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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/).

For moderate and high speed values of the sea surface current, an aliasing phenomenon, due to an under-sampling in the time-domain, can strongly affect the reconstruction of the sea surface elevation derived from X-band radar images. Here, we propose a de-aliasing strategy that exploits the physical information provided by the dispersion law for gravity waves. In particular, we utilize simplifying hypotheses and numerical tests with synthetic data are presented to demonstrate the effectiveness of the presented method.

It is well known that images collected by nautical X-band radar enclose sea state information [

Therefore, data processing is necessary to estimate the sea-wave spectrum starting from the spectrum of the radar image (image spectrum). In particular, data processing is formulated as a linear inverse problem where, starting from the radar images sequence, the sea surface elevation is obtained as a function of two spatial variables (covering to the zone investigated by the radar) and of the time. The linear inversion scheme [

Being the inversion procedure strongly based on Fourier transform techniques, an aliasing phenomenon may arise in presence of an inappropriate time steps adopted to sample the radar image. In fact, due to the slow repetition time

In this paper, by considering the simpler case of a 2D problem,

Therefore, the problem at hand is relevant to the naval context, where the characterization of the sea state with X-band radar system becomes a challenge, when a semi-displacement or a planning vessel is considered, even with head sea conditions. The high speed of the ship emphasizes the aliasing phenomenon of the data in the time domain, making it necessary to tackle the problem.

The aliasing problem has been already addressed in [

As general comment, it is worth noting that, here, de-aliasing is possible since we exploit information on the physical model of the phenomenon and therefore the case at hand is completely different from the classical one reported in the text books [

In this paper, we move in the same framework of [

Therefore, the paper is organized as follows. Section 2 reports briefly the data processing approach for sea state monitoring from X-band radar data. In Section 3, the aliasing problem is sketched by referring to head waves whereas a strategy to address this problem is proposed in Section 4. Reconstruction results against synthetic data are shown in Section 5 and finally, the Conclusions follow.

This Section briefly presents the scheme commonly exploited to reconstruct the time-spatial evolution of the sea surface elevation starting from the X-band radar images. For sake of simplicity, the inversion approach is presented in the 2D case with a sea surface elevation that is considered as a function of the time

The initial step consists in Fourier transforming (2D Fast Fourier Transform, 2D-FFT) the raw data images sequence so to obtain the 2D image spectrum

In the second step the expected linear gravity wave components are extracted from the HP filtered image spectrum _{I}

In order to properly apply the filtering process based on the dispersion relation, an accurate estimation of the current value

Once the current _{I}_{W}_{I}^{2} is applied to the filtered spectrum _{I}^{2} = ^{β}

Finally, the knowledge of the sea wave spectrum _{W}_{W}

This Section is devoted to investigating the effect of the aliasing phenomenon on the performance of the inversion procedure described in the Section 2. This analysis provides the basis to propose an effective de-aliasing strategy that will be presented in the Section 4. Let us first consider the case of absence of sea surface current,

By defining _{B}_{B}_{B}

For typical X-band radar images, the spatial step Δ

When relation (4) does not hold, the replicas of the dispersion relation centered at ±_{s}

In particular, the undesired folding phenomenon occurs for the range of the wave-number _{m}_{B}_{m}

According to _{B}_{m}

Let us turn now to consider the case of a non null sea surface current. In particular, we consider a surface current moving in the same direction of the waves (head waves). In this case, we have to consider the dispersion law (see

In fact, the bandwidth _{BP}_{B}_{B}_{B}_{B}

When relation (11) holds, the collected data will be affected by aliasing and folded (aliased) shells arise in the second and fourth quadrant; this phenomenon is shown in the upper panel of

Accordingly, for a value of the surface current _{m}_{B}_{m}

_{m}_{m}_{m}

This Section is devoted at presenting a strategy with the aim of mitigating the effect of the spectrum folding on the sea-state reconstruction results. The proposed strategy moves within the same framework of the techniques presented in [

The

The

This second step is preparatory to the

As said above, the first step aims at assembling the sea wave spectrum in the allowable Nyquist band [−_{s}_{s}_{v}”_{v}

In particular, the choice of the surface virtual current _{v}_{k∈[0,kB]}|_{v}_{k∈[0,kB]}|

With the aim of addressing the problem of the global minimization of _{v} = [0,3.05, 6.1, 12.2, 15.25, 18.3] m/s.

For small values of the virtual current _{v}_{v} equal to 3.05 and 6.1 m/s, respectively), the function _{B}_{v}

The behavior mentioned above (_{B}_{max} the value where the function _{v} equal to 11 m/s is used.

From the considerations above, it seems that it could be possible to increase _{B}_{v} = 12.2 m/s, 15.25 m/s and U_{v} = 18.3 m/s of

By exploiting the above considerations about the behavior of the function _{v}_{v} =12.2 m/s in

The solution of the

The choice of the virtual current according to (17) is numerically justified from _{k∈[0,kb]}|

Let us turn now to exploit the above reasoning to set-up the de-aliasing strategy. First of all, we note that, for the values of the current _{B}

The aim of the proposed mitigation strategy is to exploit the virtual surface current so to constrain the unfolded spectrum in the allowable band

The lower panel of

In this way, we can state that the unfolding spectrum technique is successfully applied when the inequality
_{s}

Once the change of variables in (14) has been performed, a zero padding on the image spectrum

The final step consists in the addition, by means of a change of variables, of the (inverted) virtual surface current _{v}

As highlighted in _{N}

This Section aims at presenting results conforming the effectiveness of the strategy described in Section 4. To this end, synthetic data, generated by a fully 2D numerical wave-maker [_{SL}_{c}

Once the velocity potential is computed on the boundary domain, the nonlinear free-surface equations are stepped forward by a fourth-order Runge-Kutta scheme and the motion of the wavemaker is updated.

A domain decomposition technique has been used [

A JONSWAP sea spectrum with H1/3 = 0.094 m and T0 = 1.97 s has been simulated in the numerical wave tank; a scale factor of 20 has been used to reconstruct the wave elevations corresponding to the full scale sea state (H1/3 = 1.88 m and T0 = 8.8 s). Here, H1/3 represents the significant sea surface elevation, and T0 the modal period associated with the prescribed spectrum. The JONSWAP sea spectrum is used in this paper, but the proposed approach is also suitable for other kinds of sea spectra.

The sea-state data have been generated by performing an average of the spectra of three sea states with the duration of 6 m. The data are sampled with a time-step of 0.34 s and a spatial step of 0.6 m. In particular, a total number of Nx = 6,306 spatial samples has been used leading to an extent of 3,750 m. For the time discretization Nt = 1,066 time-samples have been considered by achieving an overall acquisition time of 6 min.

The averaged data has been decimated; in fact, the samples actually used to perform the reconstruction are Nx = 630 and Nt = 32 with a step of Δx = 5.9 m and Δt = 2.4 s, according to the spectral parameters of

The corresponding radar data have been generated by exploiting the procedure proposed in [

The upper panel of _{B} (see

The change of variable in (14) is performed by exploiting the value of the virtual current m/s U_{v} = 12.2m/s m/s that allows us to achieve the value

As further assessment of the effectiveness of the proposed method, the panels of _{NA}_{NA}

In order to give a quantitative evaluation of the effectiveness of the proposed strategy the normalized quadratic norm error:
_{NA}_{A}_{NA}_{A}

Finally, we present a comparison between the proposed procedure and the strategy implemented according to the scheme in [

This paper has dealt with the question of the aliasing problem that may arise in the reconstruction of the sea surface elevation starting from the images collected by a X-band radar system. In particular, the effect of a non appropriate time-step, adopted in radar data acquisition, has been thoroughly analyzed for the progressive waves. The investigation was concerned with both the cases of the absence and presence of the surface current and we have shown, by simple theoretical considerations, how the effect of an increasing value of the surface current badly affects the aliasing problem.

Based on these theoretical considerations, a strategy was proposed to mitigate the aliasing problem and its effectiveness has been shown by an analysis of synthetic data. The proposed method has been presented for unimodal sea states. As future development of this research activity, more challenging cases where the aliasing problem occurs, such as regressive and mixed waves, should be addressed and where the further question of the ambiguities arises [

Part of the work of Nieto Borge has been supported by Ministerio de Ciencia e Innovacion under project no. TEC2009-14217. The authors would like to thank the anonymous reviewers for their very useful suggestions and comments that have permitted to improve the quality of the work.

Block diagram of the inversion procedure.

Folded spectrum related to the parameters of

Dispersion relation for the progressive waves when the surface current

Plot of _{m}

Behaviour of _{v} = [0,3.05, 6.1, 12.2, 15.25, 18.3] m/s.

Behaviour of _{v} = [11, 12.2] m/s. The dotted point accounts for the maximum point (_{max}, _{max}).

Behavior of the spectral bandwidth

Zero padded non-folded spectrum _{N}

Pictorial sketch of the mathematical problem. The geometrical parameters are h_{0} = 3.6 m, h1 = 1.8 m, L = 230 m.

Upper panel: folded image spectrum

Wave spectrum _{N}

True (non-folded) sea-wave spectrum sampled with correct space and time steps.

Reconstructed function _{NA}_{A}

Zoom of middle panel (t = 36 s) of

The assembled spectrum according to the procedure in [

The assembled spectrum according to the procedure proposed herein.

Parameters for generation of the folded spectrum.

Sampling frequency (_{s} |
1.05 rad/meter |

Sampling frequency (_{s} |
2.61 rad/s |

Frequency step (Δ |
1.7E-3 rad/meter |

Frequency step (Δ |
0.08 rad/s |

Bandwidth (_{B} |
0.52 rad/meter |

Bandwidth (_{B} |
2.3 rad/s |