Wave-by-Wave Forecasting via Assimilation of Marine Radar Data
1. Introduction
A real-time, wave-by-wave forecasting system has a number of potential applications. For example, the performance of wave energy converters (WEC) can be improved when the device’s behavior is tuned via adaptive control on a wave-by-wave basis (Brekken 2011; Li et al. 2012; Korde 2014; O’Sullivan and Lightbody 2017). Also, maritime operation tasks such as cargo and personnel transfer, helicopter landing, high speed navigation, and small craft recovery would benefit from the prediction of workable time windows and/or extreme waves. The short-term (30–90 s) phase-resolved prediction of ocean waves is often referred to as deterministic sea wave prediction (DSWP) and has gathered increasing attention in recent years (e.g., Belmont et al. 2014; Naaijen and Wijaya 2014). The various proposed approaches typically fall under two schema: prediction by mathematical tools that utilize single point measurements, and predictions by models that utilize multipoint measurements at a distance and subsequent reconstruction of the surrounding wave field. The current study falls under the latter category, where reconstruction and prediction are attempted using wave physics capable of capturing refraction, diffraction, and shoaling.
In the design of WEC control systems, DSWP approaches that use only single-point observations at the WEC location (which is also the point of forecast) are attractive because they negate the need for observations of the surrounding wave field. Methods have been proposed using linear autoregressive filters (Fusco and Ringwood 2010), fast Fourier transform decomposition (Halliday et al. 2011), Kautz models (Schoen et al. 2011), artificial neural networks (Hesam et al. 2014; Gillespie 2015), and Kalman filters (Ling and Batten 2015). Though these approaches may be convenient from a logistics standpoint, the forecast performance typically breaks down after 1–1.5 wave periods, while it has been argued that a minimum 2–4-wave-period (30–60 s) forecast is needed for WEC control applications (Fusco and Ringwood 2012). Additionally, these methods are likely not sufficient for the prediction of extreme or rogue waves, which may induce extraneous loads on WEC components or dangerous conditions for maritime operations.
An alternate approach to DSWP uses measurements throughout the wave field to reconstruct the two-dimensional sea and propagate toward a location of forecast interest. In this case, the predict-ahead time is a balance between computation time and wave group (Wu 2004) or phase velocities (Edgar et al. 2000), which can be demonstrated using space–time diagrams (Abusedra and Belmont 2011; Naaijen and Wijaya 2014; Qi et al. 2018b). There are several approaches to this type of DSWP, each of which requires some present knowledge of the wave field. The most common approach relies on the Fourier Transform to obtain free wave components via decomposition of measured wave elevation time series (Morris et al. 1998; Blondel et al. 2010; Belmont et al. 2014; Qi et al. 2018a). An additional approach is to obtain periodic boundary conditions via iterative procedures on measured data (Zhang et al. 1999), or through data assimilation (Wu 2004; Aragh and Nwogu 2008; Nouguier et al. 2014). In some cases, nonlinear models are used to better capture nonlinear wave–wave interactions (Prislin et al. 1997; Zhang et al. 1999; Blondel et al. 2008; Blondel-Couprie et al. 2013; Simanesew et al. 2017; Qi et al. 2018a; Nouguier et al. 2014); however, given that the predict-ahead time is limited by model computation time, effective linear models may be more appealing. The linear approaches have shown various levels of success through numerical simulations or wave tank testing, but the challenges of field implementation mean that a robust approach has yet to be demonstrated.
Wave field measurements are a key component of DSWP. In situ observations have the advantage of being direct measurements of the sea surface elevation, although typically the available data are spatially sparse and deployment of in situ sensors is not possible in mobile operations. Remote sensing techniques are attractive in that they record thousands of point measurements throughout a wave field; the challenge with these data is that they are an indirect representation of sea surface elevation (lidar being an exception). Various remote sensing techniques for phase-resolved wave measurements include noncoherent marine radar (e.g., Dankert and Rosenthal 2004), coherent or Doppler marine radar (e.g., Støle-Hentschel et al. 2018), or lidar (e.g., Nouguier et al. 2014).
The present study utilizes a noncoherent marine radar, which will be referred to herein simply as radar. Methods for retrieving wave information from radars date back to the 1960s. Originally, images were inspected visually to obtain estimates of wave direction, wavelength and period (Ijima et al. 1964; Wright 1965). Since then, notable improvements have been made in computing wave spectra for retrieval of statistical wave parameters (Ziemer and Gunther 1994; Borge et al. 1999; Liu et al. 2015). It is only within recent years that radar has been investigated for phase-resolved wave measurements (Dankert and Rosenthal 2004; Belmont et al. 2014; Naaijen and Wijaya 2014; Wijaya et al. 2015; Qi et al. 2016).
Attempts to retrieve phase-resolved wave information from radar images rely inherently on principles of radar backscatter. The dominant interaction between radar electromagnetic waves and the sea surface is Bragg scattering, by which water waves of twice the electromagnetic wavelength contribute to backscatter. Longer gravity waves are visible through their modulation of the surface Bragg waves. The majority of phase-resolved reconstruction attempts consider the dominant modulation mechanisms to be 1) wave shadowing, by which areas behind a gravity wave are not directly measured by the radar signal, and 2) surface tilt, in which the intensity of backscatter return is affected by the wave slope. Phase-resolved reconstruction attempts based on wave shadowing build on bulk parameter estimations that use three-dimensional Fourier analysis to create a spectral representation of the shadowed radar images (Young et al. 1985). The spectra are filtered to retain components satisfying the linear dispersion relationship so that they are close in shape to the actual wave spectra, and may be scaled using an external calibration factor to match in magnitude (Nieto Borge et al. 2004; Naaijen and Wijaya 2014; Wijaya et al. 2015; Qi et al. 2016; Støle-Hentschel et al. 2018). The calibration factor is determined empirically from in situ measurements and can be problematic in its accuracy and versatility. From the calibrated spectrum, an inverse Fourier transform would reproduce the measured wave elevations. Coherent radars, or Doppler radars, which measure Doppler velocity from the motion of scattering elements, may not require in situ calibration due to the direct relationship of their measurements with wave orbital velocities. Stochastic and phase-resolved reconstructions from coherent radars are comparable to noncoherent approaches (Støle-Hentschel et al. 2018); however, coherent radars are typically operated along a single range line rather than a full rotational scan, thus limiting the spatial extent of reconstruction.
Alternative to modulation transfer functions, the method of Dankert and Rosenthal (2004) is an empirical inversion that relies on integrating the surface tilt, or surface slope, to obtain surface elevation, thereby not requiring in situ calibration. The surface tilt angle is said to be equivalent to the depression angle (angle of radar beam downward from horizontal), which is estimated as deviations from the mean radar cross section. Surface elevations are found from integrating the surface tilt estimations. The major benefit of this approach is that calibration is not required. However, shadowing effects in the radar domain must be minimal for the efficacy of the integration kernel. This requires that the installation height of the radar be large enough to avoid significant shadowing throughout the imaging domain, a limitation in its applicability.
Herein, we utilize the radar imaging model given by Lyzenga and Walker (2015) for retrieval of ocean wave information from radar imagery. The imaging model includes the mechanisms of both surface tilt and geometric shadowing. In this approach, calibration is not required, and there is minimal limitation on radar height. The model provides a direct relationship between the radial slope (component of wave slope in radar look direction) of ocean waves and the radar backscatter intensity that is applicable to illuminated (i.e., unshadowed) regions of the sea surface. When applied to time series of radar imagery, the radar model provides an estimate of phase-resolved radial slope in two spatial dimensions and in time. The presence of data gaps due to wave shadowing prevents the use of an integration kernel to calculate surface elevation directly. Thus, an assimilation algorithm is used whereby the estimated radial slopes are assimilated, and a physics-based wave model is used to reconstruct the observed wave field. As a first step, we have chosen the linear mild slope equations (MSE; Berkhoff 1972) as the basis for the wave model. This choice was made for its applicability in all water depths (full dispersion). However, the MSE is a linear wave model and accuracy will degrade for large amplitude waves. Nonlinear models could be substituted with potential cost in computation time. The final solution for wave field reconstruction is found iteratively by minimizing a cost function that represents the differences between the modeled and observed slope field.
This paper presents a detailed description of the algorithm including timing constraints (section 2), algorithm performance assessment using a synthetic dataset (section 3), and an assessment of the complete method (imaging model + algorithm) performance using three field datasets (section 4). The work is then summarized in section 5.
2. Description of the forecasting algorithm
The basic tasks of the forecasting algorithm are to ingest radar observations sampled approximately every 1.2 s, estimate a hindcast source function that can reproduce the observed waves, then propagate the source function in faster than real time across a polar domain to reconstruct the observed wave field and forecast the waves forward in time and space. Forecast waves can be produced at any desired location in the model domain as long as the model propagation is computed faster than the propagation of the observed waves. This process is depicted in Fig. 1, and is described in more detail in the proceeding sections.
Structure of the developed forecasting algorithm.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Structure of the developed forecasting algorithm.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Structure of the developed forecasting algorithm.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
a. Radar model
The information recorded by the X-band radar is backscatter intensity, which is a result of known and unknown modulation mechanisms of surface roughness. A recently derived radar model (Lyzenga and Walker 2015) provides a direct relationship between the backscatter intensity at each pixel in range and azimuth I(r, φ) and estimated radial component of sea surface slope ηr:
where ⟨I(r, φ)⟩ is the local mean, h is the radar height above the ocean surface, and r is the range distance from the radar antenna. The radial slope is the slope of the waves in the radar look direction; thus, when the radar look direction is aligned with the incoming wave vector, the radial slope is equal to the absolute wave slope. The estimated component of the wave slope decreases with azimuth angles away from the wave direction, until no wave information is available when the radar is looking along crest.
The backscatter model is built on perturbation theory (Valenzuela 1978), and an expression for the backscattered power received by a radar system. Additionally, geometric shadowing is considered analytically in the model through a geometric expression. Specific model assumptions are 1) radar grazing angles are small and thus incidence angle θ ≈ π/2 and depression angle θd ≈ h/r, 2) the height of the radar is much larger than the surface elevation of the waves (h ≫ η), and 3) the radar produces a Gaussian antenna pattern and has a time invariant logarithmic amplifier in the receiver. The result of assumption 3 is that the time-averaged radar backscatter intensity is expected to fall off with range at r−7/4. For the full derivation of the radar model, the reader is directed to the paper by Lyzenga and Walker (2015).
The first assumption is usually well satisfied by a shipborne or shore-based radar, but assumption 2 requires mounting the radar sufficiently high with respect to incident wave heights, such that the wave height has a negligible influence on the grazing angle, which is more challenging on small vessels. The third assumption is generally a reasonable assumption for marine radars; however, field validation of the range fall off predicted by assumption 3 thus far consists only of a limited comparison to two short datasets in Lyzenga and Walker (2015). The advantage of this radar imaging model is that it does not require an external calibration like other models.
b. Assimilation and model domain configuration
The radar images are collected in polar coordinates (range and azimuth), and thus a polar model domain of the same resolution is chosen for wave reconstruction and propagation. This provides a seamless mesh of the assimilated radar data with the model domain. Figure 2 shows a sample model domain. The gray gridlines are a coarsened representation of the range and azimuthal resolution, which in application is 3 m in range and 0.7° in azimuth. Sponge layers are drawn as red dashed lines inside of the outer bounds of the computation domain and are used for minimizing reflection off the model boundaries. A sponge-layer width of 1–2 wavelengths is sufficient to minimize wave reflection.
Schematic of the polar model domain. The annulus of data used for assimilation is outlined in yellow. The sponge layer is outlined in dashed red. The grid lines of the computation domain are shown as gray lines at 1:50 and 1:7 scales for range and azimuth, respectively. The ideal direction of wave propagation through the domain is demonstrated.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Schematic of the polar model domain. The annulus of data used for assimilation is outlined in yellow. The sponge layer is outlined in dashed red. The grid lines of the computation domain are shown as gray lines at 1:50 and 1:7 scales for range and azimuth, respectively. The ideal direction of wave propagation through the domain is demonstrated.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Schematic of the polar model domain. The annulus of data used for assimilation is outlined in yellow. The sponge layer is outlined in dashed red. The grid lines of the computation domain are shown as gray lines at 1:50 and 1:7 scales for range and azimuth, respectively. The ideal direction of wave propagation through the domain is demonstrated.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
The yellow lines in Fig. 2 indicate the region of the radar images that is used for data assimilation. Despite the radar images likely encompassing or exceeding the chosen model domain, a range subset of the radar images is chosen to reduce model computation time during data assimilation. The orientation of this annulus with respect to the location of interest for forecast waves is important. The annulus must be oriented so that it contains up-wave information; that is, the waves should be traveling toward the desired forecast location. In addition, the range distance between the inner edge of the annulus and the location of forecast interest governs the available time to produce the forecast as it will be the time it takes for wave propagation. The annulus must be placed at a distance from the forecast location that maximizes the forecast duration, while also maintaining adequate wave information available in the assimilation data due to increased wave shadowing at further ranges.
c. Estimating the source function
From measured data, the typical approach to reconstructing the wave field is through periodic functions matched across a handful of time series. Alternatively, the present assimilation routine utilizes a physics-based wave model to find a best-fit solution through an iterative technique. The major benefit of using a wave model is that evolution of the directional wave field is physically constrained. In the presented data assimilation process, a source function is computed from the assimilated data using the MSEs. The MSEs were chosen for their applicability to intermediate water depths where shoaling, dispersion, and refraction are all active processes. Originally derived by Berkhoff (1972), the MSEs are linear and in their time-dependent form are given as
Equation (2) solves for the time derivative of water surface elevation η, using knowledge of the wave phase speed C, wave group speed Cg, and wavenumber k as a function of position for a fixed angular frequency, ω. The source function, S, generates waves at the outer boundary and is added herein for the data assimilation method. Equation (3) solves for the velocity potential ϕ. While this version of the algorithm uses the linear MSEs, there is potential for inclusion of second-order Stokes nonlinear correction terms for increased regional applicability (Beji and Nadaoka 1997). This will increase computation time considerably but provides the potential for higher accuracy in shallower water depths. The nonlinear correction terms are not included in this study.
The source function S added in Eq. (2) is a time varying term that initializes the waves that are sent through the model domain, acting like a wavemaker. It is applied along the inner edge of the sponge layer at the outer edge of the domain (Fig. 2). An important aspect of the MSEs is that they are solved for a single frequency. Thus, the source function is computed for only one frequency per model solution. If the spectrum of waves is broad or bimodal, the MSEs may be solved several times, thus resulting in several source functions. The resulting single-frequency wave fields will be combined using linear superposition to form a complete description of the wave field.
Each frequency’s source function is determined through an iterative process of minimizing a simple cost function J0, which is defined as
where ηr is the radial derivative of the model estimated water surface elevation,
We wish to minimize the cost function subject to the constraint that the wave field is a solution of the MSEs [Eqs. (2) and (3)]. Hence, we augment the cost function with the constraints of the MSEs multiplied by the associated adjoint variables (Lagrange multipliers) to get
(see, e.g., Bennett 2002). We require that the first variation with respect to η, ϕ, and S vanish independently at the minimum. The first two conditions yield a set of adjoint equations:
Note that the adjoint equations are identical to the MSEs (i.e., they are self-adjoint), with (α, ψ) corresponding to (η, ϕ) in the equations above; however, the wavemaker source S in Eq. (2) vanishes in Eq. (6), and a source term based on the error in the predicted radial slope is now included in Eq. (7). The adjoint equations are solved with the error in the present prediction of the radial slope field (the source term) as the model forcing. They are solved backward in time (t′ = −t) and serve to propagate errors in the prediction back to the time and location of corresponding errors in the source term S(xs, t). The first variation of J with respect to S is
if we choose the adjustment to the source term to be δS = −bψ, where b is a positive constant, the change in the value of the cost function will be negative (i.e., the value of the cost function will get smaller), and δJ will vanish at the minimum (the final condition cited above). In this way, ψ can be considered to be proportional to the gradient of J with respect to S.
In each iteration, the wavemaker source S is updated using the gradient ψ, where b is again a positive constant:
In practice, b is determined by least squares estimation. We first compute the solution η′ using Snew for b = 1, and define
a straightforward least squares estimation problem where
Then S is updated with Snew for the next iterations. The forward and adjoint equations are solved iteratively until convergence of J. This procedure in the algorithm can be summarized in the following steps:
These steps are carried out once for every specified frequency in the model configuration. Thus, there exists a source function for each specified frequency, which may be used independently to propagate waves through the domain. The wave fields from each source function are summed linearly to produce the final wave field.
d. Model timing
The primary assessment of the algorithm’s performance presented here is on accuracy. The proceeding section defines the domain and computation constraints of making the system operational in real time. The key requirement of the system is that the data assimilation and forward propagation can be computed in faster than real time. This allows forecast waves to be computed before the actual waves arrive.
Figure 3 shows a schematic of model timing constraints, where distances and times are those used for synthetic testing, presented in section 3. The ordinate values represent distance in range (m). The ordinate where x > 0 represents the entire range of the domain, from the outer edge (x = 3800 m) to the origin (x = 0 m); Xassim is the size in range of the assimilation domain (1000 m). The distance between the assimilation region and the origin is called Xpred and is the distance across which the predicted waves travel to the origin (2800 m). The abscissa represents time in seconds. The duration over which observations are collected is called Tobs (200 s). Once the observations are recorded, the computation phase begins. The duration of the computation is depicted as Δtcomp. In this schematic, the computation time using assimilation of 200 s of synthetic data was 30 s. During the computation phase, two major steps are achieved: 1) the observed data are assimilated for computation of the source function and 2) the source function is propagated from the outer boundary of the domain, through Xassim and Xpred.
Schematic of constraints on model timing showing actual time and space scales used for synthetic validation.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Schematic of constraints on model timing showing actual time and space scales used for synthetic validation.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Schematic of constraints on model timing showing actual time and space scales used for synthetic validation.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
To provide a forecast, the computation must be performed before the actual waves arrive at the location of interest (x = 0). Thus, the velocity at which the actual waves travel (group velocity Cg) must be considered. This rate is the slope of the diagonal lines in Fig. 3. This line is drawn from the inner edge of the assimilation region because these are the first waves from the assimilation data to arrive at the location of interest. The time between the completion of the computation, and the arrival of the actual waves is the duration in advance that the waves can be forecast Δtfcst. In this sample configuration, the waves are predicted 75 s before their arrival at x = 0.
Illustrations of the sample forecasts are shown at the bottom of Fig. 3. The first forecast begins with still water, until the arrival of the first modeled waves. The available forecast waves begin at the origin once the first observed waves arrive at the inner edge of the assimilation domain. The duration of forecast time is dictated by the region and amount of assimilated data. The dashed diagonal line represents the last observed wave at the outer range of the assimilation domain. Since this is the last observed wave, it will be the last wave able to be forecast. Thus, the duration of the first forecast lasts until the first diagonal dashed line crosses the domain origin.
The second forecast is available after the computation using the second observation period. This forecast will begin at the start of the second observation. The second forecast also begins with still water, because in this example the waves from the first forecast have not yet arrived. Note that when the waves arrive, the forecasts contain overlapping waves. Configuring a domain such that forecasts contain overlapping waves is not necessary, though it will provide improved accuracy.
By the time the third forecast is available, waves have reached the origin, thus the entire forecast contains predicted waves. This will be the case for all succeeding forecasts. In this realistic schematic, each forecast can be predicted 75 s in advance. Thus, there is a continuous time series available of predicted waves 75 s before the actual waves arrive. This time horizon can be shortened or lengthened by manipulating the assimilation and domain size parameters. Additional influences on this time horizon will be the number of frequencies solved for in the MSEs (this schematic represents a solution using a single frequency), and the quality of the assimilated data. With lower quality assimilation data, for example with increased wave shadowing or noise, Δtcomp may increase slightly. However, in the synthetic trials presented in this paper, with significant wave shadowing and noise, Δtcomp was only influenced on the order of several seconds.
3. Synthetic validation
A synthetic dataset is created and used for preliminary validation of the forecasting algorithm. The synthetic dataset is generated using the forward solution to the MSEs and the radial slope is estimated with second-order centered finite differences along each range line. In addition, noise and shadowing are added to the radial slope data for a more realistic representation of radar imagery.
a. Generating a synthetic dataset
The wave spectrum used for synthetic validation is a multidirectional Texel, Marine Remote Sensing Experiment at the North Sea (MARSEN), and Atlantic Remote Sensing Land and Ocean Experiment (ARSLOE) (TMA) wave spectrum (Bouws et al. 1985). The significant wave height is 2.4 m, peak period of 10 s, mean wave direction of 270° with 20° directional spread, and uniform water depth of 65 m. The surface elevations of the wave field are generated by using the MSEs to propagate the wave components across a polar grid using a finite-differencing approach. The frequency and directional components are summed to create a synthesized multifrequency and multidirectional wave field. From the surface elevations, the radial slope is computed to mimic the radar radial slope data used for assimilation. This is done by taking the finite-differenced derivative of the surface elevation along each range line, where for each range r and azimuth φ, the radial slope ηr is
Snapshots of the surface elevation and radial slope in polar coordinates are shown in Fig. 4.
(a) Synthetically generated water surface elevation using a TMA spectrum with Hs = 2.3 m, along with Tp = 10 s and 20° directional spread. (b) Radial component of the sea surface slope computed from the surface elevations using a finite-difference derivative.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Synthetically generated water surface elevation using a TMA spectrum with Hs = 2.3 m, along with Tp = 10 s and 20° directional spread. (b) Radial component of the sea surface slope computed from the surface elevations using a finite-difference derivative.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Synthetically generated water surface elevation using a TMA spectrum with Hs = 2.3 m, along with Tp = 10 s and 20° directional spread. (b) Radial component of the sea surface slope computed from the surface elevations using a finite-difference derivative.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
To make the synthetic simulation more realistic, wave shadowing and an approximation for noise are added. Shadowing is accomplished through geometric consideration along each range line. If at any range, r, the incidence angle is less than any prior incidence angle, the sea surface at that r is shadowed (Nieto Borge et al. 2004), and this point is set equal to zero. The noise is simulated by empirically evaluating a radar dataset collected in Newport, Oregon. In the domain of the shore-mounted radar, there is a bluff that blocks the radar signal, creating a large region of shadow behind it. The intensity of backscatter noise behind the bluff is a representation of the radar noise alone. The raw intensity noise is normally distributed with a mean of 12.2 and a standard deviation of 3.5. A synthetic time series of intensity noise INt can be generated using a normally distributed random number generator; however, the intensity noise must be converted to radial slope noise ηrN for addition to the radial slope synthetic input. To do this, the radar model (Lyzenga and Walker 2015) is considered. An empirical constant ϵ is derived that, when multiplied by the synthetically generated intensity noise, generates random radial slope noise. In Eq. (11), I(r, φ, t) represents the intensity at a location where there is a wave signal (e.g., not behind bluff), IN0 is the noise mean computed in the region behind the bluff, and angle brackets are the operator for the local mean:
From considering two datasets collected offshore Newport, a best approximation for ϵ is determined to be 0.05. Using geometric shadowing considerations and this derivation of radial slope noise, synthetic datasets are generating using radar heights of 2, 3, 5 and 10 m. With increasing radar height, shadowing becomes less prominent in the domain. An example dataset using a 5 m radar height is shown in Fig. 5a. The shadowed radial slope with addition of simulated noise is shown in Fig. 5b. Transects in range of these two examples are shown in Figs. 6a and 6b, respectively.
(a) Surface elevation with shadowing applied from a 5-m-high radar. (b) Radial slopes with shadowing applied from a 5-m-high radar and synthetic thermal noise added. The outlined annulus shows the data used for assimilation.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Surface elevation with shadowing applied from a 5-m-high radar. (b) Radial slopes with shadowing applied from a 5-m-high radar and synthetic thermal noise added. The outlined annulus shows the data used for assimilation.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Surface elevation with shadowing applied from a 5-m-high radar. (b) Radial slopes with shadowing applied from a 5-m-high radar and synthetic thermal noise added. The outlined annulus shows the data used for assimilation.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Transect of shadowed surface elevation along the dashed line shown in Fig. 5a. The dashed line represents the surface elevations, and the blue represents the illuminated segments, unaffected by the imposed shadowing. (b) Transect of the shadowed and noisy radial slope along the dashed line shown in Fig. 5b. The dashed line represents the radial slope, and the red represents the segments retained after the shadow mask has been applied, where the shadow mask is determined from the surface elevations.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Transect of shadowed surface elevation along the dashed line shown in Fig. 5a. The dashed line represents the surface elevations, and the blue represents the illuminated segments, unaffected by the imposed shadowing. (b) Transect of the shadowed and noisy radial slope along the dashed line shown in Fig. 5b. The dashed line represents the radial slope, and the red represents the segments retained after the shadow mask has been applied, where the shadow mask is determined from the surface elevations.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Transect of shadowed surface elevation along the dashed line shown in Fig. 5a. The dashed line represents the surface elevations, and the blue represents the illuminated segments, unaffected by the imposed shadowing. (b) Transect of the shadowed and noisy radial slope along the dashed line shown in Fig. 5b. The dashed line represents the radial slope, and the red represents the segments retained after the shadow mask has been applied, where the shadow mask is determined from the surface elevations.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
b. Results of assimilating synthetic dataset
The synthetic radial slope data are used for data assimilation in the algorithm. Five trials are presented: four trials of varying radar height with simulated noise, and one trial with no shadowing or noise for a ground control of the best possible reconstruction. Results from these trials are presented in Table 1 and described in more detail here. The forecast waves can be assessed for accuracy by comparing the produced model time series to the original MSE surface elevation data. Six locations throughout the model domain are chosen for forecast assessment, as labeled in Fig. 7. Locations 1, 2, 5, and 6 are within the assimilation annulus, thus these waves are reconstructed from observations but are not predictions. Locations 3 and 4 are outside of the assimilation annulus, thus these time series are predictions. Locations 1–4 lie along the dominant wave direction, while locations 5 and 6 are wider azimuths, thus the angled radar-look direction will compromise the integrity of the surface elevation values.
Summary of results from synthetic trials with varying simulated radar height. Location numbers refer to the point locations in Fig. 7.
Schematic of the synthetic domain. The annulus used for data assimilation is outlined in yellow. The sponge layer is outlined in dashed red. The numbered locations represent points used for time series comparison of forecast waves with the ground-truth surface elevations.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Schematic of the synthetic domain. The annulus used for data assimilation is outlined in yellow. The sponge layer is outlined in dashed red. The numbered locations represent points used for time series comparison of forecast waves with the ground-truth surface elevations.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Schematic of the synthetic domain. The annulus used for data assimilation is outlined in yellow. The sponge layer is outlined in dashed red. The numbered locations represent points used for time series comparison of forecast waves with the ground-truth surface elevations.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Figures 8 and 9 show direct time series comparisons between assimilated and forecast waves for raw synthetic input, and realistic synthetic input (containing shadow and noise), respectively. At each comparison location, the correlation coefficient at zero lag is computed between the forecast and ground-truth time series, and presented in Table 1. When assimilation data are used without noise or shadowing (Fig. 8), the highest correlation is achieved (0.9–0.97) varying by range and azimuthal distance. The higher correlations are achieved within the assimilation domain along the dominant wave direction (locations 1 and 2), with a decrease in correlations at wider azimuths (locations 5 and 6) and finally the weakest correlations outside of the assimilation domain where waves are purely predicted (locations 3 and 4). With synthetic noise and shadowing from a relatively low height (Fig. 9) correlations for reconstructed and predicted waves are quite high, between 0.7 and 0.88. With the lowest tested radar height (2 m) much of the domain is shadowed. Despite, correlations of reconstructed waves are still high (0.79 and 0.9) with a decrease in correlation (0.41) for the forecast waves. This is an unrealistically low radar height; with a more realistic though still fairly low radar (10 m) the correlations are decent (0.74 forecast and 0.91 reconstructed).
(a) Time series comparison between forecast waves and synthetic ground-truth surface elevations with no noise or shadowing. These time series lie in the direction of wave propagation at various range locations 2–4, as seen in the schematic in Fig. 7. (b) Time series comparison between forecast waves and synthetic ground-truth surface elevations with no noise or shadowing. These time series lie along the same range at three azimuthal locations: locations 1, 5, and 6 seen in the schematic in Fig. 7.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Time series comparison between forecast waves and synthetic ground-truth surface elevations with no noise or shadowing. These time series lie in the direction of wave propagation at various range locations 2–4, as seen in the schematic in Fig. 7. (b) Time series comparison between forecast waves and synthetic ground-truth surface elevations with no noise or shadowing. These time series lie along the same range at three azimuthal locations: locations 1, 5, and 6 seen in the schematic in Fig. 7.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Time series comparison between forecast waves and synthetic ground-truth surface elevations with no noise or shadowing. These time series lie in the direction of wave propagation at various range locations 2–4, as seen in the schematic in Fig. 7. (b) Time series comparison between forecast waves and synthetic ground-truth surface elevations with no noise or shadowing. These time series lie along the same range at three azimuthal locations: locations 1, 5, and 6 seen in the schematic in Fig. 7.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
As in Fig. 8, but the comparisons are between forecast waves and synthetic ground-truth surface elevations with simulated radar noise and shadowing from a 5-m-tall radar.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
As in Fig. 8, but the comparisons are between forecast waves and synthetic ground-truth surface elevations with simulated radar noise and shadowing from a 5-m-tall radar.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
As in Fig. 8, but the comparisons are between forecast waves and synthetic ground-truth surface elevations with simulated radar noise and shadowing from a 5-m-tall radar.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
4. Field validation
Three field datasets are used in algorithm testing. Each dataset includes radar image time series and wave profiling buoy(s) located within the radar image domain. Two field campaigns used a shipboard radar (Newport and Monterey, California), and one field campaign used a shore-based radar (Guadalupe, California). The radar used for each field campaign is a commercial, horizontally polarized, marine radar that operates at 9.45 GHz with a peak 25-kW transmitted power. It has a 2.74-m open array antenna rotating at 44 revolutions per minute, such that the ocean surface is imaged every 1.25 s (0.80 Hz). For these analyses, sets of 512 rotations were collected, spanning approximately 11 min each. Further information about the radar system can be found in Catalan et al. (2011).
a. Shipboard radar
1) Newport
For the shipboard field campaigns, the radar was mounted on a small [<50 ft (15 m)] vessel. The first campaign was conducted in December 2015 off Newport aboard the fishing vessel Umatilla II. In situ surface elevation data were recorded by a “TRIAXYS” Directional Wave Buoy, owned and operated by the Pacific Marine Energy Center [formerly Northwest National Marine Renewable Energy Center (NNMREC)] at Oregon State University. The radar and TRIAXYS are shown in Figs. 10a and 10b. A schematic illustrating the positioning of the Umatilla II during the deployment is shown in Fig. 11. Vessel GPS information was collected via a handheld device made by Garmin, Ltd., but proved to be insufficient for image rectification because of low heading accuracy; thus, a postprocessing technique was developed to stabilize image rectification before use in the assimilation algorithm. However, geolocation of the TRIAXYS buoy was not available so only a spectral assessment of performance is possible using this dataset.
(a) Radar mounted on the Umatilla II. (b) TRIAXYS buoy as seen from on board the Umatilla II.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Radar mounted on the Umatilla II. (b) TRIAXYS buoy as seen from on board the Umatilla II.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Radar mounted on the Umatilla II. (b) TRIAXYS buoy as seen from on board the Umatilla II.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Schematic of field data collection offshore Newport.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Schematic of field data collection offshore Newport.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Schematic of field data collection offshore Newport.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
An additional challenge to this campaign was that the relative wind and wave directions were not ideal for wave imaging. Optimal radar imaging requires looking in the upwind direction and, ideally, the wind and waves are in a similar direction. Data from the National Data Buoy Center Station 46050, which is approximately 25 km from the field site, indicate that during the data collection period the wind direction was perpendicular to the 11-s-period waves. In addition, the wave conditions were bimodal; both an 11- and 20-s peak were present as seen in Figs. 12a and 12b. Given that the MSEs give separate solutions per frequency, a bimodal spectrum is a more challenging initial test case than is a narrow-banded spectrum.
(a) Frequency spectrum recorded by the TRIAXYS during field data collection. (b) Frequency–direction spectrum recorded by the TRIAXYS during field data collection.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Frequency spectrum recorded by the TRIAXYS during field data collection. (b) Frequency–direction spectrum recorded by the TRIAXYS during field data collection.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Frequency spectrum recorded by the TRIAXYS during field data collection. (b) Frequency–direction spectrum recorded by the TRIAXYS during field data collection.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
For image stabilization and rectification, a two-part correction scheme was developed that utilizes ocean waves and the signatures of two marker buoys. In the first step of the correction scheme, the vessel position and the wave field were both assumed stationary from image to image (~1.2 s). A vessel heading rotation angle was found that maximized the lagged cross correlation (i.e., wave field alignment) between each sequential pair of radar images. Subtracting the mean from these heading corrections offset a nearly constant azimuthal bias incurred due to azimuthally asymmetric imaging of the propagating wave field. After this first rough stabilization step, the intensity signatures of two moored (corner marker) buoys in the footprint became visible in time-averaged (~12 s) radar images (Fig. 13), which was a good indication of a stable georectification. In the second step of the correction scheme, a similar lagged cross-correlation maximization routine was applied to pairs of these time-averaged radar images with a search window focused on the two buoys. This step provided a second, more slowly varying heading rotation angle, and improved stabilization. The images were then georeferenced using the GPS coordinates of the “corner marker” buoys (known to within 50 m).
(a) Raw intensity collected from the ship-mounted radar aboard the Umatilla II. (b) A zoom-in showing a 10-frame running average of stabilized ship-mounted radar data collected on the Umatilla II. Since the frames have been stabilized, signatures of corner marker buoys become visible in the running average.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Raw intensity collected from the ship-mounted radar aboard the Umatilla II. (b) A zoom-in showing a 10-frame running average of stabilized ship-mounted radar data collected on the Umatilla II. Since the frames have been stabilized, signatures of corner marker buoys become visible in the running average.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Raw intensity collected from the ship-mounted radar aboard the Umatilla II. (b) A zoom-in showing a 10-frame running average of stabilized ship-mounted radar data collected on the Umatilla II. Since the frames have been stabilized, signatures of corner marker buoys become visible in the running average.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
This stabilized radar dataset is used for assimilation, and assimilation results were compared with spectra from the TRIAXYS buoy. The regions of the radar data used for assimilation and the computation domain are shown in Fig. 14. The domain is chosen carefully such that 1) the computation domain contains the location of the TRIAXYS, 2) the assimilation domain contains up-wave data, and 3) the assimilation domain contains wave data that are not too severely influenced by wave shadowing. In the chosen configuration, the TRIAXYS is within the assimilation domain, therefore the modeled waves at this location have been observed by the radar, rather than being a pure forecast.
Radial slope estimated from the ship-mounted radar data collected on the Umatilla II. The domain used for assimilation and prediction is shown on the image, where the black outline represents the entire model domain, the yellow line represents the assimilated data, and the red dashed line shows the sponge layer. The location of the TRIAXYS buoy is indicated by a white diamond.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Radial slope estimated from the ship-mounted radar data collected on the Umatilla II. The domain used for assimilation and prediction is shown on the image, where the black outline represents the entire model domain, the yellow line represents the assimilated data, and the red dashed line shows the sponge layer. The location of the TRIAXYS buoy is indicated by a white diamond.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Radial slope estimated from the ship-mounted radar data collected on the Umatilla II. The domain used for assimilation and prediction is shown on the image, where the black outline represents the entire model domain, the yellow line represents the assimilated data, and the red dashed line shows the sponge layer. The location of the TRIAXYS buoy is indicated by a white diamond.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
From the intensity values, radial slopes are estimated using the radar model of Lyzenga and Walker (2015)—see Eq. (1). Four hundred seconds of radial slope data are used for data assimilation, and solutions to the MSEs are solved for both peak frequencies in the bimodal spectra. A spectral comparison between the reconstructed waves and the in situ buoy observations is shown in Figs. 15a and 15b. In Fig. 15a the significant wave height Hsig is computed from the results of the data assimilation at each location in the domain. The TRIAXYS buoy recorded an Hsig of 1.8 m. The maximum Hsig reached in the assimilation output is 1.6 m, with an average Hsig throughout the domain of 0.77 m. Figure 15b shows a comparison between the TRIAXYS spectrum and the spectrum of the assimilation algorithm output at the estimated location of the TRIAXYS (as indicated by the points shown in the southwest corner of the domain in Fig. 15a). The two peaks in the spectrum are resolved; however, only 44% of the spectral energy is reconstructed by the algorithm, which is determined by comparing the integrals of the buoy spectrum and the reconstructed spectrum. When single-frequency solutions are used, the total reconstructed spectral energy is 20% for the longer peak period (Tp = 20 s) solution and 33% for the shorter peak period (Tp = 11 s) solution. While this result is disappointing, this field effort was helpful in identifying the challenges in implementation with respect to vessel (radar) positioning and daily environmental conditions (wind and wave directions). In addition, the novel postprocessing image rectification method has utility for future radar collections from moving platforms on the ocean.
(a) Significant wave height computed from model results via assimilation of the field dataset. (b) Power spectral density comparison of the TRIAXYS time series with a model time series at the estimated location of the TRIAXYS buoy.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Significant wave height computed from model results via assimilation of the field dataset. (b) Power spectral density comparison of the TRIAXYS time series with a model time series at the estimated location of the TRIAXYS buoy.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Significant wave height computed from model results via assimilation of the field dataset. (b) Power spectral density comparison of the TRIAXYS time series with a model time series at the estimated location of the TRIAXYS buoy.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
2) Monterey
The second shipboard field campaign was conducted in Monterey Bay, where an array of eight wave buoys (“Spotter” buoys; Raghukumar et al. 2019) were temporarily deployed in ~40 m water depth, the locations of which can be seen in Fig. 16a. Because of their small size (38-cm diameter; <5.5 kg) and ease of deployment (Raghukumar et al. 2019), the Spotter buoys were a convenient approach for this type of short-term field study. These buoys use GPS signals to record horizontal displacements, which are then related to vertical displacements using linear wave theory. The directional moments of the wave spectrum are determined from the cross-spectra of these signals and used to reproduce the full wave directional spectrum (Raghukumar et al. 2019). The Research Vessel Shana Rae, on which the radar was mounted, is shown in Fig. 16b.
(a) Array of Spotter buoys, with an image of a buoy shown on the left; (b) research vessel with mounted X-band radar.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Array of Spotter buoys, with an image of a buoy shown on the left; (b) research vessel with mounted X-band radar.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Array of Spotter buoys, with an image of a buoy shown on the left; (b) research vessel with mounted X-band radar.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
For this shipboard field campaign, the vessel GPS was upgraded to a Hemisphere GPS compass. Using the GPS compass, the location of the vessel was recorded with enough accuracy for spatial rectification. An additional stabilization scheme was implemented on the radar images to remove heading jitter in which each frame was cross correlated to a single frame, utilizing stationary shoreline features to find a rotational offset using image-to-image cross correlations. Once the offsets were applied, a more stable dataset was generated with high enough rectification accuracy to locate the Spotter buoys within each image. The radial slopes were then estimated from the georectified intensity dataset. Sample intensity and radial slope data are shown in Figs. 17a and 17b.
(a) Georectified radar backscatter intensity collected in Monterey Bay, with Spotter buoy locations shown as cyan dots; (b) radial slopes computed from the intensity shown in (a), with buoy locations again shown as cyan dots.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Georectified radar backscatter intensity collected in Monterey Bay, with Spotter buoy locations shown as cyan dots; (b) radial slopes computed from the intensity shown in (a), with buoy locations again shown as cyan dots.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Georectified radar backscatter intensity collected in Monterey Bay, with Spotter buoy locations shown as cyan dots; (b) radial slopes computed from the intensity shown in (a), with buoy locations again shown as cyan dots.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
An initial assessment of the radar-estimated radial slopes can be made by comparing to the buoy measurements. From measured water surface elevation, a spatial slope value ηx can be estimated using the relationship between the time derivative of the measured surface elevation and wave celerity computed from the peak period and wavelength, where propagation direction is from the buoy toward the radar:
The comparison between radial slope estimated from radar, and spatial slope estimated from buoy is shown in Fig. 18. Theoretical wave shadowing is added to the buoy slopes for a more representative comparison. The radar-derived radial slope is underestimated by approximately 70%. This underestimate is thought to be due to two main factors: 1) the radar height, which is a key variable in the radar imaging model, is highly variable due to the ship’s motion over the waves, and 2) the derivation of the radar imaging model predicts a relationship between the intensity and range, which this dataset does not show. Specifically addressing item 2, the radar imaging model expects that the intensity has an r−7/4 rolloff, but the present dataset deviates from this, potentially due to the relatively low radar height on board the ship. Because the underestimates in radial slope, the assimilation algorithm also produces an underestimate in water surface elevation. Future work would need to better account for imaging characteristics from small vessels with low antenna heights.
Radar-derived radial slope compared with wave slope estimate from in situ surface elevations. A shadow mask is applied in which segments of waves not visible to the radar are removed.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Radar-derived radial slope compared with wave slope estimate from in situ surface elevations. A shadow mask is applied in which segments of waves not visible to the radar are removed.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Radar-derived radial slope compared with wave slope estimate from in situ surface elevations. A shadow mask is applied in which segments of waves not visible to the radar are removed.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
b. Land-based radar
Guadalupe
The land-based field campaign used a marine radar mounted on a 30-m tower approximately 0.5 km inland from the shoreline of an open beach, again with Spotter buoys in the image footprint. This dataset was collected as part of a larger experiment studying inner shelf processes in Guadalupe, near Vandenberg Air Force Base (Lerczak et al. 2019). The major advantage of this shore-based dataset is that the radar platform was stationary and the antenna elevation was significantly larger than the wave heights. Images showing raw intensity and computed radial slope are shown in Figs. 19a and 19b, respectively. The Spotter buoy locations are indicated for the buoys within the radar domain (Spotters 20 and 28).
(a) Raw radar intensity, with Spotter buoys 20 and 28 indicated; (b) radial slopes computed from (a).
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Raw radar intensity, with Spotter buoys 20 and 28 indicated; (b) radial slopes computed from (a).
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Raw radar intensity, with Spotter buoys 20 and 28 indicated; (b) radial slopes computed from (a).
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
Two limitations exist in this dataset: 1) the wave conditions are nonlinear (Stokes second order at Spotter 28 and nearly cnoidal at Spotter 20) thus violating the assumptions of the linear mild slope equations, and 2) the intensity shows the expected range rolloff at midranges for the imaging model, but not at the close and far ranges that overlap with the buoy locations. The ramifications of this are 1) the phase speed is underestimated in the reconstruction and 2) the slopes are not correctly estimated at the buoy locations.
The region of the radar imagery used for assimilation is outlined in Fig. 20a with the Spotter 20 location indicated as a white dot. The radial slopes used for assimilation are shown in Fig. 20b, and a time series comparison is shown in Fig. 20c. The still water at the beginning of the reconstructed time series is due to the fact that the reconstructed waves propagate from the outer edge of the domain inwards. It is seen that the first 4–5 waves are predicted well; however, after that the phases deviate to a nearly 180° phase shift. One possible explanation for this is that the model is approximating the water surface elevations using a single-frequency solution to the MSEs. In the buoy data, there appears to be a wave beat caused by two closely spaced frequencies, which the model would not be able to recreate. Despite this fact, the first several waves are predicted well. In a realistic implementation of the algorithm, a recurring series of assimilations would be used to achieve a longer record of predicted waves. Regarding model timing, the computation time when using a single frequency is approximately 100 s for an assimilation period of 400 s and a prediction period of 400 s. This means that, in theory, approximately 300 s of predict-ahead time is possible.
(a) Snapshot of shore-based radar intensity with assimilation domain outlined in white, and Spotter 20 location indicated as a white dot; (b) radial slope data used for assimilation; (c) time series comparison between bandpass-filtered buoy heave and reconstructed surface elevations.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Snapshot of shore-based radar intensity with assimilation domain outlined in white, and Spotter 20 location indicated as a white dot; (b) radial slope data used for assimilation; (c) time series comparison between bandpass-filtered buoy heave and reconstructed surface elevations.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
(a) Snapshot of shore-based radar intensity with assimilation domain outlined in white, and Spotter 20 location indicated as a white dot; (b) radial slope data used for assimilation; (c) time series comparison between bandpass-filtered buoy heave and reconstructed surface elevations.
Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0127.1
5. Summary
The presented work outlines the structure and initial testing of a recently developed wave-by-wave forecasting algorithm. The system assimilates radial slope data estimated from X-band marine radar imagery. The radial slope is the component of the surface slope in the radar look direction and is computed using a recently derived radar model (Lyzenga and Walker 2015). A data assimilation scheme built upon the linear mild slope equations is applied to an annulus of radial slope data at a distance from the radar. The algorithm minimizes a cost function to determine the best-fit wave source function to reconstruct the observed waves. Waves generated by the source functions are propagated from the outer edge of the assimilation annulus inward across the polar computation domain toward the origin. The final wave field is a summation of the reconstructed wave components computed from each of the source functions for specific frequencies.
The first phase of validating the algorithm uses a synthetic radial slope dataset determined from a water surface elevation simulation created using the MSEs. The synthetic dataset incorporates wave shadowing and synthetic noise to simulate the expected shadowing and noise present in radar imagery. The results of assimilating synthetic radial slopes are very good. With a simulated radar antenna height of 5 m, the algorithm is able to predict waves with a correlation coefficient between 0.84 and 0.95 with the input data, varying by range and azimuthal location in the domain. The strongest correlations occur along a transect in the direction of wave travel, with decrease in correlation at outer azimuths of the domain due to the effects of obliquely observing the waves at these wider angles. It was also demonstrated that the algorithm, as formulated, is able to produce wave-by-wave forecasts in real time (considering computation time required by the assimilation algorithm only).
Field testing of the algorithm was less successful; however, challenges to field implantation were identified and future work can attempt to address these. For field testing, two shipboard radar datasets and one land-based radar dataset were used. Specific challenges with field implementation were as follows: 1) radar image rectification from a moving platform, 2) assumptions of the rate of radar intensity falloff with range and antenna height to wave height ratio as assumed in the radar imaging model of Lyzenga and Walker (2015), and 3) the validity of a linear wave model (MSEs) in an energetic sea state. Use of the shore-based tower effectively mitigated items 1 and 2, such that a forecast 4–5 waves into the future was possible. However, future versions of the algorithm should consider nonlinear corrections to the MSEs as well as radar imaging models of increased sophistication.
Acknowledgments
We thank Dean Jordan (SRI International); Pieter Smit and Tim Janssen (Sofar); Randall Pitman (Oregon State University); Sam McWilliams, Grace Chang, Frank Williams, Craig Jones, and Kaus Raghukumar (Integral Consulting); and the crews of the Umatilla II and Shana Rae. This work was funded through U.S. Department of Energy (Award DE-EE0006789) with additional funding for author Simpson from the Link Foundation Ocean Engineering and Instrumentation Fellowship.
REFERENCES
Abusedra, L., and M. R. Belmont, 2011: Prediction diagrams for deterministic sea wave prediction and the introduction of the data extension prediction method. Int. J. Shipbuild. Prog., 58, 59–81, http://doi.org/10.3233/ISP-2011-0069.
Aragh, S., and O. Nwogu, 2008: Variation assimilating of synthetic radar data into a pseudo-spectral wave model. J. Coastal Res., 10052, 235–244, https://doi.org/10.2112/1551-5036-52.sp1.235.
Beji, S., and K. Nadaoka, 1997: A time-dependent nonlinear mild slope equation for water waves. Proc. Roy. Soc. London, 453A, 319–332, https://doi.org/10.1098/rspa.1997.0018.
Belmont, M. R., J. Christmas, J. Dannenberg, T. Hilmer, J. Duncan, J. M. Duncan, and B. Ferrier, 2014: An examination of the feasibility of linear deterministic sea wave prediction in multidirectional seas using wave profiling radar: Theory, simulation, and sea trials. J. Atmos. Oceanic Technol., 31, 1601–1614, https://doi.org/10.1175/JTECH-D-13-00170.1.
Bennett, A. F., 2002: Inverse Modeling of the Ocean and Atmosphere. Cambridge University Press, 256 pp.
Berkhoff, J. C. W., 1972: Computation of combined refraction-diffraction. Proc. 13th Coastal Eng. Conf., Vancouver, Canada, ASCE, 471–490, https://icce-ojs-tamu.tdl.org/icce/index.php/icce/article/view/2767.
Blondel-Couprie, E., G. Ducrozet, F. Bonnefoy, and P. Ferrant, 2008: Deterministic reconstruction and prediction of a non-linear wave field using probe data. 27th Int. Conf. on Offshore Mechanics and Arctic Engineering, Estoril, Portugal, American Society of Mechanical Engineers, 379–386, https://doi.org/10.1115/OMAE2008-57558.
Blondel-Couprie, E., F. Bonnefoy, and P. Ferrant, 2010: Deterministic non-linear wave prediction using probe data. Ocean Eng., 37, 913–926, https://doi.org/10.1016/j.oceaneng.2010.03.002.
Blondel-Couprie, E., F. Bonnefoy, and P. Ferrant, 2013: Experimental validation of non-linear deterministic prediction schemes for long-crested waves. Ocean Eng., 58, 284–292, https://doi.org/10.1016/j.oceaneng.2012.10.014.
Bouws, E., H. Günther, W. Rosenthal, and C. L. Vincent, 1985: Similarity of the wind wave spectrum in finite depth water: 1. Spectral form. J. Geophys. Res., 90, 975–986, https://doi.org/10.1029/JC090iC01p00975.
Brekken, T. K., 2011: On model predictive control for a point absorber wave energy converter. Proc. Trondheim PowerTech, Trondheim, Norway, IEEE, 1–8, https://doi.org/10.1109/PTC.2011.6019367.
Catalan, P. A., M. C. Haller, R. A. Holman, and W. J. Plant, 2011: Optical and microwave detection of wave breaking in the surf zone. IEEE Trans. Geosci. Remote Sens., 49, 1879–1893, https://doi.org/10.1109/TGRS.2010.2095864.
Dankert, H., and W. Rosenthal, 2004: Ocean surface determination from X-band radar-image sequences. J. Geophys. Res., 109, C04016, https://doi.org/10.1029/2003JC002130.
Edgar, D. R., J. M. K. Horwood, R. Thurley, and M. R. Belmont, 2000: The effects of parameters on the maximum prediction time possible in short term forecasting of the sea surface shape. Int. Shipbuild. Prog., 47, 287–301.
Fusco, F., and J. V. Ringwood, 2010: Short-term wave forecasting with AR models in real-time optimal control of wave energy converters. Int. Symp. on Industrial Electronics, Bari, Italy, IEEE, 2475–2480, https://doi.org/10.1109/ISIE.2010.5637714.
Fusco, F., and J. V. Ringwood, 2012: A study of the prediction requirements in real-time control of wave energy converters. IEEE Trans. Sustain. Energy, 3, 176–184, https://doi.org/10.1109/TSTE.2011.2170226.
Gillespie, A., 2015: Wave-by-wave forecasting of sea surface elevation for WEC applications utilizing NARX neural networks. M.S. thesis, Dept. of Civil Engineering, Oregon State University, 78 pp.
Halliday, J. R., D. G. Dorrell, and A. R. Wood, 2011: An application of the fast Fourier transform to the short-term prediction of sea wave behaviour. Renewable Energy, 36, 1685–1692, https://doi.org/10.1016/j.renene.2010.11.035.
Hesam, E. S. J., B. Ling, and B. A. Batten, 2014: Use of artificial neural networks for real-time prediction of heave displacement in ocean buoys. Third Int. Conf. on Renewable Energy Research and Applications, Milwaukee, WI, IEEE, 907–912, https://doi.org/10.1109/ICRERA.2014.7016517.
Ijima, T., T. Takahashi, and H. Sasaki, 1964: Application of radars to wave observations. Coastal Eng. Proc., 30, 10–22.
Korde, U. A., 2014: On a near-optimal control approach for a wave energy converter in irregular waves. Appl. Ocean Res., 46, 79–93, https://doi.org/10.1016/j.apor.2014.02.002.
Lerczak, J., and Coauthors, 2019: Untangling a web of interactions where surf meets coastal ocean. Eos, Trans. Amer. Geophys. Union, 100, https://doi.org/10.1029/2019eo122141.
Li, G., G. Weiss, M. Mueller, S. Townley, and M. R. Belmont, 2012: Wave energy converter control by wave prediction and dynamic programming. Renewable Energy, 48, 392–403, https://doi.org/10.1016/j.renene.2012.05.003.
Ling, B., and B. Batten, 2015: Real time estimation and prediction of wave excitation forces on a heaving body. 34th Int. Conf. on Ocean, Offshore and Arctic Engineering, Newfoundland, Canada, American Society of Mechanical Engineers, OMAE2015-41087, https://doi.org/10.1115/OMAE2015-41087.
Liu, X., W. Huang, and E. W. Gill, 2015: Shadowing-analysis-based wave height measurement from ship-borne X-band nautical radar images. OCEANS 2015—Genova, Genoa, Italy, IEEE, https://doi.org/10.1109/OCEANS-Genova.2015.7271581.
Lyzenga, D. R., and D. T. Walker, 2015: A simple model for marine radar images of the ocean surface. IEEE Geosci. Remote Sens. Lett., 12, 2389–2392, https://doi.org/10.1109/LGRS.2015.2478390.
Morris, E. L., H. K. Zienkiewicz, and M. R. Belmont, 1998: Short term forecasting of the sea surface shape. Int. Shipbuild. Prog., 45, 383–400.
Naaijen, P., and A. P. Wijaya, 2014: Phase resolved wave prediction from synthetic radar images. Proc. 33rd Int. Conf. on Ocean, Offshore and Arctic Engineering, San Francisco, CA, American Society of Mechanical Engineers, OMAE2014-23470, https://asmedigitalcollection.asme.org/OMAE/proceedings-abstract/OMAE2014/45509/V08AT06A045/279048.
Nieto Borge, J. C., K. Reichert, and J. Dittmer, 1999: Use of nautical radar as a wave monitoring instrument. Coastal Eng., 37, 331–342, https://doi.org/10.1016/S0378-3839(99)00032-0.
Nieto Borge, J. C., G. Rodríguez Rodríguez, K. Hessner, and P. I. González, 2004: Inversion of marine radar images for surface wave analysis. J. Atmos. Oceanic Technol., 21, 1291–1300, https://doi.org/10.1175/1520-0426(2004)021<1291:IOMRIF>2.0.CO;2.
Nouguier, F., S. T. Grilli, and C. A. Guérin, 2014: Nonlinear ocean wave reconstruction algorithms based on simulated spatiotemporal data acquired by a flash lidar camera. IEEE Trans. Geosci. Remote Sens., 52, 1761–1771, https://doi.org/10.1109/TGRS.2013.2254494.
O’Sullivan, A. C. M., and G. Lightbody, 2017: Co-design of a wave energy converter using constrained predictive control. Renewable Energy, 102, 142–156, https://doi.org/10.1016/j.renene.2016.10.034.
Prislin, I., J. Zhang, and R. J. Seymour, 1997: Deterministic decomposition of deep water short-crested irregular gravity waves. J. Geophys. Res., 102, 12 677–12 688, https://doi.org/10.1029/97JC00791.
Qi, Y., W. Xiao, and D. K. P. Yue, 2016: Phase-resolved wave field simulation calibration of sea surface reconstruction using noncoherent marine radar. J. Atmos. Oceanic Technol., 33, 1135–1149, https://doi.org/10.1175/JTECH-D-15-0130.1.
Qi, Y., G. Wu, Y. Liu, M. H. Kim, and D. K. P. Yue, 2018a: Nonlinear phase-resolved reconstruction of irregular water waves. J. Fluid Mech., 838, 544–572, https://doi.org/10.1017/jfm.2017.904.
Qi, Y., G. Wu, Y. Liu, and D. K. P. Yue, 2018b: Predictable zone for phase-resolved reconstruction and forecast of irregular waves. Wave Motion, 77, 195–213, https://doi.org/10.1016/j.wavemoti.2017.12.001.
Raghukumar, K., G. Chang, F. Spada, C. Jones, T. Janssen, and A. Gans, 2019: Performance characteristics of “Spotter,” a newly developed real-time wave measurement buoy. J. Atmos. Oceanic Technol., 36, 1127–1141, https://doi.org/10.1175/JTECH-D-18-0151.1.
Schoen, M. P., J. Hals, and T. Moan, 2011: Wave prediction and robust control of heaving wave energy devices for irregular waves. IEEE Trans. Energy Convers., 26, 627–638, https://doi.org/10.1109/TEC.2010.2101075.
Simanesew, A., K. Trulsen, H. E. Krogstad, and J. C. Nieto Borge, 2017: Surface wave predictions in weakly nonlinear directional seas. Appl. Ocean Res., 65, 79–89, https://doi.org/10.1016/j.apor.2017.03.009.
Støle-Hentschel, S., J. Seemann, J. C. Nieto Borge, and K. Trulsen, 2018: Consistency between sea surface reconstructions from nautical X-band radar Doppler and amplitude measurements. J. Atmos. Oceanic Technol., 35, 1201–1220, https://doi.org/10.1175/JTECH-D-17-0145.1.
Valenzuela, G. R., 1978: Theories for the interaction of electromagnetic and oceanic waves—A review. Bound.-Layer Meteor., 13, 61–85, https://doi.org/10.1007/BF00913863.
Wijaya, P., P. Naaijen, Andonowati, and E. van Groesen, 2015: Reconstruction and future prediction of the sea surface from radar observations. Ocean Eng., 106, 261–270, https://doi.org/10.1016/j.oceaneng.2015.07.009.
Wright, F. F., 1965: Wave observations by shipboard radar. Ocean Sci. Ocean Eng., 1, 506–514.
Wu, G., 2004: Direct simulation and deterministic prediction of large-scale nonlinear ocean wave-field. Ph.D. thesis, Massachusetts Institute of Technology, 258 pp.
Young, I. R., W. Rosenthal, and F. Ziemer, 1985: A three-dimensional analysis of marine radar images for the determination of ocean wave directionality and surface currents. J. Geophys. Res., 90, 1049–1059, https://doi.org/10.1029/JC090iC01p01049.
Zhang, J., J. Yang, J. Wen, I. Prislin, and K. Hong, 1999: Deterministic wave model for short-crested ocean waves: Part I. Theory and numerical scheme. Appl. Ocean Res., 21, 167–188, https://doi.org/10.1016/S0141-1187(99)00011-5.
Ziemer, F., and H. Gunther, 1994: A system to monitor ocean wave fields. Proc. Second Int. Conf. on Air–Sea Interaction and Meteorology and Oceanography of the Coastal Zone, Lisbon, Portugal, Amer. Meteor. Soc., 18–19.