New Strategy and Numerical Results

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data Processing Approach

_{1}(k,ω) is performed. To this aim, a filtering is performed through the gravity dispersion relation that relates the wave number k to the pulsation ω(k) [1,3,16]:

^{2}], U is the sea surface current [m/s] and h is the sea depth [m].

_{1}(k,ω) so as to produce the spectrum F̃

_{I}(k, ω).

_{I}(k, ω) to the sea-wave spectrum F

_{W}(k,ω). This step is implemented by resorting to the Modulation Transfer Function (MTF) [3]. In particular the Modulation Transfer Function (MTF) |M (k)|

^{2}is applied to the filtered spectrum F̃

_{I}(k, ω) according to:

^{2}= k

^{β}; an empirical analysis provided β = −1.2 [3] as a reliable estimation.

_{W}(k, ω) allows us to determine the main parameters of the sea state; finally, also the time-space evolution of the wave height η(x,t) can be estimated by performing an inverse 2D-FFT of the function F

_{W}(k, ω).

## 3. Sea Depth Determination

_{I}(k, ω)| and the characteristic function G(k,h,ω,U) defined as:

_{F}and P

_{G}are the power associated to the image spectrum |F

_{I}(·)| and G(·), respectively. The effectiveness of the proposed strategy is analyzed in the Section below by considering only the sea depth estimation problem.

## 4. Validation of the Approach by Synthetic Data

_{i}is the circular frequency and the amplitude A(ω

_{i}) is chosen according to a prescribed sea spectrum S(ω), k(ω

_{i}) is the wave-number which satisfies the relation dispersion for the fixed value of the sea-depth h; the phase shift φ(ω

_{i}) is randomly generated through a suitable algorithm.

_{e}) as function of the encounter circular frequency ω

_{e}defined as ω

_{e}= ω − kUcos β with β representing the direction of the sea current with respect to the direction of propagation of the wave system; finally, S′(ω

_{e}) = S(ω) / |1–2ωU cos(β) / g |. The Equation (5) is used with ω

_{e}instead of the absolute circular frequency ω to generate the wave field. In the following β = 0, π is assumed for the cases at hand.

^{−3}being the Phillips constant and V is the wind speed, assumed equal to 19.5 m.

_{1/3}= 3.25 m and T

_{0}= 6.25 s has been generated. Here, H

_{1/3}represents the significant wave height, and T

_{0}the modal period associated with the prescribed spectrum. The second synthesized sea spectrum is a Pierson-Moskowitz (PM) one, with H

_{1/3}= 3.25m and T

_{p}= 7.5 sec.

## 5. Conclusions

## Acknowledgments

## References

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**Figure 2.**Behaviour of the tanh(|k|h) function at variance of the sea depth (sea depth values 2, 5, 10, 15, 20, 40 and 100 m).

**Figure 6.**Sea depth reconstruction for the different true values equal to 5 m (red line), 10 m (green line), 15 m (blue line), 20 m (orange line), 25 m (black line), when the current varies (JONSWAP sea spectrum).

**Figure 7.**Sea depth reconstruction for the different true values equal to 5 m (red line), 10 m (green line), 15 m (blue line), 20 m (orange line), 25 m (black line), when the current varies (Pierson-Moskowitz sea spectrum).

**Figure 8.**Panels (a) and (b): Comparison between the true wave height (black) and the reconstructed wave height obtained by supposing a deep water case [tanh(kh) = 1]. Panels (c) and (d) same as (a) and (b), but with a reconstructed wave height (red line) obtained by using the correct sea depth equal to 6 m.

Parameter | Value |
---|---|

Time step (Δt) | 0.2 s |

Spatial Step (Δx) | 2 m |

Number of time steps (N_{t}) | 2,500 |

Number of spatial steps (N_{x}) | 1,000 |

JONSWAP | PIERSON-MOSKOWITZ | |||
---|---|---|---|---|

True Value | Max error | Mean Error | Max error | Mean Error |

5.000 | 0.6000 | 0.1927 | 0.6000 | 0.1632 |

6.000 | 0.4000 | 0.1799 | 0.3000 | 0.1543 |

7.000 | 0.4000 | 0.1988 | 0.5000 | 0.2400 |

8.000 | 0.6000 | 0.2672 | 0.5000 | 0.2645 |

9.000 | 0.6000 | 0.3147 | 0.7000 | 0.3450 |

10.000 | 1.3000 | 0.4561 | 1.2000 | 0.4477 |

11.000 | 0.9000 | 0.4191 | 1.7000 | 0.6690 |

12.000 | 1.7000 | 0.5944 | 2.0000 | 0.7656 |

13.000 | 2.7000 | 0.7807 | 2.2000 | 1.1073 |

14.000 | 2.9000 | 1.2112 | 3.1000 | 1.0151 |

15.000 | 1.8000 | 0.8050 | 3.5000 | 1.6874 |

16.000 | 3.5000 | 1.4607 | 3.3000 | 1.8299 |

17.000 | 4.6000 | 1.6897 | 3.0000 | 1.8938 |

18.000 | 5.20000 | 1.4912 | 5.2000 | 2.3053 |

19.000 | 6.60000 | 2.0715 | 4.7000 | 2.0073 |

20.000 | 5.00000 | 2.0422 | 7.2000 | 2.8224 |

21.000 | 8.10000 | 2.8181 | 5.7000 | 3.2339 |

22.000 | 7.80000 | 2.8181 | 5.4000 | 3.2171 |

23.000 | 5.90000 | 1.7681 | 7.6000 | 3.5082 |

24.000 | 10.4000 | 4.3861 | 6.2000 | 3.5211 |

25.000 | 7.90000 | 2.3711 | 9.1000 | 4.2168 |

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**MDPI and ACS Style**

Serafino, F.; Lugni, C.; Nieto Borge, J.C.; Zamparelli, V.; Soldovieri, F.
Bathymetry Determination via X-Band Radar Data: A New Strategy and Numerical Results. *Sensors* **2010**, *10*, 6522-6534.
https://doi.org/10.3390/s100706522

**AMA Style**

Serafino F, Lugni C, Nieto Borge JC, Zamparelli V, Soldovieri F.
Bathymetry Determination via X-Band Radar Data: A New Strategy and Numerical Results. *Sensors*. 2010; 10(7):6522-6534.
https://doi.org/10.3390/s100706522

**Chicago/Turabian Style**

Serafino, Francesco, Claudio Lugni, Jose Carlos Nieto Borge, Virginia Zamparelli, and Francesco Soldovieri.
2010. "Bathymetry Determination via X-Band Radar Data: A New Strategy and Numerical Results" *Sensors* 10, no. 7: 6522-6534.
https://doi.org/10.3390/s100706522