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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This work deals with the question of sea state monitoring using marine X-band radar images and focuses its attention on the problem of sea depth estimation. We present and discuss a technique to estimate bathymetry by exploiting the dispersion relation for surface gravity waves. This estimation technique is based on the correlation between the measured and the theoretical sea wave spectra and a simple analysis of the approach is performed through test cases with synthetic data. More in detail, the reliability of the estimate technique is verified through simulated data sets that are concerned with different values of bathymetry and surface currents for two types of sea spectrum: JONSWAP and Pierson-Moskowitz. The results show how the estimated bathymetry is fairly accurate for low depth values, while the estimate is less accurate as the bathymetry increases, due to a less significant role of the bathymetry on the sea surface waves as the water depth increases.

Sea state monitoring using obtained marine radar data is of timely interest due to the fact that X-band radar systems provide the opportunity to scan the sea surface with high temporal and spatial resolution [

These radar signatures are considered clutter when the radar is exploited for the usual aim of the navigation control. Conversely, these radar signatures can be processed to achieve information about sea state conditions, resulting in a useful tool for regular monitoring. The intensity of clutter depends on the wind and sea state [

The backscattering by the sea arises due to the Bragg resonance [

As a result, the radar image is not a direct representation of the sea state and thus a reconstruction procedure is needed. In general, data processing is cast as an inversion problem where, starting from a time series of spatial radar images collected at different time-instants, one aims at determining the elevation

In this paper we focus the attention to the problem of the determination of the sea depth starting from the images collected by an X-band radar system. This problem has significant practical motivations since coastal monitoring of the sea state, and in particular changes in water depth near the coast, is a topic of timely and great interest. In fact, the possibility of continuously measuring the evolution of sea state and bathymetry represents a key point for many applications such as: coastal erosion; control of coastal areas affected by the anomalous wave hazards; support to navigation in zones close to ports and coasts.

This problem has been already tackled in the literature in [

The above drawbacks have been overcome in [

In [

Here, we address the problem of sea depth estimation by exploiting the technique already proposed in [

In principle the approach presented here is able to simultaneously determine both the surface current and sea depth; however, here we focus the attention only on the problem of the water depth estimation. The reliability of this strategy is tested against synthetic data in the simplified case of the sea wave function with only a spatial variable

Therefore, the paper is organized as follows. Section 2 is devoted to presenting briefly the data processing approach while in Section 3 the problem of the sea depth estimation is analyzed and the reconstruction strategy is presented. Section 4 deals with the numerical analysis of the proposed reconstruction strategy and finally the conclusions follow.

This section briefly describes the solution scheme usually exploited to extract the behavior of the wave elevation in space and time from the X-band radar images. The inversion approach is here presented in a 2D (space and time) domain; therefore the sea wave elevation is a function of the time

The starting step consists in applying a 2D Fast Fourier Transform, (2D-FFT) to obtain a 2D image spectrum

In the second step, the extraction of the desired linear gravity wave components from the HP filtered image spectrum _{1}(k,ω)^{2}],

The filtering procedure dictated by

The estimation of the surface current is made possible by different strategies such as the ones given in [_{1}(k,ω)_{I}

The successive step is to pass from the filtered radar image spectrum _{I}_{W}(k,ω)^{2} is applied to the filtered spectrum _{I}^{2} = ^{β}

The determination of the sea wave spectrum _{W}_{W}

This Section is devoted to presenting the problem of the sea depth determination and describing a strategy for the determination of such a quantity

Here, we perform the sea depth estimation by means of a technique similar to the one already presented in [_{I}

The NSP (as function of the current components _{F}_{G}_{I}

This Section aims at showing the effectiveness of the proposed strategy against synthetic data. Synthetic data have been generated using the linear theory for wave propagation in finite depth condition [_{i}_{i}_{i}_{i}

In particular, fixed the sea spectrum

The effect of the surface current is taken into account by reformulating the sea spectrum _{e}_{e}_{e}_{e}_{e}

For the presented results, we have considered two theoretical models of scalar spectral density, namely the Pierson-Moskowitz sea spectrum (PM) [

The Pierson-Moskowitz sea spectrum is the typical parameterization of the scalar spectrum of the waves. For this spectrum an important hypothesis is made, that is if the wind blows constantly for a “long time” on a “wide area” then the waves are in equilibrium with the wind. This is the concept of a fully developed sea. “Long-time” means ten thousand wave periods and “wide area” means five thousand wavelengths.

To this assumption, the spectral density takes the following form:
^{−3} being the Phillips constant and

The second spectrum is the JONSWAP one, where JONSWAP is the acronym of “Joint North Sea Wave Project”. For the JONSWAP model it is expected that the sea continues to develop through nonlinear wave-wave interactions, even after a long time and over long distances. Therefore, the spectrum corresponds to a sea wind spectrum partially developed,

Let us turn now to present the sea-depth estimation results. First, a JONSWAP sea spectrum with H_{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 _{1/3} = 3.25_{p}

The main parameters exploited in the sea wave simulation are reported in

The effects of both the surface current and sea depth were added to the data; in particular, we have considered the sea depth values in a range [5 m, 25 m] with a step equal to 1 m and surface current values in a range [−5 m/s, 5 m/s] with a 0.5 m/s step. For each of the pair of values (

Finally, starting from each radar image set, the sea depth has been evaluated via the NSP procedure presented in the previous Section. The results presented below refer to the cases when the surface current is assumed known and therefore the NSP in (4) is maximized only with respect to the sea depth

First, we present in detail some results of the large numerical analysis. The first test case is concerned with the JONSWAP case and a true sea depth

The same analysis has been performed for the sea depth values equal to 15 m and 30 m and

The outcomes of the overall numerical analysis are summarised by

Similar performances of the estimation procedure hold also for the Pierson-Moskowitz spectrum as reported in

The considerations above can be summarised in

Finally,

The paper has dealt with the problem of the sea depth estimation starting from X-band radar measurements. First, a simple analysis of the mathematical features of the problem was performed and after we have presented an estimation strategy. Then numerical analysis has provided results coherent with the theoretical expectations and pointed out how the intrinsic ill-conditioning of the problem makes it inapplicable for large values of the sea depth. In addition, the results showed that the proposed method is accurate and independent from the type of input data, which is captured by the radar and good performances of the approach were observed for a range of sea-depths up to about twenty meters. The main contribution of the work was the adoption of a “correlation” procedure to estimate the sea depth. Such a procedure has been already compared with the classical Least Square approach, largely used in literature, when we aimed at determining the sea surface current and the better performances of the correlation approach have been outlined (for such a comparison see [

Despite of the encouraging results, some factors have to be considered to reach a full assessment of the proposed approach. An estimation with real data is also necessary in order to analyse the effect of some limiting factors such as the presence of breaking waves in shallower water (6–10 m depth) and other non-linear wave behaviours that are not accounted for by the dispersion relation.

First, we aim at addressing one of the main factors limiting the effectiveness of proposed approach, and in general for the overall sea state monitoring, which is concerned with passage from the radar images to the sea state. This is a timely topic of significant interest that however until now has been tackled mostly on an empirical basis. The presented analysis was focused on the estimation of bathymetry considered constant in all the area seen by the radar. This assumption was necessary to conduct a preliminary study of the problem, although it represents a strong limit. Future developments will extend the analysis to the more realistic case of non-uniform bathymetry. Finally, it is noteworthy that the presented analysis and the proposed strategy hold in cases more general with respect to the radar data and are of interest in the case also of video images and other kind of sensors.

We would like to thank the two anonymous reviewers whose very helpful comments and suggestions have allowed us to improve the quality of the work.

Block diagram of the inversion procedure.

Behaviour of the tanh(|

Behavior of the NSP function in the case of

Behavior of the NSP function in the case of

Behavior of the NSP function in the case of

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).

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).

Panels (a) and (b): Comparison between the true wave height (black) and the reconstructed wave height obtained by supposing a deep water case [tanh(

Parameters of the numerical analysis.

Time step (Δ |
0.2 s |

Spatial Step (Δ |
2 m |

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

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

Estimation errors for the JONSWAP and Pierson-Moskowitz spectrum.

| ||||
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0.6000 | 0.1927 | 0.6000 | 0.1632 | |

0.4000 | 0.1799 | 0.3000 | 0.1543 | |

0.4000 | 0.1988 | 0.5000 | 0.2400 | |

0.6000 | 0.2672 | 0.5000 | 0.2645 | |

0.6000 | 0.3147 | 0.7000 | 0.3450 | |

1.3000 | 0.4561 | 1.2000 | 0.4477 | |

0.9000 | 0.4191 | 1.7000 | 0.6690 | |

1.7000 | 0.5944 | 2.0000 | 0.7656 | |

2.7000 | 0.7807 | 2.2000 | 1.1073 | |

2.9000 | 1.2112 | 3.1000 | 1.0151 | |

1.8000 | 0.8050 | 3.5000 | 1.6874 | |

3.5000 | 1.4607 | 3.3000 | 1.8299 | |

4.6000 | 1.6897 | 3.0000 | 1.8938 | |

5.20000 | 1.4912 | 5.2000 | 2.3053 | |

6.60000 | 2.0715 | 4.7000 | 2.0073 | |

5.00000 | 2.0422 | 7.2000 | 2.8224 | |

8.10000 | 2.8181 | 5.7000 | 3.2339 | |

7.80000 | 2.8181 | 5.4000 | 3.2171 | |

5.90000 | 1.7681 | 7.6000 | 3.5082 | |

10.4000 | 4.3861 | 6.2000 | 3.5211 | |

7.90000 | 2.3711 | 9.1000 | 4.2168 |