Method for determining a free surface elevation of an area of a body of water

Abstract

The invention relates to a method for determining a free surface elevation (ESL) of an area of a body of water: 1) forming a training database of pairs of images (W, R), each pair consisting of an FSE image (W) and a radar return image (R), the two images of a single pair corresponding to a single area and an identical or similar point in time; 2) adding, to each pair of images (W, R), spatial characteristics of the area, comprising the position with respect to the radar and to the main direction(s) of the waves; 3) training deep neural networks with this database, using the spatial characteristics to specialise the neural networks and obtain an FSE reconstruction model; 4) acquiring real return images by means of a radar; and 5) applying the model to the real radar return images to obtain the FSE.

Description

Technical field The present invention relates to the field of characterizing waves, in particular for monitoring, operating and commanding (controlling) a system subjected to waves, whether or not it is floating. In order to characterize waves within a body of water (for example: a sea, an ocean, a lake), time series of the free surface elevation of the water are notably used. This free surface elevation provides the height of the water (therefore the height of the waves), compared to the situation where the surface is not disturbed (calm sea, for example), at any point on the water surface. Measuring the free surface elevation is a widespread issue that needs to be addressed in order to tackle several challenges. One of the first challenges is the precise characterization (at the level of individual waves) of the sea conditions, which is most often characterized by descriptions of the statistical distribution of the heights of individual waves, such as the spectrum, the significant height and the mean period. These descriptors are commonly used for navigation tracking, for monitoring operations at sea, for monitoring an energy generation platform, for site monitoring for marine energy exploitation. Having a free surface elevation measurement not only allows more accurate statistical characterizations to be carried out, but also allows short-term predictions (from a few seconds to a few minutes) to be made of the height of the swell and the resulting movements of a floating system. These predictions allow, for example, the arrival of a series of waves to be detected that are potentially dangerous for a given operation (for example, transferring personnel aboard a wind turbine or a boat) or, on the contrary, a calm period, which would allow the relevant operation to be carried out. Another challenge of wave and swell predictions that are carried out based on free surface elevation is the control of floating systems (notably by means of predictive control): for example, to stabilize a vessel, to compensate for a movement, for example, heave compensation (which can be advantageous for a floating hydrocarbon generation platform), to ensure real-time control of floating wind turbines or wave-energy systems (notably with the aim of maximizing the generated energy and / or of reducing the fatigue of the components of such systems). Prior art Several technologies have been developed to measure and determine the free surface elevation. Some of these solutions allow a feature resulting from waves or swell to be measured at only one measurement point, for example, by means of an instrumented buoy, which does not allow an indication to be provided of the free surface elevation at any point of an area of a body of water. Marine radar, for example, X-band radar, found on all large vessels and many offshore facilities, are a particularly interesting technology for measuring waves from a distance, although they were initially primarily intended for navigation and collision prevention. The images produced by maritime radar detect not only targets or obstacles, such as vessels and coastlines, but also reflections from the surface of the sea, known as “sea clutter”, caused by the interaction between the radar waves and small surface ripples caused by the wind. Backscattering of this rough surface reveals the underlying shape of the waves. With appropriate processing, in principle it is possible to carry out wave measurements over a wide field of view (up to a range of approximately 5 km, depending on the installation height of the radar antenna and the sea conditions), with good spatial resolution (of the order of 5 m by distance and 1° by azimuth) and sufficient temporal resolution (one image every 1 to 3 seconds) in order to monitor the waves individually. More specifically, the principle of wave measurement using a radar is as follows: the electromagnetic waves emitted by the radar interact, by virtue of Bragg’s law, with the ripples (height variations) of the water surface, whose wavelength is centimetric (like those of radar waves). The result is a backscattered signal, sea clutter or even sea echo, which returns to the radar receiver, and whose intensity is modulated by the waves (the length of which can range between ten and several hundred meters) via a set of mechanisms that are not yet fully understood. The radar images therefore have patterns that resemble the waves, and from which they theoretically can be reconstructed, with suitable processing. The radar images can provide estimates of the sea conditions, i.e., statistical information concerning the wave height, and their energy content according to the frequency and the direction. However, they should also allow much more to be achieved, with a genuine reconstruction of the “wave-to-wave” free surface elevation (FSE), like a three-dimensional film of the sea surface. Such wave-to-wave reconstruction, with the vast range and resolution of the radar, is suitable for a very large number of applications. It is particularly ideal for monitoring purposes in the field of renewable marine energies, and paves the way for realtime prediction of waves, or the movement of a vessel, over horizons of several minutes, thus improving the feasibility and the safety of a large number of offshore operations. However, despite their promising perspectives, radar only provide a very indirect measurement of the waves, via images, or intensity signals, of the sea clutter. The intensity of the sea clutter is modulated by the orbital velocity of the fluid on the surface and by the angle of incidence of the radar beam over the surface of the sea, which can be linearly related to the free surface elevation to be measured. Other wave-related factors complicate this modulation, in particular shading effects (when certain areas of the sea surface are geometrically hidden by waves closer to the radar) or even the presence of micro-surf. In addition to these wave-induced modulations, there are other factors that alter the signal due to the waves, in particular the amplification function of the signal received by the radar, the presence of speckle noise, or even meteorological factors such as rain or spray, that can generate backscatter. In order to overcome this problem, wave reconstruction from radar images is generally based on the “standard method”, which is notably described in the following documents: Young, I.R., Rosenthal, W., & Ziemer, F. (1985). “A three-dimensional analysis of marine radar images for the determination of ocean wave directionality and surface currents”. Journal of Geophysical Research: Oceans, 90(C1), 1049-1059; Nieto Borge, J., Rodriguez, G. R., Hessner, K., & Gonzalez, P. I. (2004). “Inversion of marine radar images for surface wave analysis”. Journal of Atmospheric and Oceanic Technology, 21(8), 1291-1300. This method dispenses with an explicit model for forming (and inverting) radar images: in this approach, the field of view of the radar is divided into rectangular areas, in which a threedimensional Fourier transform of the signal is applied. The wave-related signal components are then identified and filtered using the dispersion relationship for gravity waves, which connects the frequency to the wavelength. The resulting components are then considered to be linearly connected to the components of the free surface elevation signal. Finally, the amplitude of the reconstructed signal simply needs to be calibrated in order to obtain images of the sea conditions with the correct energy. As can be seen, the standard method avoids the need for detailed knowledge of the image formation mechanisms, but relies on a considerable number of empirical parameters, such as the filtering parameters, those of the modulation transfer function, and the amplitude calibration parameters. The optimization or verification of these parameters can be based on the use of additional sensors, such as instrumented buoys, the inertial unit of a vessel, laser remote sensing (LiDAR sensor) or even a microseismic wave sensor on the shore. Another approach for improving the standard method involves using coherent radar, which also detect the Doppler signal due to surface motion. Indeed, the Doppler speed in principle can be used to reconstruct the free surface elevation without completing the calibration step via a modulation transfer function; however, in order to obtain stable Doppler signals, the radar must observe the same points over a relatively long period of time, which makes the proposed measurement procedures quite complex. In addition, the vast majority of current maritime radar do not have this feature. The standard method is none other than linear filtering. However, the relationship between surface elevation and sea clutter is highly non-linear. The performance levels that can be achieved by the standard method therefore are intrinsically limited. Finally, this filtering is performed a posteriori, thereby limiting its real-time applicability. It would be preferable to carry out an “image-by-image” inversion of the sea clutter, which would open the way for a real-time application. In addition, the sea clutter is highly inhomogeneous, depending on the considered area of the field of view of the radar. Close to the radar, the sea clutter is mainly modulated by the radial pitch of the free surface elevation, while far away from the radar, it is the occlusion (or masking) of troughs and small waves by the larger waves in front that dominates the formation of the image. In addition, the patterns of the sea clutter are highly pronounced along the main propagation axis of the waves, but much less so along the axis perpendicular thereto. Thus, a method for reconstructing the free surface elevation (FSE) taking into account the location in the field of view of the radar is desirable. With a view to overcoming the limitations of the standard method, the following studies focus on the inversion of sea clutter image-by-image, using deep learning methods to solve highly non-linear problems: - Ehlers, Svenja; Klein, Marco; Heinlein, Alexander; Wedler, Mathies; Desmars, Nicolas; Hoffmann, Norbert; Stender, Merten (2023), “Machine learning for phase-resolved reconstruction of non-linear ocean wave surface elevations from sparse remote sensing data”. In: Ocean Engineering, Vol. 288, p. 116059. DOI: 10.1016 / j.oceaneng.2023.116059. - Zhao, Mingxu; Zheng, Yaokun; Lin, Zhiliang (2023), “Sea surface reconstruction from marine radar images using deep convolutional neural networks”. In: Journal of Ocean Engineering and Science, Vol. 8, no. 6, p. 647-661. DOI: 10.1016 / j.joes.2023.09.002. However, neither of these two studies adjusts the inversion model based on the location in the field of view of the radar. In addition, the first of these two references only performs an inversion along a radial section of the field of view of a radar, not on a surface. Furthermore, the method described in patent FR 3108152 (corresponding to patent applications WO 2021 / 180502 and US 2023 / 0167796) implements one or more sensors (for example, a radar, a LiDAR, an accelerometer, a movement sensor, a pressure sensor, etc.) measuring the free surface elevation of swell or resulting features at one or more points and deriving swell predictions or resulting swell features therefrom by means of transfer functions. This method provides a good prediction of a resultant of the swell but does not allow the free surface elevation at any point of an area to be determined from a radar signal. Therefore, the aim of the present invention is to develop an improved method for determining the free surface elevation of a body of water using a radar. It notably involves overcoming the disadvantages of the previous solutions by taking into account the spatial variability of the sea clutter in the field of view of the radar. Summary of the invention The initial aim of the invention is a method for determining a free surface elevation (FSE) of an area of a body of water, such that, according to said method, 1) - at least one learning database of a plurality of pairs of images (W, R) is formed, each pair being formed by a first image (W) of the free surface elevation (FSE) and of a second radar return image (R), the two images of the same pair corresponding to the same area of the surface of the body of water and to an identical or close instant in time, said images (W) and (R) being real or simulated; 2) - each pair of images (W, R) is associated with spatial features of said area of the surface of the body of water corresponding to this pair, said spatial features comprising at least the position relative to the radar and to the one or more main directions of waves, said spatial features being directly known when acquiring the images, or being deduced from the analysis of the images themselves, or even being provided by an external source; 3) - at least one deep neural network, notably of the convolutional type, or of the “transformer” type or a Fourier neural operator, is trained on at least said learning database, by using said spatial features for the specialization of said one or more deep neural networks, and for obtaining a model for reconstructing the free surface elevation (FSE) as follows: 4) - real return images are acquired by a radar; 5) - said reconstruction model is applied to said acquired real return radar images to obtain the free surface elevation (FSE). Within the meaning of the invention, “close” time is understood to mean a time that is different but that is shifted by an instant less than the rotation period of a radar, notably by an instant less than three seconds. Advantageously, the first image (W) of each pair of images (W, R) can be obtained by discretizing three-dimensional time series of the free surface elevation (FSE) generated by a numerical wave field simulation code, and the second image (R) of each pair of images (W, R) by discretizing radar returns generated by a numerical radar simulation code. The first images (W) of each pair of images (W, R) can be derived from real measurements of free surface elevation (FSE) over at least some of the surface area covered by the radar. They notably can be carried out with at least one remote sensor of the LiDAR (Light Detection and Ranging, which is a remote measurement technique based on the analysis of the properties of a light beam returned to its emitter) type, or with a system of stereoscopic cameras or a set of measurement buoys. The second radar return images (R) can originate from real measurements taken by at least one radar, notably an X-band or S-band maritime radar. The learning database can comprise a first database of pairs of images (W, R) of surface elevation and radar return images (R), all the images of which are simulated, and a second database of pairs of images (W, R) of surface elevation and radar return images, all the images of which are real, with the relative weight of said two databases optionally being adjustable. Said spatial features can include a distance to an origin, said origin notably corresponding to the position of the radar, and an angular orientation, notably an azimuth. The one or at least one of the deep neural networks can be an encoder-decoder network, for example, a U-Net architecture. In order to obtain the model for reconstructing free surface elevation (FSE), the following can be carried out: - each pair of images (W, R) is divided into n distinct areas, according to spatial features, with n being greater than or equal to 2, and notably being equal to 4; - n deep neural networks are used, with each deep neural network specializing in a dedicated area of the pair of images; - each neural network is trained in its dedicated area of the pair of images (W, R). In this case, said pairs of images (W, R) can be divided into n areas as a function of the azimuth and of the distance to the radar. In order obtain said reconstruction model, the following can be carried out: - at least one first guide image (Rg1) is formed for each radar return image (R) from the learning database by means of a distance map to the radar, and the deep neural network is trained by means of said surface elevation image (W) and of said at least one first associated guide image (Rg1). In this case, at least one second guide image (Rg2) can be formed by means of an angular orientation map, and the deep neural network is also trained by means of said associated second guide image (Rg2). The method for determining a free surface elevation (FSE) of an area of a body of water according to the invention can implement the following steps: A1) constructing a first database of images (W) obtained from three-dimensional time series of free surface elevation, notably the elevation (z), the position as Cartesian (x, y) or polar (r, ¢) coordinates, and the time (t), generated by a code for simulating wave fields corresponding to different sea conditions; A2) constructing a second database of images, called return images (R), obtained from radar feedback generated by a radar simulation code using the time series generated by the wave field simulation code, configured according to environmental conditions, notably the wind speed and direction and the presence of precipitation; A3) constructing a learning database comprising a first set of the images from the first database (W) constructed in step A1) and a first set of the corresponding return images from the second database (R) constructed in step A2), while integrating the environmental conditions used in step A2), so as to obtain pairs of images (W, R) each formed by a first surface elevation image (W) and a second radar return image (R), the two images of the same pair corresponding to the same area of the surface of the body of water and to an identical or close instant in time; A4) associating each pair of images (W, R) from the learning database with the spatial features of the area of the surface of the body of water corresponding to this pair, said spatial features comprising a distance to an origin, said origin notably corresponding to the position of the radar, and an orientation, notably by azimuth, relative to the one or more main directions of the waves; A5) constructing a model for reconstructing free surface elevation from the learning database obtained in step A3) and from the spatial features specified in step A4), by training at least one convolutional or “transformer” deep neural network, notably of the U-Net type or a transformer architecture adapted to the vision, or even a Fourier neural operator, said one or more networks specializing in the spatial features obtained in step A4); A6) acquiring real images using at least one radar, notably a maritime radar, with which their spatial features defined in step A4) are associated; A7) determining the free surface elevation by applying the model for reconstructing free surface elevation obtained in step A5) at least to the real images acquired with the radar in step A6) with their spatial features. Preferably, in step A1) of constructing the first image database, the images are generated with digitization parameters comprising a distance discretization (range) △’’, an azimuth angle discretization △< / > in a polar coordinate system centered around the radar, and a temporal discreteation At = ^60 , where Nr is the speed of rotation of the radar in rpm. Preferably, in step A2) of constructing the second database of return images, the radar simulation code is configured according to environmental conditions selected from among at least: the presence or the absence of wind, the wind speed, the wind direction, the presence or the absence of precipitation such as rain, the intensity of precipitation, the presence or the absence of fog, the temperature and the sea conditions. Advantageously, in step A2) of constructing the second database of return images, it is possible to generate each image with spatial A and A temporal discretizations A according to the same digitization parameters as those used in step A1) of constructing the first image database. Advantageously, in step A5) of constructing the model for reconstructing free surface elevation, said model can be validated with a validation database made up of a second set of images from the first database constructed in step A1) and of a second set of corresponding return images from the second database constructed in step A2), with said sets being different from those used for learning. Advantageously, in step A6) of acquiring real images using at least one maritime radar, said images can be recorded, and, in step A7) of determining the free surface elevation (FSE), a communication means, notably a computer means, can be used to access said recorded radar images. Steps A1) to A5) can be carried out at least without connecting to a communication network, such as the intranet or the Internet. Step A7) can be carried without connecting to a communication network, on real radar images that are acquired in step A6) and are pre-recorded. Alternatively, steps A6) and A7) can be performed by connecting to a communication network, involving acquiring and recording the radar images in step A6) and using said radar images in step A7) as they are recorded. The method according to the invention can also implement the following additional steps, which are carried out between step A5) and step A6): B0) taking measurements of the free surface elevation (FSE) over at least some of the area covered by the radar, notably a maritime radar, simulated in step A1), notably by means of a stereoscopic camera system, a LIDAR or a set of measurement buoys; B1) constructing a database of obtained images by digitizing the real wave measurements obtained in step B0); B2) constructing a database of obtained images from the real return signals obtained with a maritime radar, time-synchronized with the image database constructed in step B1); B3) constructing a real learning database from a set of measurement images of the free surface elevation (FSE) of the database obtained in step B1) and a set of images of the real radar returns from the database obtained in step B2), in the form of pairs of images; B4) associating each pair of images from step B3) associating real measurements of free surface elevation (FSE), and the real radar returns with the spatial features of the corresponding area of the body of water, notably in terms of the distance and the direction from the one or more main directions of waves; B5) improving the reliability of the model for reconstructing free surface elevation constructed in step A5) from the real learning database constructed in step B3) and the spatial features determined in step B4), using machine learning or deep learning methods, on areas that match between real measurements and real radar returns. (The term “improving the reliability” means that by applying this reconstruction algorithm to real radar data, the errors in reconstructing the free surface elevation FSE are less than those that would have been obtained with the algorithm trained only on simulated data). B6) acquiring real images using at least one radar, notably a maritime radar; B7) replacing step A7) with step B7) by applying the improved model for reconstructing free surface elevation obtained in step B5) at least to the real images acquired with the radar in step B6). In order to carry out step B5), the real data from step B3) can be used directly with the simulated data from step A3) and learning can be carried out from the beginning in the same way as in step A5), or a “transfer learning” approach even can be used, in which approach the network pre-trained in step A5) is adjusted based on real data constructed in step B4). Steps B0) to B5) can be carried out at least off-line, without connecting to a communication network, after having completed steps A1) to A5). A further aim of the invention is a method for monitoring, operating, or controlling a system subjected to waves within a body of water, wherein the following steps are implemented: - determining the free surface elevation of said body of water by means of the method for determining free surface elevation as described above; and - monitoring, using or controlling said system subjected to waves as a function of the free surface elevation of said body of water thus determined. A further aim of the invention is a PC, a server or a computer configured to implement the method for determining free surface elevation as described above. A further aim of the invention is a computer program product that can be downloaded from a communication network and / or stored on a medium that can be read by a PC, a server or a computer and / or that can be executed by a processor, comprising program code instructions for implementing the method for determining free surface elevation described above, when said program is executed on a PC, a server or a computer. A further aim of the invention is a storage medium that can be read by a PC, a server or a computer and that stores instructions which, when they are executed by a PC, a server or a computer, cause the PC, the server or the computer to implement the method for determining free surface elevation described above. Therefore, the invention allows swell fields to be reconstructed from radar images using deep neural networks, for example, convolutional neural networks, which specialize in terms of direction and distance: the one or more main directions of the swell are determined based on the raw radar signals (images), then deep neural networks (or another model derived from machine or deep learning) are applied, specializing in the directions derived from the previous analysis and the distance between the radar measurement and the radar itself. In fact, since deep neural networks provide a prediction that is intrinsically translationally invariant, information indicating how far from the origin the data is located, as well as its orientation, improves the prediction quality: integrating this type of spatial information, which is not explicitly contained in the data, into neural network training had never been proposed until now. In summary, the invention thus determines, in real-time, the free surface elevation at any point of an area of a body of water, in a precise and simple manner, from images of the area of a body of water that are provided by a radar. To this end, the invention implements a learning campaign that is to be carried out, notably off-line. This learning campaign is carried out based on data from a radar simulator, notably synthetic (simulated) free surface elevations and synthetic (simulated) radar returns computed by the simulator. It allows a model for reconstructing free surface elevation to be reconstructed from radar images, based on deep neural networks (or another machine learning or deep learning method), on networks that are preferably specialized or guided by the direction and the distance of the radar return (echo or sea clutter). The invention proposes integrating information for indicating how far from the origin the data is located, as well as its orientation, in the training of neural networks. Several implementations / variants of the invention are possible, notably: - using at least n (n > 2) deep neural networks on as many different areas of the image; or - using a single neural network, modified compared to the standard formulation, by guiding it based on spatial maps informing it of the distance to the position of the radar and / or the direction (or directions) of propagation of the waves; or - by combining the two approaches (several spatially guided neural networks). Other features and advantages of the method and the system according to the invention will become apparent upon reading the following description of non-limiting embodiments, with reference to the appended figures described below. List of figures Figure 1 is an example of a radar image obtained by an X-band radar system on board a vessel. Figure 2 is an image obtained from a simulated wave field. Figure 3 is an image obtained from simulated radar returns. Figure 4a is a section of a radar image for an azimuth of 10°. Figure 4a is a section of a radar image for an azimuth of 90°. Figure 5 illustrates the relationship between the waves V and the radar images R. Figure 6 shows a synoptic of the reconstruction of free surface elevation by a deep neural network RN. Figure 7 is an application of the invention with four different neural networks RN1 to RN4, specializing according to the azimuth range and the range (distance). Figure 8a is a representation of the reconstruction obtained with a single deep neural network (in this case a U-Net convolutional neural network). Figure 8b is a representation of the reconstruction obtained with four specialized deep neural networks (in this case U-Net convolutional neural networks) per area of the image. Figure 9 uses a block diagram to show the method of the invention according to a first embodiment. Figure 10 uses a block diagram to show the method of the invention according to a second embodiment. Figure 11 shows radar and guide images, with the initial radar image (as a polar representation) in the top left, with the corresponding guide on the right (distance function, intensity from black to white indicating short to long distances, respectively), and at the bottom the radar image as a Cartesian representation on the left, and the corresponding guide on the right. The references retain the same meaning from one figure to another. Description of the embodiments The invention relates to a method for determining the free surface elevation of an area of a body of water. By way of a reminder, the free surface elevation is the height of water at a point of a body of water relative to the water surface without any height variation (which would not be disturbed in any way). Thus, the free surface elevation reflects the height of the waves or swell. A body of water can be a sea, an ocean, a lake, a river, etc. The area considered by the method according to the invention is an area of interest, notably for monitoring, operating or controlling a system subjected to waves. The extent of this area of interest can depend on the system subjected to the waves. As a variant, the extent of this area of interest can correspond to a measurement area of the radar implemented in the method. Figure 5 illustrates the relationship between the waves V and the radar images R that correspond to the sea clutter FM, with the reconstruction model H according to the invention aiming to obtain sequences of images or a 3D film of the sea surface from a sequence of radar images. The method for determining the free surface elevation of an area of a body of water according to the invention notably can be implemented by a computer and can be used for monitoring, operating or controlling a system subjected to waves within said body of water. The method according to the invention notably has two non-limiting embodiments. The first embodiment is schematically shown in figure 9, and the second embodiment is shown in figure 10. The references in both figures have the following meanings: A: Simulation of wave fields B: Simulation of radar returns C: Construction of the database of free surface elevation images D: Construction of the database of radar return images E: Construction of the learning database E’: Expansion of the learning database F: Acquisition of radar images from a real radar G: Recognition of the spatial features of the images H: Construction of a model for reconstructing free surface elevation H’: Improvement of the model for reconstructing free surface elevation I: Determining the free surface elevation J: Acquisition of direct measurements of free surface elevation over at least some of the area covered by the radar K: Construction of the database of directly measured free surface elevation images 1: Discretization parameters derived from the digitization features of the radar 2: 3D time series of free surface elevation 4: Simulated radar return images 5: Simulated environmental conditions 6: Pairs of images (simulated) 6’: Pairs of images (real) 7: Pairs of images, conditions 8: Real radar return images 9: Spatial features 10: Reconstruction model 10’: Improved reconstruction model 11: Real environmental conditions In marine X-band radar systems, signals corresponding to radar backscatter, after postprocessing, are sent to a screen as the radar beam rotates. Most modern radar can digitize and store these signals, making them accessible. Generally, the backscattered signals stored during a rotation of the antenna are consolidated into a single image, like a “snapshot” of the sea clutter. Figure 1 shows such an example of a radar image obtained by an X-band radar system on board a vessel. In terms of image processing, the challenge of reconstructing the surface elevation of swell involves finding, from a “snapshot” radar image R of the sea clutter, a “snapshot” image of the wave field W of the free surface elevation in the area covered by the radar. A first embodiment of the invention will be described with reference to figure 9: based on discretization parameters resulting from the digitization features of a radar 1, by simulating wave fields A, series of images 2 of simulated 3D time series of free surface elevation are obtained, and, based on these series of images, and taking into account simulated environmental conditions 5, the radar returns B are simulated: associated simulated radar return images 4 are obtained. A database C of images of free surface elevation and a database D of images of radar returns are then constructed. By combining the two databases C and D and by using simulated environmental conditions 5, a learning database E is formed, comprising pairs of images 6 (preferably at least 3 pairs) associating simulated images and their simulated radar returns and the environmental conditions 7 corresponding to the pairs of images. The spatial features 9 of the images are recognized G, which is used to construct H a model 10 for reconstructing free surface elevation, which will allow, with the acquisition F of radar images from a real radar resulting in real radar return images 8 being obtained, the free surface elevation to be determined I, taking into account the real environmental conditions 11. A second embodiment of the invention will be described with reference to figure 10. This second embodiment improves the obtained results by expanding the learning database with real images as follows: direct measurements of free surface elevation are acquired J over at least some of the area covered by the radar, the images 12 of direct measurements of free surface elevation are digitized for the construction K of a database of directly measured images of free surface elevation. Furthermore, real radar images 8 are acquired F from a real radar F, and a database of images of real radar returns synchronized with the direct measurements is formed. The pairs of real images that are obtained 6’ expand E’ the initial learning database E obtained according to the previous embodiment (see figure 9). Based on the pairs of images 6 and 6’ and the corresponding environmental conditions 7, the spatial features of the images 9 are recognized G, which allows the model 10’ for reconstructing free surface elevation to be improved H’, in order to acquire F, as in the previous embodiment, real radar images 8 from a real radar, and to be able to determine the free surface elevation based on the improved model 10’ and the real return images 8, under real environmental conditions 11. The system for determining free surface elevation applying the method according to the invention can comprise means for communicating between various devices. Advantageously, the free surface elevation determined by the system can be shared normally over a computer communication bus, notably a computer communication bus connected to a system using this measurement in real-time or otherwise (for example, a system for monitoring, operating or controlling a system subject to waves). Furthermore, the invention relates to a method for monitoring (supervising), operating or controlling (commanding), preferably an operating or controlling method, a system subjected to waves (such as a floating or non-floating platform, a vessel, a wave-energy converter system, for example), in which the following steps are implemented: a) determining the free surface elevation of an area of a body of water by means of the method for determining free surface elevation according to any one of the variants / embodiments or combinations of variants / embodiments described above; and b) controlling the system subjected to waves as a function of the determined free surface elevation. Thus, the method according to the invention allows a system subjected to waves to be monitored, operated or controlled, in order to increase its performance, and / or to limit the fatigue of the system subjected to waves and / or to improve the feasibility and the safety of a large number of operations at sea. According to one embodiment of the invention, the monitoring, operating or controlling method can comprise a step of predicting (with a phase lead) the free surface elevation of the area of the body of water by means of the determined free surface elevation, and controlling is carried out according to the prediction of the free surface elevation so as to anticipate controlling at the predicted future free surface elevation, before it reaches the system to be monitored, operated or controlled. The system subjected to the waves notably can be selected from a vessel, an aircraft carrier, an energy generation platform (for example, an oil platform), a renewable energy generation system (for example, an offshore wind turbine, a wave-energy converter system), or a device for transferring personnel or equipment at sea (gangway, crane), or any similar system. The monitoring, operating or controlling step can notably involve: - a vessel navigating in an area where the free surface elevation is minimal, so the vessel can avoid severe sea conditions such as strong waves; - stabilizing a floating platform or a vessel according to the free surface elevation; - controlling (for example, predictive control) a renewable energy generation system in order to maximize the generated energy or to reduce the fatigue of the generation system based on the free surface elevation. It can involve controlling an electric machine fitted to a wave-energy converter system in order to optimize the power that is generated depending on the height of the swell. It can also involve controlling the angle of orientation of the blades, and / or the nacelle of a wind turbine depending on the waves and the swell. The monitoring, operating or controlling step can also involve carrying out an operation at sea when the free surface elevation is minimal, for example: - rescuing a shipwrecked individual from a vessel; - recovering an object on the surface of the water from a vessel; - installing a marine or underwater object from a vessel or a platform (for example, a hydrocarbon pipeline); - an aircraft taking-off or landing on an aircraft carrier, etc.; - transferring personnel from a vessel or a platform (for example, maintaining an offshore wind turbine) Examples The features and advantages of the method according to the invention will become more clearly apparent from reading the following application examples. Example 1 Considering a radar image R and the image W of free surface elevation of a corresponding wave field. These two images, which are made up of pixels, have the same width and the same height. Figure 2 shows the image W generated with a code for simulating wave fields, for given sea conditions, characterized by an average period and a significant height of the waves and a directional spectrum, and represents a “snapshot” of the surface of the sea over a circle with a radius of 2,000 centered around the radar, extracted from the generated elevation time series that is at least as long as a rotation of the radar (2 seconds in this case). On the image, which is 1024 x 1024 pixels, the horizontal axis represents the azimuth , which ranges from 0 to 360°, and the vertical axis represents the distance to the radar , which ranges from 48 to 2,000 meters. Each pixel in the image represents the free surface elevation, normalized so as to have the same order of magnitude as the intensity in the radar image. On the abscissa, the figure represents the angle 0 in degrees, on the ordinate the left shows the range in meters and the right shows the normalized free surface elevation. The image R, shown in figure 3, is obtained from radar returns generated by a radar simulation code using the time series generated by the wave field simulation code, configured according to different environmental conditions (presence or absence of wind, rain, fog, etc.). Each image is generated with the same spatial discretizations △’’ and A^ as the wave field image (same units shown on the abscissa and on the ordinate as in figure 2, with the intensity of the radar return on the right). By analyzing the image R, it can be seen that the features of the radar return are quite different in the propagation direction of the waves (which corresponds to azimuths of 90° -100° and 270° - 280°, in this example) notably relative to the perpendicular direction. They are also different depending on the distance to the radar, between the lower part and the upper part of the image. These differences also can be seen by means of the sections at different azimuths of the radar image shown in figures 4a (section with an azimuth of 10°) and 4b (section with an azimuth of 90°): differences can be seen between the close range, up to 1,000 m, and the far range, beyond 1,000 m, but also between the two sections, especially in the far range. A learning database is created from pairs of images (R, ) corresponding to the same situation. At least three pairs of images need to be provided in order to form this learning database. However, it is preferable for the database to contain more pairs in order to obtain a model capable of reconstructing the free surface elevation of the swell in several scenarios. During the learning phase, a deep neural network of the “transformer” type or of the convolutional f type can be used, for example, a U-Net encoder / decoder network, which is the non-limiting example used for the examples, preferably with a set of weights to be optimized (as is known, each neural network is associated with a set of parameters (weights) to be optimized). During this phase, a radar image R is provided at the input of the convolutional neural network f and the latter computes a result image W. This output is compared with the expected wave field image W by a loss function L. This loss function is, for example, a structural similarity index measure (please refer to the publication by Wang et al., 2004: Zhou Wang, A.C. Bovik, H.R. Sheikh, and E.P. Simoncelli entitled, “Image quality assessment: from error visibility to structural similarity”, IEEE Transactions on Image Processing, Vol. 13, No. 4, pp. 600-612, 2004). The result of is used to update all the weights of the network G by a stochastic gradient backpropagation method and thus obtain optimal weight values 0 . For example, one possible optimization method is described by Adam in the publication by Diederik & Ba, 2015 (Diederik, P. K., & Ba, J. (2015). Adam: “A method for stochastic optimization”. CoRR, abs / 1412.6980), with 50 epochs: all the pairs of images in the learning database are seen by the network 50 times, with a learning rate of 3.10-4. Formally: 0 = argmine[E(R)[L(lV / e ( / ?))]] L^) = 1 — SSIM(^) = 1 — 22 ' '' '  +^^+ ' v* w + 2w + ci) (y2v + 2w + c 2) with being mean, being variance, 1 = 0.01and 2 = 0.03. This process for reconstructing free surface elevation using a convolutional neural network is schematically shown in figure 6, which is a synoptic of the process for reconstructing free surface elevation using a deep neural network RN, such as a U-Net network: it shows the image W processed by the radar model RM, in order to obtain the radar return R, then the use of a neural network like U-Net to obtain the result image MZ. It can be seen that if several standard convolutional neural networks are used, then by training them on different areas of the pair of images, rather than just one, much better results are obtained. In this example, four convolutional neural networks are considered that are specialized according to the azimuth range and the range (distance): - Network No. 1 RN1 (figure 7) learns on the close range, up to 1,000 m, and on the “inwave” azimuth range, i.e., two intervals of 90° centered around the main direction of the waves (~95°) and its complement to 360° (~275°). - Network No. 2 RN2 learns on the close range, up to 1,000 m, and on the “off-wave” azimuth range, i.e., two intervals of 90° centered around directions perpendicular to the main direction of the waves (~5° and ~185°). - Network No. 3 RN3 learns on the “far range”, beyond 1,000 m, and on the “in-wave” azimuth range. - Network No. 4 RN4 learns on the far range, and on the “off-wave” azimuth range. These four neural networks RN1 to RN4 are shown in figure 7: they are thus specialized in the image that is divided into four areas, both according to the azimuth range and according to the range (distance). This selection of four neural networks is clearly an example, with the invention being able to be applied to a lower or higher number of neural networks, depending on the number of areas to be distinguished in the image. Figures 8a and 8b show reconstruction results (with the same units on the abscissa and on the ordinate as for figures 2 and 3): - with a single neural network (U-Net standard): figure 8a; - and with 4 specialized neural networks (U-Net) per area of the image: figure 8b. The metric that is used is the local correlation between V / and W in both cases: values close to 1 indicate a strong similarity between the images, area-by-area. Clearly, a better reconstruction is obtained with the specialized U-Nets, especially in the “off-wave” azimuth ranges. In this example, the standard convolutional neural network (U-Net type) is modified by guiding it from a distance map indicating the distance to the radar position. This information, which is not initially present in the data, can be constructed from a distance map, i.e., an image the same size as the radar image, indicating the distance to the radar at each point. Examples of these guides, as a polar representation and a Cartesian representation, are provided in figure 11, which shows radar images and guides, with the initial radar image (as a polar representation) in the top left, with the corresponding guide on the right (distance function, intensity from black to white indicating short to long distances, respectively), and at the bottom the radar image as a Cartesian representation on the left, and the corresponding guide on the right. The information corresponding to the orientation can be constructed in the same way. The training and prediction from the network is carried out using the one or more guides. In practice: - a guide G is formed based on the problem to be solved. In this case, it involves constructing an image that is the same size as the radar image indicating the distance to the radar at each point. Another guide also can be an image providing an angular indication; - the input data is modified into a pair of images, a radar image and a distance map, and issued to the convolutional neural network. It also would be possible to use a trio or more guide images. Formally: 9 = argmme[Ew [r( W / gX^))]] ,()=   (,) When applied to wave height estimation, on a learning database of only 5 pairs of images, the following improvements were obtained, both as an initial data representation (polar representation) and as a Cartesian representation. Table 1 below is a comparison of an estimation by a standard or guided U-net network, as a polar or Cartesian representation. The closer the SSIM score is to 1, the greater the similarity. In order to quantify the results, the network estimates are compared to the expected results by a structural similarity index measure [Wang et al., 2004]. [Table 1] Representation Polar Cartesian Network U-net Guided U-net U-net Guided U-net SSIM score 0.8608 0.8639 0.7504 0.8279 This data is checked in order to determine that with a U-Net type network good similarity is already achieved, and that with a guided U-Net network the results are even better (SSIM scores even closer to 1). By way of a reminder, SSIM is an acronym for the term “Structural Similarity Index Measure”. In conclusion, the invention is therefore very effective, and proposes different performance levels depending on the selected embodiment. It should be emphasized that, while the examples use, by way of an illustration, U-Net neural networks, the invention similarly applies to other types of deep neural networks, notably those called "transformers" (in a vision adapted “transformer” architecture), notably like those described in the publication by: Tianyang Lin, Yuxin Wang, Xiangyang Liu, Xipeng Qiu entitled, “A survey of transformers”, AI Open, Volume 3, 2022, Pages 111-132, ISSN 26666510. (https: / / doi.org / 10.1016 / j.aiopen.2022.10.001). It also similarly applies to at least one Fourier neural operator, notably as described in the publication by Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., & Anandkumar, A. (2020) entitled, “Fourier neural operator for parametric partial differential equations”. arXiv preprint arXiv:2010.08895. It also should be emphasized that the method according to the invention comprises various steps, which can be implemented with computing means, notably a PC, a server or a computer. The relevant steps can be carried out on-line (computer means connected by computer communication means, Internet / intranet, for example) or off-line: thus, notably, the acquisition of real images can be carried out on-line or off-line (by being pre-recorded and made accessible).

Claims

1. A method for determining a free surface elevation FSE of an area of a body of water, characterized in that, according to said method,1) - at least one learning database (E) of a plurality of pairs (6) of images (W, R) is formed, each pair being formed by a first image (W) of the free surface elevation FSE and of a second radar return image (R), the two images of the same pair corresponding to the same area of the surface of the body of water and to an identical or close instant in time, said images (W) and (R) being real or simulated;2) - each pair of images (W, R) is associated with spatial features of said area of the surface of the body of water corresponding to this pair, said spatial features (9) comprising at least the position relative to the radar and to the one or more main directions of waves, said spatial features being directly known when acquiring the images, or being deduced from the analysis of the images themselves, or even being provided by an external source;3) - at least one deep neural network, notably of the convolutional type, or of the “transform” type or a Fourier neural operator, is trained on at least said learning database (E, E’), by using said spatial features (9) for the specialization of said one or more deep neural networks, and to obtain a model (H) for reconstructing the free surface elevation FSE as follows:- each pair of images (W, R) is divided into n distinct areas, according to spatial features, with n being greater than or equal to 2, and notably being equal to 4;- n deep neural networks (RN1, RN2, RN3, RN4) are used, with each deep neural network specializing in a dedicated area of the pair (6) of images (W, R);- each neural network is trained in its dedicated area of the pair of images (W, R);4) - real return images (8) are acquired by a radar;5) - said reconstruction model (H, H’) is applied to said acquired real return radar images (8) in order to obtain the free surface elevation FSE (I).

2. The method as claimed in the preceding claim, characterized in that the first image (W) of each pair (6) of images (W, R) is obtained by discretizing three-dimensional time series (2) of the free surface elevation FSE generated by a numerical wave field simulation code (A), and the second image (R) of each pair of images (W, R) by discretizing radar returns generated by a numerical radar simulation code (B).

3. The method as claimed in any of the preceding claims, characterized in that the first images (W) of each pair (6) of images (W, R) originate from real measurements of free surface elevation FSE, notably produced with at least one LiDAR type remote sensor or astereoscopic camera system or a set of measurement buoys, over at least part of the surface area covered by the radar, and in that the second radar return images (R) originate from real measurements performed by at least one radar, notably an X-band or S-band maritime radar.

4. The method as claimed in any of the preceding claims, characterized in that the learning database (E, E’) comprises a first database of pairs of images (W, R) of surface elevation and radar return images (R), all the images of which are simulated, and a second database of pairs of images (W, R) of surface elevation and radar return images, all the images of which are real, the relative weight of said two databases optionally being adjustable.

5. The method as claimed in any of the preceding claims, characterized in that said spatial features (9) include a distance to an origin, said origin notably corresponding to the position of the radar, and an angular orientation, notably an azimuth.

6. The method as claimed in any of the preceding claims, characterized in that the one or at least one of the deep neural networks (RN1, RN2, RN3, RN4) is an encoder-decoder network, for example, with a U-Net architecture.

7. The method as claimed in any of the preceding claims, characterized in that said pairs (6) of images (W, R) are divided into n areas as a function of the azimuth and of the distance to the radar.

8. The method as claimed in any of the preceding claims, characterized in that, in order to obtain said reconstruction model (H, H’), at least one first guide image Rg1 is formed for each radar return image (R) from the learning database (E, E’) by means of a distance map to the radar, and the deep neural network is trained by means of said surface elevation image (W) and of said at least one first associated guide image Rg1.

9. The method as claimed in the preceding claim, characterized in that, in order to obtain said reconstruction model (H, H’), at least one second guide image Rg2 is formed by means of an angular orientation map, and the deep neural network is also trained by means of said associated second guide image Rg2.

10. The method for determining a free surface elevation FSE of an area of a body of water as claimed in any of the preceding claims, characterized in that the following steps are implemented:A1) constructing a first database of images (W) obtained from three-dimensional time series (2) of free surface elevation, notably the elevation (z), the position as Cartesian (x, y) or polar (r, ¢) coordinates, and the time (t), generated by a code for simulating wave fields (A) corresponding to different sea conditions;A2) constructing a second database of images, called return images (R), obtained from radar returns generated by a radar simulation code (B) using said time series generated by the wave field simulation code, configured according to environmental conditions (5), notably the wind speed and direction and the presence of precipitation;A3) constructing a learning database (E, E’) comprising a first set of the images from the first database (W) constructed in step A1) and a first set of the corresponding return images from the second database (R) constructed in step A2), while integrating the environmental conditions used in step A2), so as to obtain pairs of images (W, R) each formed by a first surface elevation image (W) and a second radar return image (R), the two images of the same pair corresponding to the same area of the surface of the body of water and to an identical or close instant in time;A4) associating each pair of images (W, R) from the learning database (E, E’) with the spatial features (9) of the area of the surface of the body of water corresponding to this pair, said spatial features comprising a distance to an origin, said origin notably corresponding to the position of the radar, and an orientation, notably by azimuth, relative to the one or more main directions of the waves; A5) constructing a model (H, H’) a model for reconstructing free surface elevation from the learning database (E, E’) obtained in step A3) and from the spatial features specified in step A4), by training at least one convolutional or “transformer” deep neural network, notably of the U-Net type, or a Fourier neural operator, said one or more networks specializing in the spatial features obtained in step A4);A6) acquiring real images (F) using at least one radar, notably a maritime radar, with which their spatial features defined in step A4) are associated;A7) determining the free surface elevation FSE by applying the model (H, H’) for reconstructing free surface elevation obtained in step A5) at least to the real images (8) acquired with the radar in step A6) with their spatial features.

11. The method as claimed in the preceding claim, characterized in that, in step A2) of constructing the second database of return images, the radar simulation code is configured according to environmental conditions (5) selected from among at least: the presence or the absence of wind, the wind speed, the wind direction, the presence or the absence of precipitation such as rain, the intensity of precipitation, the presence or the absence of fog, the temperature and the sea conditions.

12. The method as claimed in any of claims 10 or 11, characterized in that, in step A5) of constructing the model (H, H’) for reconstructing free surface elevation, said model is validated with a validation database made up of a second set of images from the first database constructed in step A1) and of a second set of corresponding return images fromthe second database constructed in step A2), with said sets being different from those used for learning.

13. The method as claimed in any of claims 10 to 12, characterized in that, in step A6) of acquiring (F) real images (8) using at least one maritime radar, said images are recorded, and in that, in step A7) of determining the free surface elevation FSE, a communication means, notably a computer means, is used to access said recorded radar images.

14. The method as claimed in any of claims 10 to 13, characterized in that the following additional steps that are carried out between step A5) and step A6) are also implemented: B0) taking measurements of the free surface elevation FSE over at least some of the area covered by the radar, notably a maritime radar, simulated in step A1), notably by means of a stereoscopic camera system, a LIDAR or a set of measurement buoys;B1) constructing a database of obtained images by digitizing the real wave measurements obtained in step B0);B2) constructing a database of obtained images from the real return signals obtained with a maritime radar, time-synchronized with the image database constructed in step B1);B3) constructing a real learning database from a set of measurement images of the FSE of the database obtained in step B1) and a set of images of the real radar returns from the database obtained in step B2), in the form of pairs of images;B4) associating each pair of images from step B3 associating real measurements of free surface elevation FSE, and the real radar returns with the spatial features of the corresponding area of the body of water, notably in terms of the distance and the direction from the one or more main directions of waves;(B5) improving the reliability of the model for reconstructing free surface elevation constructed in step A5) from the real learning database obtained in step B3) and the spatial features determined in step B4), using machine learning or deep learning methods, on areas that match between real measurements and real radar returns;B6) acquiring (F) real images (8) using at least one radar, notably a maritime radar;B7) replacing step A7) with step B7) by applying the improved model (H’) for reconstructing free surface elevation obtained in step B5) at least to the real images (8) acquired with the radar in step B6).

15. A method for monitoring, operating, or controlling a system subjected to waves within a body of water, wherein the following steps are implemented:- determining the free surface elevation of said body of water by means of the method for determining free surface elevation as claimed in any of the preceding claims; and- monitoring, using or controlling said system subjected to waves as a function of the free surface elevation of said body of water thus determined.

16. A PC, server or computer configured to implement the method as claimed in any of claims 1 to 15.5   17. A computer program product downloadable from a communication network and / or storedon a medium that can be read by a PC, a server or a computer and / or that can be executed by a processor, comprising program code instructions for implementing the method as claimed in any of claims 1 to 15, when said program is executed on a PC, a server or a computer.10   18. A storage medium that can be read by a PC, a server or a computer and storinginstructions which, when they are executed by a PC, a server or a computer, cause the PC, the server or the computer to implement the method as claimed in any of claims 1 to 15.

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