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... this point, the schematic representation may have outlived its usefullness because of the complex (and not here defined) meaning of the convolution-like operator ⊗ , and because the expression for G in terms of P has created an even more schematic view in terms of an exponentiated P . So for now we will leave the schematic representation, and journey on with more concrete quantities the rest of the way through. The formal schematic discussion put forward here does have a mathematically and physically precise counterpart. The field of study in Radiative Transfer has been applied for some time to water optics, by a large number researchers. The references cited are excellent reading for further information. As mentioned, there is one additional radiosity scenario that can be important to ocean rendering under certain circumstances, but which we will not consider. The situation is illustrated in figure 2. Following the trail of the arrows, which track the direction light is travelling, we see that sometimes light coming to the surface (from above or below), can reflect and/or transmit through the surface more than once. The conditions which produce this behavior in significant amounts are: the wave heights must be fairly high, and the direction of viewing the waves, or the direction of the light source must be nearly grazing the surface. The higher the waves are, the less grazing the light source or camera need to be. This phenonmenon has been examined experimentally and in computer simulations. It is reasonably well understood, and we will ignore it from this point on. In this section we take a look at the mathematical problem we are trying to solve. We simplify the mathematics considerably by applying a series of approximations. How do we know these approximations are any good? There are decades of oceanographic research in which the ocean surface motion has been ...
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... mentioned, there is one additional radiosity scenario that can be important to ocean rendering under certain circumstances, but which we will not consider. The situation is illustrated in figure 2. Following the trail of the arrows, which track the direction light is travelling, we see that sometimes light coming to the surface (from above or below), can reflect and/or transmit through the sur- face more than once. ...
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... 19 shows the wave profiles of both surfaces along a slice through the surfaces. Finally, the profile of the choppy wave is plotted together with the value of the minimum eigenvalue in figure 20, showing the clear connection between folding and the negative value of J−. ...
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... is not yet clear just how quanti- tatively accurate these interactions are. Figure 22 shows a rendered scene with a high resolution calculation of waves interacting with the hull of the ship. ...
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... we could build a collection of kernels over a range of depth values D. At each point on the grid, a custom kernel is constructed for the actual depth at that grid point by interpolating from the set of prebuilt values. Figure 23 is the wave height from a simulation using this technique. The bottom slopes from deep on the right hand side, to a depth of 0 on the left edge. ...
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... derivation of the expressions for R and T is based on the elec- tromagnetic theory of dielectrics. We will not carry out the deriva- tions, but merely write down the solution Figure 24 is a plot of the reflectivity for rays of light traveling down onto a water surface as a function of the angle of incidence to the surface. The plot extends from a grazing angle of 0 degrees to per- pendicular incidence at 90 degrees. ...
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... is the quantitative explanation of why the When the incident ray comes from below the water surface, there are important differences in the reflectivity and transmissivity. Fig- ure 25 shows the reflectivity as a function of incidence angle again, but this time for incident light from below. In this case, the re- flectivity reaches unity at a fairly large angle, near 41 degrees. ...
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... shader was used to render the image in figure 27 using the BMRT raytrace renderer. For reference, the exact same image has been rendered in 26 with a generic plastic shader. ...
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... is a mathematically precise result that the ratio K/c depends only on b/c, the single scatter albedo, and some details of the angular distribution of individual scattering distributions. Figure 28 is an example of a model of K/c for reasonable water conditions. Models have been gener- ated for the color dependence of K, most notably by Jerlov. ...
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... quantity Ref is the scaling factor that varies with position on the fictitious plane due to focussing and de- focussing of waves, and is called a caustic pattern. Figure 29 shows an example of the caustic pattern Ref . Notice that the caustic pat- tern exhibits filaments and ring-like structure. ...
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... two PDF's were collected for different surface roughness conditions: rougher wa- ter tended to suppress more of the high magnitude fluctuations in intensity. Figure 32 shows the pdf at two depths from a simulation of the ocean surface. These two sets do not match Dera's measurements because of many factors, but most importantly because we have not simulated the environmental conditions and instrumentation in Dera's experiments. ...
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... After prescribing specific environmental parameters, combining a wave spectrum with Fast Fourier Transform (FFT) techniques facilitates efficient simulation of realistic sea surface states [26], [34], [66]. This procedure, which is briefly discussed in Appendix B, allows spatially detailed simulations of the sea surface, as shown in Fig. 2, to be rendered from statistical models of sea surface parameters and vice versa. ...
... In this study, discrete, 2-D random elevation grids are generated using the FFT technique and the wave spectrum model introduced above, as illustrated in Fig. 2. The process, which follows Mobley [26] and Tessendorf [66], allows for efficient generation and analysis of simulated sea surfaces. Analysis of these simulated sea surfaces bypasses issues of measurement uncertainty and inconsistency in generating robust statistics for model fitting and validation. ...
... EAC4 aerosol optical depth reanalysis products are available from the Copernicus Atmosphere Data Store (ADS) at https://ads.atmosphere.copernicus.eu/cdsapp#!/dataset/camsglobal-reanalysis-eac4. The method for simulating sea surfaces used in this study is publicly available from Mobley [26] and Tessendorf [66]. meteorology, natural hazard and applications of the ...
A generalized probability density function (PDF) is introduced to enhance sea surface slope modeling for remote sensing applications. This new PDF, which incorporates the anisotropy index to better capture the direction and tilt of surface waves relative to the classical Cox & Munk model proposed 70 years ago, is then applied to sun glint correction in satellite imagery. Sixteen different mean square slope (MSS) models are reviewed to establish both the strengths and limitations of the classical model and the stability and adaptability of the anisotropy index. The new sea surface model applies to a wider range of sea surface states, including those in coastal environments, and provides a stable quantitative description of sea surface topography. Application of the generalized PDF to sun glint correction in satellite imagery demonstrates its overall accuracy and improved efficacy compared to the Cox & Munk model, particularly in maintaining the integrity of sea surface and cloud features in complex weather environments. This initial study provides a promising approach to improve the accuracy and reliability of sun glint correction in remote sensing of water surfaces, with applications to improving both historical and future satellite-based climate data records.
... Fluid simulation based on solving complex partial differential equations (PDEs) is ubiquitous in science and engineering problems such as simulating the weather [23], ocean currents [33], the water flow in a pipe [13], and the airflow around an airplane [5], to name but a few. Despite the massive increase in computational power and the development of acceleration techniques [7,11], traditional numerical methods such as finite element [1] or finite difference [34] remain unsuitable for time-critical scenarios such as realtime graphics simulation [17]. ...
Learning-based fluid simulation has proliferated due to its ability to replicate the dynamics with substantial computational savings over traditional numerical solvers. To this end, graph neural networks (GNNs) are a suitable tool to capture fluid dynamics through local particle interactions. Nonetheless, it remains challenging to model the long-range behaviors. To tackle this, this paper models the fluid flow via graphs at different scales in succinct considerability and physical constraints. We propose a novel multi-scale GNN for physics-informed fluid simulation (MSG) by introducing a nonparametric sampling and aggregation method to combine features from graphs with different resolutions. Our design reduces the size of the learnable model and accelerates the model inference time. In addition, zero velocity divergence is explicitly incorporated as a physical constraint through the training loss function. Finally, a fusion mechanism of consecutive predictions is incorporated to alleviate the inductive bias caused by the Markovian assumption. Extensive experiments corroborate the merits over leading particle-based neural network models in terms of both one-step accuracy (+6.7%) and long trajectory prediction (+16.9%). This comes with a run-time reduction by 2.8% over the best baseline method.
... Research into ocean rendering dates back to the work of Fournier and Reeves. Tessendorf [16] presented a sophisticated lighting model for realistic reproduction of ocean waves, while Ashikhmin et al. [17,18] presented a light transport approach for the complex lighting effects of the ocean. However, traditional physically-based rendering is timeconsuming and impractical for real-time applications. ...
Learning and inferring underlying motion patterns of captured 2D scenes and then re-creating dynamic evolution consistent with the real-world natural phenomena have high appeal for graphics and animation. To bridge the technical gap between virtual and real environments, we focus on the inverse modeling and reconstruction of visually consistent and property-verifiable oceans, taking advantage of deep learning and differentiable physics to learn geometry and constitute waves in a self-supervised manner. First, we infer hierarchical geometry using two networks, which are optimized via the differentiable renderer. We extract wave components from the sequence of inferred geometry through a network equipped with a differentiable ocean model. Then, ocean dynamics can be evolved using the reconstructed wave components. Through extensive experiments, we verify that our new method yields satisfactory results for both geometry reconstruction and wave estimation. Moreover, the new framework has the inverse modeling potential to facilitate a host of graphics applications, such as the rapid production of physically accurate scene animation and editing guided by real ocean scenes.
... With regard to the numerical simulations, sea waves propagating first on a constant depth seabed and later on a spatially varying bathymetry are considered. For the constant sea depth scenario, a synthetic sea wave simulator based on the fast Fourier transform (FFT) [37,38] and the Joint North Sea Wave Observation Project (JONSWAP) model [39] for the sea wave spectrum is implemented. As for the variable depth scenario, a hydrodynamic solver for coastal dynamics is used to describe wave propagation over the reference bathymetry [40][41][42]. ...
... The models adopted for generating the synthetic sea wave field (SWF) are described in this section. Specifically, a SWF over a constant bathymetry is first produced by adopting the Fourier domain approach [38]. Then, random waves are generated and propagated over a planar-sloping beach according to the model for coastal dynamics described in [40]. ...
... Synthetic sea images for a constant bathymetry scenario are generated in a very straightforward and computationally efficient manner by using the FFT algorithm. According to [38], the complex Fourier amplitudes of a wave elevation field at time zero are given by: ...
This paper provides an assessment of a 24 GHz multiple-input multiple-output radar as a remote sensing tool to retrieve bathymetric maps in coastal areas. The reconstruction procedure considered here exploits the dispersion relation and has been previously employed to elaborate the data acquired via X-band marine radar. The estimation capabilities of the sensor are investigated firstly on synthetic radar data. With this aim, case studies referring to sea waves interacting with a constant and a spatially varying bathymetry are both considered. Finally, the reconstruction procedure is tested by processing real data recorded at Bagnoli Bay, Naples, South Italy. The preliminary results shown here confirm the potential of the radar sensor as a tool for sea wave monitoring.
... The commonly used directional extension functions are those recommended by ITTC or ISSC, but these two only include the effect of angle and not the effect of frequency. The directional expansion function obtained from the Stereo Wave Observation Project (SWOP) [27,28] is applied to this paper, which contains the effects of angle and frequency. ...
... This underscores the superior applicability of the numerical model introduced in this study in addressing engineering problems. (27) where, xx I represents the moment of inertia of roll; xx I Δ denotes the added mass moment of inertia; ...
A variety of floating structures at sea play a vital role in the exploitation and utilization of marine resources. The study about interactions between waves and structures is necessary for the impact of the harsh marine environment on the motion and service life of structures. Currently, most studies about the seakeeping of structures are based on simplified regular waves. Because the regular waves do not truly restore the actual wave conditions at sea, the simulation of irregular waves has great practical importance to the study of interactions between waves and structures. Based on the potential flow theory and high-order boundary element method (HOBEM), a Fortran code is developed in this paper and named as SWBI (Solver for Wave–Body Interactions). This program consists of the following two parts: a time–domain numerical model about interactions between waves and 3D structures is based on weakly nonlinear method, and a numerical model about simulation of the nonlinear regular waves, the long-crested irregular waves, and the short-crested irregular waves. This Fortran code is used to simulate the motion of Floating Production Storage and Offloading (FPSO) under three different ocean wave spectra (including ITTC two-parameters spectrum, JONSWAP spectrum and the most likely spectral form of Ochi-Hubble) and found that: To a certain extent, the difference in the motion of FPSO under different wave spectra have a connection with different type of wave, sea conditions and incident angle. The difference in roll of FPSO is quite significant in short-crested irregular waves. The range of FPOS’s roll under the JONSWAP spectrum is the largest when the incident angle is 30°, and range of FPOS’s roll under the most likely spectral form of Ochi-Hubble is the largest when the incident angle is 90°.
... We make use of the techniques employed by Tessendorf [21] and consequently Horvath [22] and Gamper [23] to model our ocean surface waves and the interested reader is highly encouraged to peruse these sources. ...
... In summary, we use a wave spectral density function (or wave spectrum, ) and random numbers drawn from a Gaussian distribution to first create the initial wave state which is in a 2D spectral domain. As time increases, the dispersion relationship [21] is used to propagate these waves forward with a specific wavelength and phase velocity. Inverse Fast Fourier Transforms are then used to reconstruct the spatial form of the waves as 2D height maps, with the capability to control horizontal spatial warping for choppy waves and whitecap regions. ...
... In the PPO algorithm, equation (26), and thus (21) are not used directly. The reason for this is that if an update to a policy is too large, the policy can become unstable, leading to a performance collapse. ...
Heading towards navigational autonomy in unmanned surface vehicles (USVs) in the maritime sector can fundamentally lead towards safer waters as well as reduced operating costs, while also providing a range of exciting new capabilities for oceanic research, exploration and monitoring. However, achieving such a goal is challenging. USV control systems must, safely and reliably, be able to adhere to the international regulations for preventing collisions at sea (COLREGs) in encounters with other vessels as they navigate to a given waypoint while being affected by realistic weather conditions, either during the day or at night. To deal with the multitude of possible scenarios, it is critical to have a virtual environment that is able to replicate the realistic operating conditions USVs will encounter, before they can be implemented in the real world. Such "digital twins" form the foundations upon which Deep Reinforcement Learning (DRL) and Computer Vision (CV) algorithms can be used to develop and guide USV control systems. In this paper we describe the novel development of a COLREG-compliant DRL-based collision avoidant navigational system with CV-based awareness in a realistic ocean simulation environment. The performance of the trained autonomous Agents resulting from this approach is evaluated in several successful navigations to set waypoints in both open sea and coastal encounters with other vessels. A binary executable version of the simulator with trained agents is available at https://github.com/aavek/Aeolus-Ocean
... To build a 3D sea state representation, knowledge of mathematics, physics, and computer graphics is required. Fortunately, virtual representations of the ocean surface have been extensively used in the film and video game industries [9]. For these industries, the sea surface and its evolution over time must be realistic, attractive, and interactive. ...
... To render the light properties of the surface, the normal to the surface must be calculated. For reproducibility, the formulas to obtain the normal direction N of a single wave using the tangent T and binormal directions B are given by (please refer to Jasper Flick (2018) 9 for a practical implementation ). ...
... Here, only a combination of 16 waves was used for an equilibrated performance in terms of realism and computation resources. A better approach to increase the number of wave modes would be to use the inverse fast Fourier transform (IFFT) in the GPU to calculate the vertex's positions and normals according to thousands of waves, as described in [9] [22]. An advantage of using the IFFT is that the resulting textures can be used as tiles, making it possible to create an endless sea surface (in this work, the sea surface was limited to the location of the seafloor observatory). ...
Oceanographic data such as wave conditions (height, period, direction), wind, and sea currents are often difficult to interpret. What is the sea state given a certain wave height, wave directional spreading, and wind speed (e.g., 2 m, 29º, 18 m/s)? An expert user might be able to imagine the sea conditions with such information, but this will be almost impossible for a non-expert user. The common approach for visualizing oceanographic data and its variability is usually through tables and 2D graphs, for example plots, bar diagrams, and latitude-longitude maps. These visualizations are often limited to displaying raw data values, which still require user interpretation. With the purpose of providing a more intuitive
view of the marine environment and sea conditions to a widespread audience, this work presents an experimental web
application. The open-source application represents in a realistic
and intuitive way the observations from a meteo-oceanographic
and seafloor observatory located in the Western Mediterranean;
the OBSEA observatory. The user can visualize the marine observatory facilities within a 3D virtual environment that changes and evolves according to the data acquired. This work
aims at a digital twin of the seafloor observatory using near-real-
time and historical data.
... As regards the numerical simulations, a synthetic sea wave simulator is developed by applying the Fourier domain approach [35], [36] and considering the Pierson-Moskowitz model [37] for the sea wave spectrum. Then, a 3D simulator of FMCW MIMO radar data is developed and implemented by taking into account the backscattering from the capillary waves and modulation phenomena (shadowing and tilt modulation) affecting the radar echoes, which depend on the generated sea model and radar antenna geometry. ...
... where and are independent outputs from a Gaussian random number generator with zero mean and unitary standard deviation [36]. Moreover, the function defines the spectral properties of the WF and it is set in accordance with the 2D Pierson-Moskowitz model, suitable for a fully developed wind sea [37] ( ) = 2 4 − ( 8) is calculated at the radar boresight (i.e. ...
... Equation (10) preserves the complex conjugate property and allows computing the sea surface elevation ( ,) as real quantity in eq. (6) via an inverse Fast Fourier Transform (IFFT) [36]. Note that, herein, the sea WF is generated in deep-sea water conditions, i.e., when tanh( ) → 1 ( → ∞). ...
This paper investigates the capabilities of 24 GHz Frequency Modulated Continuous Wave (FMCW) Multiple-Input Multiple-Output (MIMO) radar technology to retrieve sea surface currents and directional wave spectra. A procedure based on the dispersion relation, which was previously applied to process X-band marine radar data is herein exploited. The estimation performance of the radar sensor is first assessed by numerical tests in the case of synthetic sea wave fields with known characteristics in terms of wave direction and surface currents. Finally, the estimation procedure is assessed on real data collected at San Vincenzo quay in the port area of Naples, Italy. The achieved results are encouraging and highlight that 24 GHz FMCW MIMO radar is a viable technology for sea wave monitoring.
... Mastin et al. (1987) introduced such a technique into water body rendering. FFT-based methods use ocean wave spectrum derived from theoretical or measured statistical data, such as the Phillips spectrum (Tessendorf 2001(Tessendorf , 2004(Tessendorf , 2008, to describe the ocean surface, combining a large number of sine waves to generate wave distribution in the frequency domain. An inverse FFT is then performed to transfer the data to the spatial domain. ...
This paper proposes a novel particle-based scheme for simulating interactive water waves. We first modify the implementation of the wave packet so that each packet can carry two wavenumbers. As a consequence, the wavelength-dependent behaviors can be accurately simulated. We then optimize the approximation technique of diffraction for simulating partial diffraction and promoting rendering efficiency. Lastly, we provide a novel evaluation module based on wave patterns generated by solving wave functions. Specifically, we introduce the singular boundary method (SBM) to serve as an analytical solution of the Laplace equation. We tested our scheme and other state-of-the-art approaches on scenes with regular-shaped, complex-shaped, and user-designed obstacles. Various results indicate that, compared to the state of the arts, our scheme can achieve higher physical accuracy and more satisfying computational efficiency.
... Gerstner Waves (1802) It is based on Navier-Stokes equation by describing a particle's motion on the surface as a circular motion to provide an approximate to simulate the air-water interface. [12,25] Phillips (1954) A fully developed sea is considered deep water. It is widely used in real-time simulation of oceanic waves. ...
... It is widely used in real-time simulation of oceanic waves. [9,25,26] Neumann (1955) It is valid for only fully developed sea, and it is valid for only gravity waves regime. ...
... Since oceanic waves and our laboratory waves have random/stochastic, but not necessarily isotropic motions, the statistical measurements of the air-sea interface are different under differing experimental configurations. Therefore, the illumination of the water facets depends on the incident angle, θ i , of the diffused light source and the refraction from the interface into the camera to obtain the wave spectrum [25]. Thus, locating the high-speed camera above the wave tank with wave direction (forward) produces different spectra from when the camera is located in the opposite direction (backward). ...
The roughness of the ocean surface significantly impacts air-to-sea imaging, oceanographic monitoring, and optical communication. Most current and previous methods for addressing this roughness and its impact on optical propagation are either entirely statistical or theoretical, or are 'mixed methods' based on a combination of statistical models and parametric-based physical models. In this paper, we performed experiments in a 50-foot-wave tank on wind-generated waves, in which we varied the wind speed to measure how the surface waves affect the laser beam propagation and develop a geometrical optical model to measure and analyze the refraction angle and slope angle of the laser beam under various environmental conditions. The study results show that the laser beam deviations/distortions and laser beam footprint size are strongly related to wind speed and laser beam incidence angle.
