1. Introduction
Under second-order stationary and homogeneous conditions, an ocean wave field
Data from heave–pitch–roll measurement systems [heave–slope (HS)] will be central in the following analysis. Such systems provide data with few restrictive assumptions about the wave field, apart from homogeneity and stationarity. The standard output from the system is collocated time series of surface elevation and surface slopes in two orthogonal directions. Data from a spatially compact array may be interpolated into an HS setting and analyzed as triplet data.
It is well known, since the pioneering work of Longuet-Higgins (1976), that nonlinear energy transfer tends to move wave energy along directions ±35° away from a narrow peak in the wavenumber vector spectrum. The result has been verified using the full solution to nonlinear interaction equations (Banner and Young 1994) and nonlinear Schrödinger equations based on narrowband Gaussian initial spectra (Dysthe et al. 2003). In a recent study, Toffoli et al. (2010) have investigated the temporal evolution of directional wave spectra based on the potential Euler equations, applying a higher-order spectral method (Dommermuth and Yue 1987) that is free of bandwidth constraints and without any external forcing, such as wind input or breaking dissipation. For a fairly narrowbanded spectrum, it is observed that the energy of short waves redistributes in agreement with Longuet-Higgins (1976). However, for broadbanded waves the energy redistribution direction is shifted, to some extent, away from directions ±35° depending on the shapes of the initial directional distributions. Moreover, Morland (1996) has suggested that energy transfer from the wind to the waves as a result of inviscid critical layer mechanisms may give rise to double-peaked distributions for high frequencies.
Despite all these indications from the theory and observations of bimodal distributions in the short-wave region, uncertainties concerning peak-splitting tendency of some of the estimation methods make definite conclusions about the shape difficult. Estimation of ocean wave spectra mostly applies data-adaptive methods, and the resulting interpretations depend on analysis techniques and their underlying assumptions and principles. Comparisons of several directional analysis methods based on directional wave basin experiments obtained from three different measuring systems were reported by Benoit and Teisson (1994). They observed that the maximum likelihood (ML) method (see, e.g., Isobe et al. 1984; Krogstad 1988) provides broader directional distributions compared to the target, and that it performs poorly for sea states with narrow-peaked or bimodal distributions. On the other hand, the iterative maximum likelihood (IML) method (Oltman-Shay and Guza 1984) provides well-resolved directional distributions, but it is less reliable for estimates obtained from wave probe array when compared to Shannon maximum entropy (SME) method (Hashimoto and Kobune 1986) or Bayesian directional methods (Benoit and Teisson 1994). However, the internal iteration method used in the SME method to determine the Lagrangian multipliers experiences convergence problems if estimated Fourier coefficients are near the edge of the infeasible set (Krogstad 2012). An alternative estimation technique based on the Burg maximum entropy (BME) principle was introduced by Lygre and Krogstad (1986). When applied to certain theoretical distributions, the differences between the BME and SME algorithms range from the ability to match both broad and narrow peaks to a strong tendency for peak splitting (Krogstad 2012). With Fourier coefficients of certain theoretical unimodal distributions as input [e.g., the
Nevertheless, several recent studies indicate that the tendency to split spectral peaks may not be a problem for the BME method when applied to real wave data. Comparisons of HF radar spectra with the BME spectra estimated from Fourier coefficients determined from the HF radar spectra and with directional Waverider BME estimates are reported in Kahma et al. (2005, p. 313). The shapes of the spectra estimated from these three cases were consistent, including the bimodal features. During the study of frequency–wavenumber spectra of young wind waves, Leckler et al. (2015) compared a BME-processed spectrum from HS time series computed from 3D surface data with the original spectrum. The observed bimodal spectra in both BME and the original estimates are in reasonable agreement, confirming that the bimodality is real and consistent with Ewans (1998) observations. Moreover, since the spectra are measured by a stereo video system, any doubts concerning spurious peaks as a result of the analysis technique may be ruled out. Recently, Simanesew et al. (2016) reported estimates of ocean wave spectra from four different directional analysis methods. In many of the spectra reported, they found a striking agreement between the BME estimates and both the IML and SME estimates, suggesting that the bimodality in the BME distributions indeed is real.
Young (2010) examined both wavenumber and frequency spectra of finite-depth wind-generated waves using the wavelet directional method (Donelan et al. 1996). He observed bimodality in the
Our investigation, based on Ekofisk laser array data, and comprehensive with respect to sea states, shows that the BME estimates are generally consistent with the IML and SME estimates. In particular, for the bimodal sea states, we find strong similarity in bimodal features between the BME and IML distributions. These two methods simultaneously display the bimodal features in the whole range of frequencies, while the SME directional distributions show these bimodal features for relatively high frequencies.
2. Theory review
a. Wave spectra
























Slices through the spectrum showing
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1















b. Directional distributions



















1) Some properties of ME distributions








2) Unimodal and bimodal regions for symmetric ME distributions
In the following we shall limit ourselves further to the important class of symmetric distributions with respect to some direction. To ensure that all Fourier coefficients
(i) Burg ME distributions





Equation (23) is identical to Eq. (2.129) in Kahma et al. (2005). From Eq. (20) we have
Demonstration of stationary points for symmetric Burg ME distributions. Distribution with (a) only the first two solutions valid, (b) all four solutions valid with the last pair being maxima, and (c) all four solutions valid with the last pair being minima.
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
Figure 3 gives an overview of
Extended graph of
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
Infeasible and feasible domains. Burg ME unimodal region and some distributions within and outside of the unimodal region. Poisson distribution (green) is completely within the Burg ME unimodal region,
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
The green curve is the relationship for the unimodal Poisson distribution, whereas the magenta curve is the
With the von Mises Fourier coefficients as input, the Burg ME distribution shows two peaks for distributions with
(left) Poisson and (right)
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
(ii) Shannon ME distributions


























Relation between
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
The von Mises distribution is completely within the Shannon ME unimodal region. It is interesting to observe, however, that both the Poisson and the
c. Estimation of directional distributions
There exists a multitude of methods for estimating directional distributions, which in turn produce pages of intercomparison plots (Donelan et al. 2015). These algorithms take time series input and follow it up with multivariate spectral analysis. The spectral terms often relate to Fourier coefficients or other parameters from


The ML method was introduced by Capon (1969) in signal processing and was later modified in the context of ocean wave spectra (see, e.g., Davis and Regier 1977; Isobe et al. 1984; Krogstad 1988; Krogstad et al. 1988). This method has a known deficiency in that it tends to fail to reproduce the cross-spectra when
3. Ekofisk laser array measurements
The Ekofisk laser array is a research system mounted at the Ekofisk oil field in the North Sea since February 2003. It consists of four down-looking Optech lasers mounted on a bridge connecting the Kilo and Bravo platforms situated about 1 n mi (1 n mi = 1.852 km) northwest of the main Ekofisk complex. The lasers are placed at the four corners of a 2.6 m × 2.6 m square structure located approximately 20 m above the mean surface of a 70-m-deep sea. Raw data are collected continuously at 5 Hz with 1-mm accuracy. The data stream is stored in 20-min files with 1-min overlap.
The system was designed by the Norwegian Meteorological Institute [Meteorologisk Institutt (MET Norway)] in cooperation with the University of Miami, and the data collection is carried out under the operational responsibility of ConocoPhillips Inc. Extensive testing of the array has been performed, some of which is published (Krogstad et al. 2006; Krogstad and Trulsen 2010) and some documented in internal reports (e.g., Machado and Krogstad 2004) that are available on request from ConocoPhillips Inc.





Scatter diagram for
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
Even if the full array consists of four lasers, the results are virtually unchanged when using only three lasers because of the array’s compact size relative to the size of the waves. This is actually convenient, since we are going to simulate an HS system by interpolating elevation and slope in the center of the triangle formed by the three remaining sensors and their elevation recordings. Both the reduced spatial array consisting of three lasers and the interpolated HS system have their usual limitations. The former suffers from spatial aliasing, since the shortest leg in the array is 2.6m, resulting in a limiting wavelength
4. Data analysis
The ML and IML analyses are performed on spatial-array data, whereas the Burg and Shannon ME methods are applied to interpolated HS triplet data. As far as the Ekofisk compact array is concerned, it does not matter whether the ML/IML analysis is performed on spatial-array data or on the corresponding HS triplet data because the results are virtually identical. Both the ML and IML methods provide estimates of directional distributions from which we may determine the four leading Fourier coefficients. However, the HS system provides these Fourier coefficients directly and independent of any additional assumptions beyond the existence of a 3D spectrum. Apart from applying the same data series, the ML/IML and HS algorithms have no direct connections. Figure 8 shows plots of ML/IML directional spread against the HS triplet directional spread at four different frequencies for 1000 records. It is observed that the directional spread at the peak varies within the range 20°–65°. The IML spread is in fairly good agreement with the HS spread, while the ML spread shows a significant positive bias relative to HS.
Ekofisk
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
Figure 9 shows color plots of the directional spectrum and the directional distribution of a typical unimodal sea state based on various analysis methods. For the directional spectra, shown in the upper row, there is only a minor difference among the methods, all showing clear unimodality, which is also reflected in the directional distributions shown in the lower row. Multimodality is a persistent feature in the Ekofisk data, where a large fraction of the directional spectra have evidence of bimodality for distributions above the spectral peak. Figure 10 shows occurrences of bimodal distributions for a set of records with
Unimodal directional wave spectrum measured at 1900 UTC 23 Sep 2007,
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
Occurrences of bimodal directional distributions in the Ekofisk laser array data from a set of records with
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
Bimodal directional wave spectrum measured at 1200 UTC 12 Oct 2004,
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
In the following section, we analyze selected data from storms that occurred during the period of 18–23 January 2007. During this period the local wind condition was unstable with the wind speed fluctuating between 1 and 25 m s−1, and the significant wave height reaching up to 9 m. Figure 12 shows the development of the bimodal directional distribution from initially a unimodal sea state, observed in the evening of 22 January 2007. Estimates of directional spectra from BME are shown in the upper row with estimates of the directional distributions from various methods in the lower three rows. The IML estimates are in the upper-middle row, the BME estimates are in the lower-middle row, and the SME estimates are in the last row. Our results are qualitatively similar to Wang and Hwang (2001). At 1800 UTC, the wave energy propagates along the mean wave direction; however, as the wave field develops, the short waves are aligned along two directions oblique to the mean wave direction. Such a phenomenon in the ocean wave spectra was already observed by Wang and Hwang (2001) and attributed to nonlinear wave–wave interactions (Banner and Young 1994; Ewans 1998; Longuet-Higgins 1976; Dysthe et al. 2003; Toffoli et al. 2010). From the estimates of directional spectra in the upper row it is evident that, in this case, the bimodality does not seem to be associated with either the presence of two distinct wave systems or cases of new wave development alongside old ones. A 10% increase in steepness is observed between the spectra at 1800 and 2200 UTC.
Development of bimodal directional distributions. Estimates are from a storm measured in the evening of 22 January 2007, based on (upper middle) IML, (lower middle) BME, and (bottom) SME methods. (top) Estimates of directional spectra from the BME method. Distributions are normalized by the maximum value, and the color scale is logarithmic.
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
In many of the spectra from the Ekofisk oil field, the ML algorithm often provides broader peaks, while the BME provides a narrow distribution compensated by a wide bottom, in agreement with previous observations. The SME distributions, while being nonsmeared, look more like the ML distributions around the spectral peak. However, when there is bimodality, the SME distributions shows more detailed bimodal features for moderate to high frequencies. On the other hand, IML distributions are quite similar to the BME distributions and lie somewhere in between the Shannon and Burg ME distributions.
To further confirm that the IML and BME methods provide virtually identical directional features, we consider nearly fully developed bimodal distributions from a number of neighboring records. The scatterplots in Fig. 13 show locations of peaks for bimodal directional distributions from 31 records. The peaks are computed as the local maxima of smoothed distributions. When there are several peaks, which is often the case for ML and IML distributions, we simply choose the two dominant ones. The overall shape of the distributions is in agreement, with the main difference being the existence of bimodality in the low-frequency range for the ME estimates and not for the ML and IML estimates. We have no explanation for this discrepancy; however, numerical evidence suggests that the distributions widen below the spectral peak as a result of nonlinear interactions (Simanesew et al. 2016). Near the spectral peak, the two bimodal arms coincide with the two dashed lines obtained by linear approximation of a parametric curve with respect to ω and θ, namely, the quartet resonance of the “figure of 8” for gravity waves derived by Phillips (1960). The relevance of these dashed lines for the directional redistribution of energy is captured by the two-dimensional cubic Schrödinger equation as discussed by Dysthe et al. (2003) in the
Locations of peaks for bimodal directional distributions from measurements recorded between 0840 and 1840 UTC 12 Oct 2004 based on the ML, IML, BME, and SME methods. Asymptotic orientation of quartet resonance of the “figure of 8” for gravity waves in the
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
Figure 14 shows scatterplots of angular separations between the two peaks for frequencies above the spectral peak, in the range
As in Fig. 13, but angular separations of the two peaks as a function of dimensionless frequency. Continuous line is Ewans’s symmetric double Gaussian parameterization.
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
As in Fig. 13, but comparisons are between the BME peak separation, and the ML, IML, and SME angular separations; a 1-to-1 regression (solid line).
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
Examples of the relationship between
Scatterplot of
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
Moreover, a stringent criterion has been imposed on the data based on the directions of
As in Fig. 16, but for the limits of the axes.
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
In summary, the data analysis and the numerical experiments in section 2 suggest that there is peak-splitting tendency in both the Burg and Shannon maximum entropy algorithms for certain Fourier coefficients. However, many of the spectra, based on data from the Ekofisk array, show evidence of bimodality for distributions above the spectral peak. Since the bimodality is consistently revealed in all four estimation methods, it is less likely to be associated with spurious peaks generated by the analysis methods.
In addition, estimates of the directional spread obtained from IML and HS algorithms are virtually identical as shown in Fig. 8. For the IML and BME distributions, the bimodal features are also nearly identical. These two methods are dissimilar, yet they produce quite similar distributions. This similarity grants a certain confidence in the results. Thus, it may be recommendable to include both methods in the estimation procedure when additional information is needed.
5. Numerical simulations
We have employed the Dysthe equation, also known as the modified nonlinear Schrödinger (MNLS) equation, for the temporal evolution of a directional wave field (Dysthe 1979; Trulsen and Dysthe 1996; Trulsen et al. 2000). We aim to study the evolution of the wave spectrum, similar to the work of Dysthe et al. (2003), with a focus on the development of bimodal distributions. The MNLS equation accounts for nonlinear interactions that can lead to frequency-dependent directional distributions (Simanesew et al. 2016).
We employ the numerical method of Lo and Mei (1985, 1987) with periodic boundary conditions in both the x and y directions. Evolution in the temporal direction is achieved with a splitting scheme in which the linear part is integrated exactly in Fourier space and the nonlinear part is integrated by finite differences. A spatial grid with
All the simulations presented here are initialized by a two-dimensional Gaussian spectrum, as given in Eq. (3) of Dysthe et al. (2003), with given values of spectral bandwidths in the x and y directions denoted by
Numerical simulation with MNLS: Evolution of directional spectra from initially Gaussian spectra with initial spectral widths
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
The corresponding directional distributions have been extracted from the spectra above and are shown in Fig. 19. In addition to generating the bimodal directional distributions seen above the spectral peak, nonlinear interactions are also responsible for the widening of the directional distributions below the spectral peak and hence the increase in the directional spread toward the low frequencies. This is observed as a gradual widening of the directional distributions from minimum near the spectral peak to maximum both at lower- and upper-frequency tails. While the simulation directly produces the wavenumber spectrum
Numerical simulation with MNLS: Evolution of directional distribution from initially Gaussian spectra with initial spectral widths
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
The maximum entropy methods are also applied here to construct the directional spectrum from the three-dimensional MNLS simulation within the time interval from
Numerical simulation with MNLS: Estimates of Burg and Shannon ME directional spectrum from a three-dimensional MNLS simulation of an initially Gaussian spectrum with initial spectral widths
Citation: Journal of Atmospheric and Oceanic Technology 35, 2; 10.1175/JTECH-D-17-0007.1
6. Conclusions
In this study we present ocean surface wave directional analysis results from Ekofisk laser array measurements and from nonlinear numerical simulations. We employ four directional analysis methods for the estimation of directional spectra, namely, maximum likelihood (ML), iterative maximum likelihood (IML), Burg maximum entropy (BME), and Shannon maximum entropy (SME).
Previous studies suggest that BME directional estimates have a tendency to split peaks, which, to the best of our knowledge, has not been reported for the SME estimates. Our data analysis and numerical experiments, based on synthetic Fourier coefficients, suggest that there is a peak-splitting tendency in both the BME and SME estimates.
A large fraction of the directional wave spectra from the Ekofisk array shows evidence of bimodality at the higher frequencies for all four estimation methods. The consistency in the results between the various estimation methods suggests that the bimodality is real and that it is associated with the waves’ own development rather than being an artifact generated by the estimation methods. The angular separation of the two peaks starts mostly within the range of frequencies between the peak and 3 times the spectral peak and increases with frequency, reaching up to about 140° at about 4 times the spectral peak.
Acknowledgments
We would like to acknowledge the constructive comments by the referees. The field data are from the laser array at the Ekofisk oil field in the North Sea and are used by permission of ConocoPhillips Inc. This research has been funded by the Research Council of Norway (RCN) and the University of Oslo through Projects RCN 225933, RCN 214556, and RCN 256466.
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