Challenges in Description of Nonlinear Waves Due to Sampling Variability

Wave description is affected by several uncertainties, with sampling variability due to limited number of observations being one of them. Ideally, temporal/spatial wave registrations should be as large as possible to eliminate this uncertainty. This is difficult to reach in nature, where stationarity of sea states is an issue, but it can in principle be obtained in laboratory tests and numerical simulations, where initial wave conditions can be kept constant and intrinsic variability can be accounted for by changing random seeds for each run. Using linear, second-order, and third-order unidirectional numerical simulations, we compare temporal and spatial statistics of selected wave parameters and show how sampling variability affects their estimators. The JONSWAP spectrum with gamma peakedness parameters γ = 1, 3.3, and 6 is used in the analysis. The third-order wave data are simulated by a numerical solver based on the higher-order spectral method which includes the leading-order nonlinear dynamical effects. Field data support the analysis. We demonstrate that the nonlinear wave field including dynamical effects is more sensitive to sampling variability than the second-order and linear ones. Furthermore, we show that the mean values of temporal and spatial wave parameters can be equal if the number of simulations is sufficiently large. Consequences for design work are discussed.


Introduction
The sea surface is random; therefore, wave parameters derived from a temporal or spatial wave record will depend on which part of a wave record is used in the analysis, as well as on the length of a wave record and/or size of the investigated ocean area (see, e.g., References [1][2][3][4][5][6]). The error introduced by a limited wave record length/domain represents statistical uncertainty, called sampling variability, which can be reduced by increasing duration/ocean area of measurements/numerical simulations. Ideally, temporal/spatial wave registrations should be as large as possible to eliminate sampling variability. This is difficult to reach in nature, where stationarity of sea states is an issue, but it can in principle be obtained in laboratory tests and numerical simulations, where initial wave conditions can be kept constant and intrinsic variability can be accounted for by changing random seeds for each run.
Wave measurements are traditionally recorded at a single point (by buoys, wave probes, lasers, etc.) and restricted to 20 or 30 min, with the duration allowing the assumption of stationarity of a sea state. Spatial wave data are affected by the size of an instrument's footprint, and they were limited for many years. Recently, their amount increased due to the application of stereo camera systems for collecting space-time ensembles of sea surface elevation, but only a few publications were dedicated to the analysis of these data so far (see, e.g., References [7,8]). Both time and space wave measurements will be affected by sampling variability. This uncertainty brings challenges in the description of ocean

Set-Up of Numerical Simulations
The investigations carried out were based on simulations with a numerical DNV GL (Det Norske Veritas Germanischer Lloyd) solver based on the higher-order spectral method (HOSM), first proposed by Dommermuth and Yue [23] and West et al. [24]. HOSM has no limitation in terms of the spectral bandwidth of the wave field. Wave fields were simulated in a spatial domain with periodic boundary conditions. The nonlinear order M in the HOSM simulations was set to M = 3 in this study, which included the leading-order nonlinear dynamical effects, including the effect of modulational instability. The solver also included the linear and second-order wave model.
For the unidirectional simulations, the spatial domain was discretized by n x = 1024 grid points (de-aliased grid), while, in the short-crested simulations, the horizontal plane was discretized using n x × n y = 512 × 512 grid points. The domain size in the Fourier space was fixed such that k x (max) = k y (max) = 8k p in the fully de-aliased grid (k is the wavenumber and k p the peak wavenumber). For the unidirectional case, the computational domain corresponded to 64 λ p , while the directional case corresponded to 32 × 32 λ p , where λ p denotes the wavelength corresponding to the peak period T p . For example, for unidirectional waves with T p = 10 s on infinite water depth, the computational domain is about 10 km. The wave fields were simulated in time for a total duration of t max = 1800 s using a variable-step ODE (ordinary differential equation) solver. A weak dissipation of high wavenumbers was included to model the energy dissipation due to wave breaking, using the dissipation model suggested in Xiao et al. [25]. The breaking option was used only for runs where high wave steepness brought difficulties in the code convergence.
In the simulations, the initial condition was chosen as a wave system with the Pierson-Moskowitz (PM) or the JONSWAP spectrum and with a cos N (φ − φ p ) directional spreading function for the short-crested simulation, where φ is the wave direction and φ p is the peak direction of propagation.
Thus, the wave spectrum was defined as E(k) = F(k)D(φ), where k = (k x ,k y ) = k(sin φ, cos φ), and and where, Γ is the gamma function, and the parameter σ has the standard values 0.07 for k ≤ k p and 0.09 for k > k p . The other spectral parameters α, γ, k p , φ p , and N were chosen to give the desired spectral shape, significant wave height H s , and peak period T p . N denotes the directional spreading coefficient.
The following gamma peakedness parameters were used in the analysis: γ = 1, 3.3, and 6. Note that, for γ = 1, the JONSWAP spectrum reduces to the Pierson-Moskowitz spectrum. Random phases and amplitudes were assigned to the initial spectrum in all cases. Intrinsic wave variability was accounted for by changing random seeds for each run.
Sea states with increasing wave steepness, k p H s /2, in which rogue waves occurred in nature (see Reference [26]), were used in simulations, assuming a unidirectional wave field. For completion, two additional sea states with k p H s /2 = 0.13, 0.14, were also included. All considered sea states are listed in Table 1. Note that H s is derived from the zero-spectral moment, commonly denoted as H mo . The simulations were carried out for deep-water conditions. Directional numerical data developed in Reference [5], using the approach specified above, were also utilized in the study. For every sea state, the 1800-s unidirectional HOSM simulations were repeated 1000 times to provide satisfactory statistical estimators of selected wave parameters, while, for directional seas, requiring more central processing unit (CPU) time, they were repeated only 20 times.

Analyzed Wave Parameters
The present analysis is limited to investigations of the maximum surface elevation, as well as theskewness and kurtosis of water surface elevation. Note that the maximum surface elevation

Discussion and Conclusions
The inherent variability of sea waves because of sampling variabi observations. For limited wave reco which part of a wave record is tak obtained. It should be noted that 20 less data than space-time measurem (given that the sampling rates of the from space-time data will always b time series. However, more impor sampling rate of the measurements dimensions of the measured waves step alone will increase the number characteristics derived from them. The results presented in the p simulations, and they are supported The HOSM simulations are restricte order nonlinear dynamical effects, i investigations, we do not expect tha present conclusions. Furthermore, th that kx (max) = ky (max) = 8kp in the fully deaffected the size of the domain in x order to obtain "converged" results max , both in time and space, represents a wave crest height, an important parameter in design work and in the selection of rogue waves in a wave record. A common definition of a rogue wave, adopted also herein, is to apply the criteria due to Haver [27], i.e., H max /H s > 2 and/or C max /H s > 1.25, where C max and H max denote the maximum crest height and maximum zero-crossing wave height in a wave time series with significant wave height H s , defined as four times the standard deviation of the sea surface, typically calculated from a 20-min measurement of the surface elevation. Following the crest criterium, we use herein the ratio C max /H s when investigating the maximum surface elevation.
The skewness κ 3 and kurtosis κ 4 coefficients are commonly used to test deviation of population from normality (see Reference [28]). For a Gaussian distributed population, the skewness coefficient is equal to zero and the kurtosis coefficient is equal to three. Wave data include a limited number of observations, thus allowing to provide only sample coefficients of maximum surface elevation, skewness, and kurtosis.
The results shown in Figure 10b demonstrate the importance of considering area ef maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the descriptio waves because of sampling variability, i.e., the statistical uncertainty due to a limited observations. For limited wave records, whether over duration or over ocean area, dep which part of a wave record is taken in an analysis, different estimators of wave para obtained. It should be noted that 20-30-min observations/simulations at a single point w less data than space-time measurements/simulations over a restricted ocean area in the s (given that the sampling rates of the measurements are the same). Thus, wave characteristi from space-time data will always be less affected by sampling variability than those de time series. However, more important than the number of observations (which depe sampling rate of the measurements) are the dimensions of the selected ocean area rela dimensions of the measured waves (i.e., the typical wavelength and period). Increasing step alone will increase the number of data points, but not the accuracy of the estimato characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirectio simulations, and they are supported by examples with directional HOSM simulations and The HOSM simulations are restricted to the third-order of nonlinearity which includes t order nonlinear dynamical effects, including the effect of modulational instability. From investigations, we do not expect that increasing the order of nonlinearity in HOSM will present conclusions. Furthermore, the domain size in the Fourier space was fixed in the an that kx (max) = ky (max) = 8kp in the fully de-aliased grid. This means that changing the number of affected the size of the domain in x/y-space, as well as the grid-spacing in wavenumbe order to obtain "converged" results, it is important to have a sufficiently large physical well as sufficiently fine resolution of the wavenumber space. Hence, a larger number of is generally an advantage. Based on our own experience with HOSM, as well as from studies using HOSM in the scientific literature, we can say that our choice of discr sufficient to obtain "converged" results with respect to discretization. Although we carried out convergence studies, we did not do this specifically for the current paper.
Herein, we compared the temporal and spatial statistics of selected wave paramet from unidirectional numerical linear, second-order, and HOSM simulations for the gamma peakedness parameter  = 1, 3.3, and 6. The maximum surface elevation, ske kurtosis were considered. It is shown that the nonlinear wave field including dynamic more sensitive to sampling variability than the second order and linear ones. The dynam have a significant impact on the analyzed parameters, particularly on ɳmax/Hs. The di between the wave parameter estimators derived from the HOSM simulations and the line much larger than those obtained from the second-order and linear simulations. We sh mean values of temporal and spatial wave parameters can be equal if the number of sim sufficiently large. Furthermore, we proposed functional relationships between the i parameters and wave steepness.
When using a single 20-or 30-min field wave record, it is challenging to conclude the of nonlinearity of surface elevation, since the sampling variability may dominate over th effects. Therefore, numerical models and laboratory tests represent important supporti max (C max ), were calculated as averages over all random realizations of the same sea state. The 1800-s simulations, repeated for every sea state 1000 times for unidirectional waves and 20 times for directional sea, were sampled every 0.2 s.
The sample skewness and kurtosis coefficients described by Equation (3) and (4), as well as the alternative "unbiased" commonly used estimators for identically distributed independent samples found in the literature (see, e.g., References [29,30]), represent biased estimators of the real populations of κ 3 and κ 4 in cases where the samples are not independent, which is the case for wave surface oscillations. Therefore, for the numerically simulated linear Gaussian surface, the skewness is not equal exactly to zero nor is the kurtosis equal exactly to three [30]. This effect is more pronounced for small simulation domains (see Reference [4]). Due to the relatively large computation area/duration considered in this study, the presented numerical results are little affected by this bias, but we also observe it.

Comparison of Temporal and Spatial Statistics of Selected Wave Parameters
In Figures 1-3, examples of histograms (frequency of occurrence) of temporal (subscript "t") and spatial (subscript "x,y,t"), from top to bottom, skewness (κ 3 ), kurtosis (κ 4 ), and The results shown in Figure 10b demonstrate th maximum wave crest height when designing marine

Discussion and Conclusions
The inherent variability of sea surface elevatio waves because of sampling variability, i.e., the stat observations. For limited wave records, whether ov which part of a wave record is taken in an analysi obtained. It should be noted that 20-30-min observa less data than space-time measurements/simulations (given that the sampling rates of the measurements ar from space-time data will always be less affected b time series. However, more important than the nu sampling rate of the measurements) are the dimen dimensions of the measured waves (i.e., the typical step alone will increase the number of data points, characteristics derived from them. The results presented in the paper are based simulations, and they are supported by examples wi The HOSM simulations are restricted to the third-or order nonlinear dynamical effects, including the effe investigations, we do not expect that increasing the present conclusions. Furthermore, the domain size in that kx (max) = ky (max) = 8kp in the fully de-aliased grid. Thi affected the size of the domain in x/y-space, as wel order to obtain "converged" results, it is important well as sufficiently fine resolution of the wavenumb is generally an advantage. Based on our own expe studies using HOSM in the scientific literature, w sufficient to obtain "converged" results with resp carried out convergence studies, we did not do this s Herein, we compared the temporal and spatial max /H s derived from unidirectional linear, second-order, and HOSM simulations with γ = 1.0 (Figure 1), 3.3 (Figure 2), and 6.0 ( Figure 3) are shown. The sea states with the lowest (a) and highest (b) wave steepness, kpH s / 2 = 0.10 and kpH s / 2 = 0.14, are plotted in the figures (see Table 1).
Note that the spatial  The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number of observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling step alone will increase the number of data points, but not the accuracy of the estimators of wave characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirectional HOSM simulations, and they are supported by examples with directional HOSM simulations and field data. The HOSM simulations are restricted to the third-order of nonlinearity which includes the leadingorder nonlinear dynamical effects, including the effect of modulational instability. From the earlier investigations, we do not expect that increasing the order of nonlinearity in HOSM will change the present conclusions. Furthermore, the domain size in the Fourier space was fixed in the analysis such that kx (max) = ky (max) = 8kp in the fully de-aliased grid. This means that changing the number of grid points affected the size of the domain in x/y-space, as well as the grid-spacing in wavenumber space. In order to obtain "converged" results, it is important to have a sufficiently large physical domain, as well as sufficiently fine resolution of the wavenumber space. Hence, a larger number of grid points is generally an advantage. Based on our own experience with HOSM, as well as from numerous max /H s, skewness and kurtosis shown in Figures 1-3 represent the mean values over the simulation domain averaged over simulation time. The plots in the figures clearly demonstrate the effect of sampling variability manifested by the histograms' width. The histograms derived from the nonlinear HOSM simulations were much broader than those obtained from the linear and second-order ones, showing that the wave field including nonlinear dynamic effects was more sensitive to sampling variability. Both the wave steepness and the spectral parameter γ impacted the histograms' width. Furthermore, it is interesting to note also that discrepancies between the estimators obtained from the linear and second-order simulations were not as pronounced as those between the linear and HOSM simulations.
As expected, the spatial The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number of observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling step alone will increase the number of data points, but not the accuracy of the estimators of wave characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirectional HOSM simulations, and they are supported by examples with directional HOSM simulations and field data.

Discussion and Conclusions
The inherent variability of sea su waves because of sampling variability observations. For limited wave record which part of a wave record is taken obtained. It should be noted that 20-30 less data than space-time measuremen (given that the sampling rates of the me from space-time data will always be l time series. However, more importan sampling rate of the measurements) a dimensions of the measured waves (i.e step alone will increase the number of characteristics derived from them.
The results presented in the pap simulations, and they are supported by The results shown in Figure 10b demonstrate the importance of considering a maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the des waves because of sampling variability, i.e., the statistical uncertainty due to a lim observations. For limited wave records, whether over duration or over ocean are which part of a wave record is taken in an analysis, different estimators of wav obtained. It should be noted that 20-30-min observations/simulations at a single p less data than space-time measurements/simulations over a restricted ocean area in (given that the sampling rates of the measurements are the same). Thus, wave charac from space-time data will always be less affected by sampling variability than tho time series. However, more important than the number of observations (which sampling rate of the measurements) are the dimensions of the selected ocean are dimensions of the measured waves (i.e., the typical wavelength and period). Incre step alone will increase the number of data points, but not the accuracy of the es characteristics derived from them. max /H s increased with higher γ and wave steepness. This is consistent with earlier findings (e.g. References [31,32]).
The skewness (κ 3 ) is primarily a second-order effect, while the kurtosis (κ 4 ) is a third-order effect. Thus, the contribution from nonlinear dynamical effects, including the effect of modulational instability, does not significantly impact the skewness, as also shown in Figures 1-3. The mean skewness was around 0.20, as predicted by the second-order wave theory. The coefficient of kurtosis was affected, however, by nonlinear dynamical effects. Although the mean temporal skewness and kurtosis were approximately equal to the mean spatial skewness and kurtosis, the shape of the temporal and spatial histograms and their standard deviations were different. Furthermore, the temporal estimators of skewness and kurtosis reached higher values than the spatial ones for all three wave models, showing that they were more affected by sampling variability than the spatial estimators, as should be expected. temporal and spatial histograms and their standard deviations were different. Furthermore, the temporal estimators of skewness and kurtosis reached higher values than the spatial ones for all three wave models, showing that they were more affected by sampling variability than the spatial estimators, as should be expected.

Discussion and Conclusions
The inherent variability of sea surface waves because of sampling variability, i.e observations. For limited wave records, w which part of a wave record is taken in a obtained. It should be noted that 20-30-mi less data than space-time measurements/si (given that the sampling rates of the measur from space-time data will always be less a time series. However, more important tha sampling rate of the measurements) are th dimensions of the measured waves (i.e., th step alone will increase the number of dat characteristics derived from them. The results presented in the paper a simulations, and they are supported by exa The HOSM simulations are restricted to th order nonlinear dynamical effects, includin investigations, we do not expect that increa present conclusions. Furthermore, the doma that kx (max) = ky (max) = 8kp in the fully de-aliased affected the size of the domain in x/y-spac order to obtain "converged" results, it is im well as sufficiently fine resolution of the w is generally an advantage. Based on our o studies using HOSM in the scientific lite sufficient to obtain "converged" results w carried out convergence studies, we did no max /H s for unidirectional linear, second-order, and HOSM (higher-order spectral method) simulations for sea states (a) kpHs/2 = 0.10, γ = 1.0 (left column), and (b) kpHs/2 = 0.14, γ = 1.0 (right column). temporal and spatial histograms and their standard deviations were different. Furthermore, the temporal estimators of skewness and kurtosis reached higher values than the spatial ones for all three wave models, showing that they were more affected by sampling variability than the spatial estimators, as should be expected.

Discussion and Conclusions
The inherent variability of sea surface el waves because of sampling variability, i.e., th observations. For limited wave records, whet which part of a wave record is taken in an a obtained. It should be noted that 20-30-min o less data than space-time measurements/simu (given that the sampling rates of the measurem from space-time data will always be less affe time series. However, more important than sampling rate of the measurements) are the d dimensions of the measured waves (i.e., the ty step alone will increase the number of data p characteristics derived from them. The results presented in the paper are simulations, and they are supported by examp The HOSM simulations are restricted to the th order nonlinear dynamical effects, including t investigations, we do not expect that increasin present conclusions. Furthermore, the domain that kx (max) = ky (max) = 8kp in the fully de-aliased gr affected the size of the domain in x/y-space, order to obtain "converged" results, it is impo well as sufficiently fine resolution of the wave is generally an advantage. Based on our own studies using HOSM in the scientific literat max /H s for unidirectional linear, second-order, and HOSM (higher-order spectral method) simulations for sea states (a) kpHs/2 = 0.10, γ = 6.0 (left column), and (b) kpHs/2 = 0.14, γ = 6.0 (right column). The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number of observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling step alone will increase the number of data points, but not the accuracy of the estimators of wave characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirectional HOSM simulations, and they are supported by examples with directional HOSM simulations and field data. The HOSM simulations are restricted to the third-order of nonlinearity which includes the leadingorder nonlinear dynamical effects, including the effect of modulational instability. From the earlier investigations, we do not expect that increasing the order of nonlinearity in HOSM will change the present conclusions. Furthermore, the domain size in the Fourier space was fixed in the analysis such that kx (max) = ky (max) = 8kp in the fully de-aliased grid. This means that changing the number of grid points affected the size of the domain in x/y-space, as well as the grid-spacing in wavenumber space. In max /H s , skewness, and kurtosis derived from linear simulations may reach values close to nonlinear HOSM ones. Furthermore, for a given seed, a single 30-min realization of a sea state may produce a higher kurtosis coefficient for the Pierson-Moskowitz spectrum (JONSWAP spectrum with γ = 1.0) than for the JONSWAP spectrum with γ = 6.0, because of sampling variability. The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number of observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling step alone will increase the number of data points, but not the accuracy of the estimators of wave characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirectional HOSM simulations, and they are supported by examples with directional HOSM simulations and field data. The HOSM simulations are restricted to the third-order of nonlinearity which includes the leadingorder nonlinear dynamical effects, including the effect of modulational instability. From the earlier max /H s , skewness, and kurtosis, calculated as the averages over all random realizations of the same sea state, as functions of wave steepness for the linear, second-order, and HOSM simulations with γ = 1.0, 3.3, and 6.0, respectively. Both temporal and spatial estimators derived from unidirectional simulations, together with the 95% confidence intervals, are plotted in the figures. It is interesting to note that, for the linear and second-order wave model, the PM spectrum (γ = 1. The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number o observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling step alone will increase the number of data points, but not the accuracy of the estimators of wave max /H s , skewness, and kurtosis than the JONSWAP spectrum with γ = 3.3 and 6.0. This was probably due to the bias of the sample estimators in comparison to the true values, as discussed above. The increase in HOSM skewness with wave steepness was not significantly dependent on the spectral parameter γ, as particularly seen for the temporal skewness estimator.
was primarily a second-order effect; however, some reduction in growth of the skewness slope gradient for kpHs/2 > 0.14 was observed.
For HOSM simulations, Figures 4-6 clearly demonstrate the effect of spectrum peakedness related to spectral width on the investigated wave parameters. Consistent with the theory, γ = 6.0 gave the highest ɳmax/Hs, skewness, and kurtosis, followed by γ = 3.3 and γ = 1.0, both for the time and the space-time estimators.   The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number of observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling step alone will increase the number of data points, but not the accuracy of the estimators of wave characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirectional HOSM simulations, and they are supported by examples with directional HOSM simulations and field data. The HOSM simulations are restricted to the third-order of nonlinearity which includes the leadingorder nonlinear dynamical effects, including the effect of modulational instability. From the earlier investigations, we do not expect that increasing the order of nonlinearity in HOSM will change the present conclusions. Furthermore, the domain size in the Fourier space was fixed in the analysis such that kx (max) = ky (max) = 8kp in the fully de-aliased grid. This means that changing the number of grid points affected the size of the domain in x/y-space, as well as the grid-spacing in wavenumber space. In order to obtain "converged" results, it is important to have a sufficiently large physical domain, as well as sufficiently fine resolution of the wavenumber space. Hence, a larger number of grid points is generally an advantage. Based on our own experience with HOSM, as well as from numerous studies using HOSM in the scientific literature, we can say that our choice of discretization is sufficient to obtain "converged" results with respect to discretization. Although we previously carried out convergence studies, we did not do this specifically for the current paper.   The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number of observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling step alone will increase the number of data points, but not the accuracy of the estimators of wave characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirectional HOSM simulations, and they are supported by examples with directional HOSM simulations and field data. The HOSM simulations are restricted to the third-order of nonlinearity which includes the leadingorder nonlinear dynamical effects, including the effect of modulational instability. From the earlier investigations, we do not expect that increasing the order of nonlinearity in HOSM will change the present conclusions. Furthermore, the domain size in the Fourier space was fixed in the analysis such that kx (max) = ky (max) = 8kp in the fully de-aliased grid. This means that changing the number of grid points affected the size of the domain in x/y-space, as well as the grid-spacing in wavenumber space. In max /H s and kurtosis increased with the increase in wave steepness up to kpH s / 2 = 0.14 and slightly decreased afterwards. This can be attributed primarily to nonlinear dynamical effects and not to wave breaking; the breaking option was used only for some runs with kpH s / 2 > 0.14 in the HOSM simulations. Wave breaking may occur for this steepness range, as observed earlier in laboratory experiments (e.g., Reference [32]). For kpH s / 2 > 0.14, non-stationary nonlinear dynamical effects limited the growth of The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number of observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling step alone will increase the number of data points, but not the accuracy of the estimators of wave characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirectional HOSM simulations, and they are supported by examples with directional HOSM simulations and field data. The HOSM simulations are restricted to the third-order of nonlinearity which includes the leading-max and kurtosis. We did not observe this effect for skewness, which was primarily a second-order effect; however, some reduction in growth of the skewness slope gradient for kpH s /2 > 0.14 was observed.
For The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number of observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling max /H s , skewness, and kurtosis, followed by γ = 3.3 and γ = 1.0, both for the time and the space-time estimators.

Correlation between Wave Characteristics and Sampling Variability
Functional relationships between wave characteristics are of importance for design and marine operations. This topic also receives attention in connection with the on-going discussion about the prediction of rogue waves. Commonly, theoretical/semi-theoretical expressions need to be verified through field data. Regarding rogue waves, even though the theory pointed out some parameters as indicators of rogue wave occurrence, wave researchers were unable to get confirmation of the theory in the analysis of measurements. Large clouds of data did not allow drawing any regression lines between wave parameters identifying the occurrence of rogue wave (see, e.g., References [33,34]). For a long time, it was not clear why the issue was so difficult to solve. Today, we know that sampling variability brings these challenges.
To illustrate the problem, Figure 7 shows an example of continuous single-point measurements of significant wave height, kurtosis, and skewness registered on 2 January 2016. The parameters plotted in the figure were calculated from time series of sea surface elevation recorded with a sampling rate of 2 Hz by a WaveRadar REX (known also as SAAB REX radar) at the Ekofisk field in central North Sea. The gray areas around the average values of Hs, kurtosis (Ku in the figure), or skewness in the figures show the standard deviations (representing sampling variability) of the parameters, derived following the procedure proposed by Reference [3], i.e., 20-min parameters (Hs, Ku, skewness) were calculated for all 20-min periods starting at 1-min intervals within each 60 min. The black dots are these values. Below, we show how numerical simulations can help to reduce deviations between temporal and spatial statistics and allow establishing a correlation between the wave parameters responsible for occurrence of rogue waves.
In Figure 9, the averages of temporal and spatial skewness and kurtosis, over all unidirectional The effect of sampling variability on field data is also demonstrated in Figure 8, where standard deviations of skewness and kurtosis, as described for Figure 7, derived from 20-min wave records, registered continuously at a single point by the WaveRadar REX, are plotted against the 1-h average skewness and kurtosis values. The data cover the period from January 2016 to September 2019. The standard deviations of these parameters were derived following the procedure proposed by Reference [3]. The mean and median are also plotted in the figure. As expected, large spreading, due to sampling variability, can be observed. It is interesting to see consistency with the numerical simulations; temporal skewness was much more affected by sampling variability than temporal kurtosis. While the mean standard deviation of kurtosis increased with the increase in kurtosis, we did not observe this for skewness, but instead saw a cloud of data. Furthermore, as expected, the unidirectional simulations shown in Figures 4-6 had larger spreading than the field data presented in Figure 8. Below, we show how numerical simulations can help to reduce deviations between temporal and spatial statistics and allow establishing a correlation between the wave parameters responsible for occurrence of rogue waves.
In Figure 9, the averages of temporal and spatial skewness and kurtosis, over all unidirectional random realizations of the same sea state with γ = 3.3, are plotted as a function of wave steepness. As expected, the average temporal and spatial estimators of skewness and kurtosis were approximately equal, due to the large numbers of runs carried out.
For unidirectional wave fields, the mean temporal/spatial skewness is not sensitive to the spectral parameter γ, and a general expression independent of γ, as shown in Figure 9a for γ = 3.3, can be proposed for skewness as a function of wave steepness. κ3_x,y,t = 1.66(kpHs/2).

(5)
This formula can be compared to the expression suggested by Reference [35]. Below, we show how numerical simulations can help to reduce deviations between temporal and spatial statistics and allow establishing a correlation between the wave parameters responsible for occurrence of rogue waves.
In Figure 9, the averages of temporal and spatial skewness and kurtosis, over all unidirectional random realizations of the same sea state with γ = 3.3, are plotted as a function of wave steepness. As expected, the average temporal and spatial estimators of skewness and kurtosis were approximately equal, due to the large numbers of runs carried out. Although the average temporal and space-time estimators of HOSM skewness and kurtosis over all runs were approximately equal, there was significant spreading around the mean values, which was larger for the temporal data than for the spatial data. The coefficient of variation (COV) was up to 0.53 for the temporal skewness and up to 0.12 for the spatial one (the time dataset was smaller than the spatial one), while, for kurtosis, COV was up to 0.16 for temporal data and up to 0.11 for spatial data. In Figure 10a, the mean spatial kurtosis over all runs as a function of wave steepness for γ = 1.0, 3.3, and 6.0 is shown, together with the fitted to the data polynomials. The figure confirms that when the mean values over a large number of runs are considered, it is not difficult to establish a correlation between wave parameters, which is consistent with wave theory. Furthermore, it should be noted that the results presented herein refer to unidirectional wave fields, giving more conservative estimators than we would expect when analyzing directionally spread waves. To document this, for comparison, Figure 10a includes the kurtosis calculated from the directional HOSM simulations of Reference [6] for kpHs/2 = 0.11, with γ = 1.0 and N = 4 (N is the spread number), with γ = 3.3 and N = 16, and with γ = 6.0 and N = 100. As expected, the effect of modulational instability was suppressed when the wave energy spreading was accounted for.
Although small deviations between the mean over all runs of skewness and kurtosis were identified, the picture was different for the average temporal and spatial ratio of the maximum surface elevation divided by significant wave height, ɳmax/Hs. The average ɳmax/Hs over all runs as a function of wave steepness is shown together with the fitted polynomials in Figure 10b. The spatial ɳmax/Hs differed significantly from the temporal one, largely exceeding the temporal values. Furthermore, the shape of the wave spectrum affected the spatial ɳmax/Hs more than the temporal one. The observed deviations between temporal and spatial estimators were due to the limited data. For unidirectional wave fields, the mean temporal/spatial skewness is not sensitive to the spectral parameter γ, and a general expression independent of γ, as shown in Figure 9a for γ = 3.3, can be proposed for skewness as a function of wave steepness.
This formula can be compared to the expression suggested by Reference [35]. Although the average temporal and space-time estimators of HOSM skewness and kurtosis over all runs were approximately equal, there was significant spreading around the mean values, which was larger for the temporal data than for the spatial data. The coefficient of variation (COV) was up to 0.53 for the temporal skewness and up to 0.12 for the spatial one (the time dataset was smaller than the spatial one), while, for kurtosis, COV was up to 0.16 for temporal data and up to 0.11 for spatial data.
In Figure 10a, the mean spatial kurtosis over all runs as a function of wave steepness for γ = 1.0, 3.3, and 6.0 is shown, together with the fitted to the data polynomials. The figure confirms that when the mean values over a large number of runs are considered, it is not difficult to establish a correlation between wave parameters, which is consistent with wave theory. Furthermore, it should be noted that the results presented herein refer to unidirectional wave fields, giving more conservative estimators than we would expect when analyzing directionally spread waves. To document this, for comparison, Figure 10a includes the kurtosis calculated from the directional HOSM simulations of Reference [6] for k p H s /2 = 0.11, with γ = 1.0 and N = 4 (N is the spread number), with γ = 3.3 and N = 16, and with γ = 6.0 and N = 100. As expected, the effect of modulational instability was suppressed when the wave energy spreading was accounted for.
In Figure 10a, the mean spatial kurtosis over all runs as a function of wave steepness for γ = 1.0, 3.3, and 6.0 is shown, together with the fitted to the data polynomials. The figure confirms that when the mean values over a large number of runs are considered, it is not difficult to establish a correlation between wave parameters, which is consistent with wave theory. Furthermore, it should be noted that the results presented herein refer to unidirectional wave fields, giving more conservative estimators than we would expect when analyzing directionally spread waves. To document this, for comparison, Figure 10a includes the kurtosis calculated from the directional HOSM simulations of Reference [6] for kpHs/2 = 0.11, with γ = 1.0 and N = 4 (N is the spread number), with γ = 3.3 and N = 16, and with γ = 6.0 and N = 100. As expected, the effect of modulational instability was suppressed when the wave energy spreading was accounted for.
Although small deviations between the mean over all runs of skewness and kurtosis were identified, the picture was different for the average temporal and spatial ratio of the maximum surface elevation divided by significant wave height, ɳmax/Hs. The average ɳmax/Hs over all runs as a function of wave steepness is shown together with the fitted polynomials in Figure 10b. The spatial ɳmax/Hs differed significantly from the temporal one, largely exceeding the temporal values. Furthermore, the shape of the wave spectrum affected the spatial ɳmax/Hs more than the temporal one. The observed deviations between temporal and spatial estimators were due to the limited data.
Increasing the number of simulations should reduce these deviations. The results shown in Figure 10b demonstrat maximum wave crest height when designing mar

Discussion and Conclusions
The inherent variability of sea surface eleva waves because of sampling variability, i.e., the observations. For limited wave records, whether which part of a wave record is taken in an anal obtained. It should be noted that 20-30-min obse less data than space-time measurements/simulati (given that the sampling rates of the measurement from space-time data will always be less affected time series. However, more important than the sampling rate of the measurements) are the dim dimensions of the measured waves (i.e., the typic step alone will increase the number of data poin characteristics derived from them. The results presented in the paper are bas simulations, and they are supported by examples The HOSM simulations are restricted to the third order nonlinear dynamical effects, including the investigations, we do not expect that increasing t present conclusions. Furthermore, the domain siz that kx (max) = ky (max) = 8kp in the fully de-aliased grid. T affected the size of the domain in x/y-space, as w order to obtain "converged" results, it is importa well as sufficiently fine resolution of the wavenu The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number of observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling step alone will increase the number of data points, but not the accuracy of the estimators of wave characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirectional HOSM simulations, and they are supported by examples with directional HOSM simulations and field data. The HOSM simulations are restricted to the third-order of nonlinearity which includes the leadingorder nonlinear dynamical effects, including the effect of modulational instability. From the earlier investigations, we do not expect that increasing the order of nonlinearity in HOSM will change the present conclusions. Furthermore, the domain size in the Fourier space was fixed in the analysis such that kx (max) = ky (max) = 8kp in the fully de-aliased grid. This means that changing the number of grid points max /H s versus wave steepness, where red, violet, and gray lines (the three lower lines) correspond to γ = 1.0, 3.3, and 6.0, respectively.
Although small deviations between the mean over all runs of skewness and kurtosis were identified, the picture was different for the average temporal and spatial ratio of the maximum surface elevation divided by significant wave height, The results shown in Figure 10b demonstrate the importance of considering area maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the descrip waves because of sampling variability, i.e., the statistical uncertainty due to a limit observations. For limited wave records, whether over duration or over ocean area, which part of a wave record is taken in an analysis, different estimators of wave p obtained. It should be noted that 20-30-min observations/simulations at a single poin less data than space-time measurements/simulations over a restricted ocean area in th (given that the sampling rates of the measurements are the same). Thus, wave character from space-time data will always be less affected by sampling variability than those time series. However, more important than the number of observations (which de sampling rate of the measurements) are the dimensions of the selected ocean area dimensions of the measured waves (i.e., the typical wavelength and period). Increasi step alone will increase the number of data points, but not the accuracy of the estim characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirec simulations, and they are supported by examples with directional HOSM simulations The results shown in Figure 10b demonstrate the importanc maximum wave crest height when designing marine installations

Discussion and Conclusions
The inherent variability of sea surface elevation brings cha waves because of sampling variability, i.e., the statistical uncer observations. For limited wave records, whether over duration which part of a wave record is taken in an analysis, different obtained. It should be noted that 20-30-min observations/simula less data than space-time measurements/simulations over a restr (given that the sampling rates of the measurements are the same). from space-time data will always be less affected by sampling v time series. However, more important than the number of ob sampling rate of the measurements) are the dimensions of the dimensions of the measured waves (i.e., the typical wavelength step alone will increase the number of data points, but not the characteristics derived from them.
The results presented in the paper are based primarily o simulations, and they are supported by examples with directiona max /H s over all runs as a function of wave steepness is shown together with the fitted polynomials in Figure 10b

Discussion and Conclusions
The inherent variability of sea su waves because of sampling variability observations. For limited wave record which part of a wave record is taken obtained. It should be noted that 20-3 less data than space-time measuremen (given that the sampling rates of the me from space-time data will always be l time series. However, more importan sampling rate of the measurements) a dimensions of the measured waves (i.e step alone will increase the number of characteristics derived from them.
The results presented in the pap max /H s differed significantly from the temporal one, largely exceeding the temporal values. Furthermore, the shape of the wave spectrum affected the spatial The results shown in Figure 10b demonstrate the importance of considering maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the d waves because of sampling variability, i.e., the statistical uncertainty due to a observations. For limited wave records, whether over duration or over ocean a which part of a wave record is taken in an analysis, different estimators of wa obtained. It should be noted that 20-30-min observations/simulations at a single less data than space-time measurements/simulations over a restricted ocean area (given that the sampling rates of the measurements are the same). Thus, wave char from space-time data will always be less affected by sampling variability than t time series. However, more important than the number of observations (whic sampling rate of the measurements) are the dimensions of the selected ocean a dimensions of the measured waves (i.e., the typical wavelength and period). Inc step alone will increase the number of data points, but not the accuracy of the max /H s more than the temporal one. The observed deviations between temporal and spatial estimators were due to the limited data. Increasing the number of simulations should reduce these deviations.
The results shown in Figure 10b demonstrate the importance of considering area effects on the maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challenges in the description of ocean waves because of sampling variability, i.e., the statistical uncertainty due to a limited number of observations. For limited wave records, whether over duration or over ocean area, depending on which part of a wave record is taken in an analysis, different estimators of wave parameters are obtained. It should be noted that 20-30-min observations/simulations at a single point will contain less data than space-time measurements/simulations over a restricted ocean area in the same period (given that the sampling rates of the measurements are the same). Thus, wave characteristics obtained from space-time data will always be less affected by sampling variability than those derived from time series. However, more important than the number of observations (which depends on the sampling rate of the measurements) are the dimensions of the selected ocean area relative to the dimensions of the measured waves (i.e., the typical wavelength and period). Increasing a sampling step alone will increase the number of data points, but not the accuracy of the estimators of wave characteristics derived from them.
The results presented in the paper are based primarily on numerical unidirectional HOSM simulations, and they are supported by examples with directional HOSM simulations and field data. The HOSM simulations are restricted to the third-order of nonlinearity which includes the leading-order nonlinear dynamical effects, including the effect of modulational instability. From the earlier investigations, we do not expect that increasing the order of nonlinearity in HOSM will change the present conclusions. Furthermore, the domain size in the Fourier space was fixed in the analysis such that k x (max) = k y (max) = 8k p in the fully de-aliased grid. This means that changing the number of grid points affected the size of the domain in x/y-space, as well as the grid-spacing in wavenumber space. In order to obtain "converged" results, it is important to have a sufficiently large physical domain, as well as sufficiently fine resolution of the wavenumber space. Hence, a larger number of grid points is generally an advantage. Based on our own experience with HOSM, as well as from numerous studies using HOSM in the scientific literature, we can say that our choice of discretization is sufficient to obtain "converged" results with respect to discretization. Although we previously carried out convergence studies, we did not do this specifically for the current paper. Herein, we compared the temporal and spatial statistics of selected wave parameters derived from unidirectional numerical linear, second-order, and HOSM simulations for the JONSWAP gamma peakedness parameter γ = 1, 3.3, and 6. The maximum surface elevation, skewness, and kurtosis were considered. It is shown that the nonlinear wave field including dynamical effects is more sensitive to sampling variability than the second order and linear ones. The dynamical effects have a significant impact on the analyzed parameters, particularly on J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW Figure 10. Average over all runs of spatial (a) kurtosis versus wave steep wave steepness. HOSM simulations, γ = 1.0 (black line), 3.3 (blue line), an also includes the spatial kurtosis for directional waves: γ = 1.0 with spread γ = 3.3 with N = 16 (blue mark), and γ = 6.0 with N = 100 (brown mark), temporal ɳmax/Hs versus wave steepness, where red, violet, and gray l correspond to γ = 1.0, 3.3, and 6.0, respectively.
The results shown in Figure 10b demonstrate the importance of c maximum wave crest height when designing marine installations.

Discussion and Conclusions
The inherent variability of sea surface elevation brings challeng waves because of sampling variability, i.e., the statistical uncertainty observations. For limited wave records, whether over duration or ov which part of a wave record is taken in an analysis, different estim obtained. It should be noted that 20-30-min observations/simulations less data than space-time measurements/simulations over a restricted (given that the sampling rates of the measurements are the same). Thus from space-time data will always be less affected by sampling variab time series. However, more important than the number of observa sampling rate of the measurements) are the dimensions of the selec dimensions of the measured waves (i.e., the typical wavelength and p step alone will increase the number of data points, but not the accur characteristics derived from them. The results presented in the paper are based primarily on num simulations, and they are supported by examples with directional HO The HOSM simulations are restricted to the third-order of nonlinearit order nonlinear dynamical effects, including the effect of modulation investigations, we do not expect that increasing the order of nonlinea present conclusions. Furthermore, the domain size in the Fourier space that kx (max) = ky (max) = 8kp in the fully de-aliased grid. This means that chan affected the size of the domain in x/y-space, as well as the grid-spac order to obtain "converged" results, it is important to have a sufficie well as sufficiently fine resolution of the wavenumber space. Hence, a is generally an advantage. Based on our own experience with HOSM studies using HOSM in the scientific literature, we can say that o max /H s . The discrepancies between the wave parameter estimators derived from the HOSM simulations and the linear ones are much larger than those obtained from the second-order and linear simulations. We show that the mean values of temporal and spatial wave parameters can be equal if the number of simulations is sufficiently large. Furthermore, we proposed functional relationships between the investigated parameters and wave steepness.
When using a single 20-or 30-min field wave record, it is challenging to conclude the importance of nonlinearity of surface elevation, since the sampling variability may dominate over the nonlinear effects. Therefore, numerical models and laboratory tests represent important supporting tools to field data in this respect. They also allow establishing functional relationships between wave parameters, which are often not possible to derive from field data, if the number of observations is not sufficiently large. We should, however, be aware of that variability is also related to such expressions.
The results presented were developed using extensive computations with relatively large domain/duration. A reduction of the area would increase the spreading around the calculated estimators of wave parameters. Furthermore, the results of this study were derived for unidirectional wave fields; therefore, they should be applied with care in engineering applications, as they represent conservative estimators of wave parameters. The examples of the directional HOSM simulations presented in the paper confirm this.
New installations of stereo video camera systems, e.g., the Norwegian Meteorological Institute stereo video camera system based on the method developed at ISMAR (Institute of Marine Sciences, Venice, Italy) by [7], will allow obtaining further insight into temporal and space-time statistics and the associated sampling variability.
Knowledge about uncertainty due to sampling variability is essential for design work and marine operations.