## 1. Introduction

Wave-following measurement buoys are widely deployed across the oceans, as they provide a cost effective, easy-to-install alternative to their Eulerian counterparts. As a result of this, they represent an abundant source of data and metocean statistics, outnumbering Eulerian observations by an order of magnitude (Christou and Ewans 2014). However, within the oceanographic community, it is generally perceived that the measurements these devices produce are less accurate particularly when considering the measurement of extreme wave crests (Allender et al. 1989; Seymour and Castel 1998; Forristall 2000; Dysthe et al. 2008; Collins et al. 2014; Christou and Ewans 2014).

*η*

_{c}as function of normalized crest height

*η*

_{c}/

*H*

_{s}; we define significant wave height

*H*

_{s}as

*m*

_{0}is the zeroth spectral moment. Individual crest heights measured by the buoy are shown as blue markers; these observations are predicted well by the Rayleigh distribution,

*H*

_{s}, mean period

*T*

_{1}=

*m*

_{0}/

*m*

_{1}, and water depth

*d*to account for this second-order nonlinearity (Forristall 2000),

*α*(

*H*

_{s},

*T*

_{1},

*d*) and

*β*(

*H*

_{s},

*T*

_{1},

*d*) (see appendix B). For the conditions in Fig. 1 (and those we will examine experimentally herein), the Forristall distribution gives near identical results to the approximate version of the Tayfun (1980) distribution proposed by Socquet-Juglard et al. (2005). For the buoy data we examine, the crest height follows the linear Rayleigh distribution (1) and is over predicted by the second-order Forristall distribution (2). Second-order distributions such as (2) have been validated using Eulerian in situ observations, and their accuracy for Eulerian measurements is relatively well established (Forristall 2000; Christou and Ewans 2011). Moreover, for extreme experimental conditions, second-order crest height distributions have even been shown to underpredict crests (Onorato et al. 2009; Latheef and Swan 2013).

*h*as function of normalized wave height

*h*/

*H*

_{s}. The blue markers show the zero-upcrossing wave height of each wave measured by the buoy, which is well predicted by the bandwidth-corrected Rayleigh distribution shown by the dash–dotted black line (Boccotti 1981),

*T*

_{1}/2 (Tayfun and Fedele 2007). Unlike wave crests, second-order bound nonlinearity does not affect wave height, and hence wave height follows a bandwidth-corrected Rayleigh distribution in the absence of higher-order effects (Longuet-Higgins 1952).

Summarizing, the in situ buoy measurements we examine follow the expected wave height distribution [(3); Tayfun and Fedele 2007] but not the expected crest height distribution [(2); Forristall 2000]; this is also observed in Casas-Prat and Holthuijsen (2010) for a much larger dataset (containing *N* = 10^{6} waves). Several studies have suggested that a buoy’s mooring or its freedom to move horizontally and avoid large crests (Allender et al. 1989; Krogstad and Barstow 2000) may result in this apparent underestimation of extreme wave crests. However, if either of these two effects were the cause of preferential avoidance of wave crests, one may expect that the measured (crest-to-trough) wave height would also be affected, which we do not observe. In previous work, we have shown the latter of these two effects (the avoidance of large crests) is not significant when considering the Lagrangian measurement of steep directionally spread nonbreaking focused wave groups in the laboratory (McAllister and van den Bremer 2019).

An alternative explanation offered for the systematic underestimation of crest heights, which is consistent with in situ observations, is that the Lagrangian motion of a wave-following buoy cancels out second-order nonlinearity and essentially linearizes the resulting measurements (van Essen et al. 2018; Casas-Prat and Holthuijsen 2010). This is partially correct. The cancellation of second-order superharmonics that occurs as a result of Lagrangian motion is accompanied by an increase in second-order subharmonics, as identified by Srokosz and Longuet-Higgins (1986) and Herbers and Janssen (2016). Therefore, in principle, crest height measured by a wave-following buoy should not differ from that measured by an Eulerian observer as a result of second-order nonlinearity alone.

The linearization of buoy measurements occurs during the signal processing involved in converting from acceleration, which is measured by the buoy’s instrumentation, to displacement (Marthinsen and Winterstein 1992; Seymour and Castel 1998; Prevosto et al. 2000; Forristall 2000). During this process, the calculated displacements are high-pass filtered. For Waverider buoys, the resulting low-frequency cutoff is 0.037 Hz (Datawell BV 2019), which represents a nonnegligible proportion of the bandwidth where second-order subharmonics lie. Hence, this process filters out the majority of the second-order contribution to crest height for ocean waves, for which typical peak frequencies *f*_{p} lie at around 0.1 Hz and below, and the majority of second-order subharmonic energy lies at around 0–0.5*f*_{p}. Some more recent buoy designs also measure velocity through GPS-based Doppler velocimetry, which can improve low-frequency resolution (de Vries et al. 2003; Raghukumar et al. 2019). However, most off-the-shelf buoys measure displacements over a frequency band with a lower limit of around 0.037 Hz.

Figure 2 illustrates the effects of second-order nonlinearity and filtering on crest height statistics. We simulate 40 directionally spread linear wave trains with a duration of 1024 s based on a JONSWAP spectrum with randomly assigned phase and direction (these wave conditions correspond to the experiments we present later; see section 2 for details). This gives a sample of approximately 30 000 waves. The crest height distribution of these linear waves is presented in Fig. 2a, where each gray marker represents a single wave crest from our simulations. As is expected, these crests follow a Rayleigh distribution. We then calculate the corresponding second-order bound components for each wave train using exact multicomponent second-order theory for both Eulerian (Dalzell 1999) and Lagrangian (Herbers and Janssen 2016) frames of reference. The red and blue markers in Fig. 2 show second-order Eulerian and Lagrangian crest heights, respectively. Both predictions of crest height follow the second-order crest height distribution proposed by Forristall (2000) well, and there is no significant difference between the predicted behavior of the two types of measurement. In Fig. 2c, blue markers show the crest heights for the bandpass filtered (0.37*f*_{p}–6*f*_{p}) Lagrangian free surface elevation. After filtering, the Lagrangian crest height predictions follow the linear Rayleigh distribution and are essentially the same as the linear crests shown by the gray markers.

In previous work, we have experimentally examined the performance of a model wave-following buoy in steep directionally spread and crossing wave groups (McAllister and van den Bremer 2019). Under these conditions, we found that the buoy’s behavior was essentially purely Lagrangian, and we did not observe any significant effects associated with the inertia or buoyancy of the model buoy. In those experiments, focused wave groups were examined, as their form provides an approximation to extreme wave groups that may be encountered in the ocean. For a random linear process, such as dispersively focusing water waves, the shape of extreme events will tend to the autocorrelation function of the underlying power spectrum (Lindgren 1970; Boccotti 1983). This concept referred to as NewWave (Tromans et al. 1991) or quasi-determinism (Boccotti 1983; Fedele and Tayfun 2009) has been validated for Eulerian measurements (Santo et al. 2013) and buoy measurements made in shallow water (Whittaker et al. 2016).

In this paper, we experientially examine whether for stochastic irregular wave trains measured Eulerian and buoy (Lagrangian) crests height statistics and sea-state parameters are comparable. We create sea states with different degrees of directional spreading including crossing. We then investigate the average measured extreme wave form of the irregular waves we create to further assess the applicability of the quasi-deterministic or NewWave approach to measurements made by buoys. Finally, we propose a method for retrospectively correcting in situ crest height distributions produced by wave-following buoys that effectively filter out bound subharmonic waves.

This paper is laid out as follows. First, in section 2 we outline our experimental method. We then present the results of our experiments in section 3. We discuss how our findings may be used to reinterpret in situ buoy data in section 4, and draw conclusions in section 5.

## 2. Experimental method

### a. Experimental setup

The experiments presented herein were carried out in the FloWave Ocean Energy Research Facility at the University of Edinburgh. This is a circular wave basin, which is 2 m deep and surrounded by 168 actively absorbing force-feedback wavemakers, which allows for the creation of waves that travel in any direction.

Eulerian measurements of free surface elevation were made using resistive wave gauges, which were calibrated at the start of each day of testing. Eight wave gauges were installed in the tank in the positions listed in Table 1 and shown in the schematic Fig. 3. We define the coordinate system where waves propagate in the positive *x* direction, the angle *θ* is measured in the clockwise direction from the positive *x* axis, and the position *x* = 0, *y* = 0 lies at the center of wave tank.

Gauge array, mean buoy positions, and the difference between the two.

To perform a direct comparison between Eulerian and buoy measurements, each set of experiments we carried out was repeated with the wave gauges raised and model buoys placed in the same positions. Seven model buoys were located at the positions listed in Table 1 and indicated by black markers on the schematic Fig. 3. The buoy positions listed in Table 1 represent the mean position at which the buoys were located during experiments.

#### Model buoys and motion tracking

We measure the position of our model buoys using a Qualisys motion tracking system, consisting of eight calibrated infrared cameras located around the circumference of the tank, which simultaneously track the position of the buoys. To aid their detection, the spherical model buoys were covered in a coating that reflects the specific band of infrared light emitted and recorded by the cameras. The buoy’s position in each set of images is then triangulated to measure its global position within the tank.

To measure free surface elevation, wave-following buoys first measure acceleration, which is then integrated twice to give displacement. These displacements are projected onto the coordinate system of heave, north, and east displacements. In our experiments, we effectively assume that there is no error in this process [see van Essen et al. (2018) for a discussion of this assumption], and consider the position of our model buoy to be what a wave buoy in the ocean would measure.

The model buoys we use have a diameter *D* = 0.07 m and were designed to have a density approximately half that of water. This size of the buoys was chosen to provide the best possible representation of wave-following buoy operating in the ocean, in other words, to maintain similar scaling parameters *D*/*λ*_{0} and *a*_{0}/*D*, within the constraints of practicality in terms of measuring the position of the buoy. The ratio of buoy diameter to characteristic wavelength *D*/*λ*_{0} for both full scale and our model buoys lie well below the limit *D*/*λ*_{0} ≪ 1, meaning the effects of diffraction are not important. However, when considering the importance of inertial effects the ratio of characteristic wave amplitude to diameter *a*_{0}/*D* of our buoy is somewhat lower than that of a real buoy. This means that our model may be slightly too large and more likely to exhibit inertial effects. However, in McAllister and van den Bremer (2019), which contains a detailed discussion of these scaling considerations, the same size of buoy did not show any observable inertial effects under similar conditions.

To maintain positioning at the desired location, each buoy was fixed to the floor of the tank using an auxiliary mooring. These simple taut moorings were designed to be highly flexible with the intention of having minimal effect measured free surface elevation. Previous experiments showed that this flexible auxiliary mooring configuration had no observable effect on the measured Lagrangian free surface elevation for individual wave groups (McAllister and van den Bremer 2019). We emphasize that the effect of a mooring on buoy motion in a realistic configuration is not the objective of the present study. However, for long-duration irregular wave trains some mooring effects are unavoidable as a result of Lagrangian-mean displacements that will accumulate over time.

### b. Matrix of experiments and input parameters

*β*is a scaling parameter, which was selected to give the desired significant wave height

*H*

_{s}. For all of our experiments, the peak frequency

*f*

_{p}was 0.6 Hz; this frequency was chosen to optimize wave generation and absorption by the wavemakers, and ensure that resulting waves propagate on deep water (

*k*

_{p}

*d*= 2.91). The shape parameters were defined using typical values:

*n*= −5,

*γ*= 3.3, and

*σ*(

*f*≤

*f*

_{p}) = 0.07 and

*σ*(

*f*>

*f*

_{p}) = 0.09.

To measure the true spectrum of a given sea state in the ocean, the duration of the observation *T* would have to be infinitely long, hence any finite duration realization of this spectrum will not have the exact form of the underlying spectrum. To correctly reproduce this natural variability when prescribing amplitudes to discrete wave components in a wave tank, for each discrete amplitude, a value should be drawn from a normal distribution with variance *S*(*f*). As we are interested in changes to spectral shape between equivalent Eulerian and buoy measurements, we define the amplitudes at each frequency deterministically

*T*=1024 s, which gives a frequency resolution of

*δf*= 1/1024 Hz. The wavemakers generate in the frequency band 0.2–2 Hz, which allows for the creation of 1844 discrete wave components. To ensure that the waves we create within the tank are ergodic and spatially homogeneous using this limited number of waves, we implement directional spreading using a single summation method (Miles and Funke 1989),

**k**

_{n}= [

*k*

_{n}cos(

*θ*

_{n}),

*k*

_{n}(

*θ*

_{n})], and the absolute wavenumber

*k*

_{n}= |

**k**

_{n}| is related to frequency

*f*

_{n}=

*ω*

_{n}/2

*π*by the linear dispersion relation

*μ*

_{n}is sampled from a uniform distribution between 0 and 2

*π*. The desired directional spreading distribution is achieved by sampling the direction of each discrete wave component

*θ*

_{n}from a probability distribution of the same form. In our experiments, we aim to produce a wrapped normal spreading distribution centered around

*θ*=

*θ*

_{0},

*θ*

_{n}is randomly sampled from a normal distribution with characteristic spreading width

*σ*

_{θ}. To generate crossing sea states, we sample

*θ*

_{n}from two superimposed normal distributions with mean directions separated by Δ

*θ*.

We carry out experiments with input significant wave height *H*_{s} = 0.225 m for four different directional conditions (see Table 2): for following or unimodal conditions with characteristic spreading widths of *σ*_{θ} = 10°, 20°, and 30°, and for crossing conditions with characteristic spreading width *σ*_{θ} = 20° and crossing angle Δ*θ* = 90°. For each of these conditions, three repeat experiments with different random phases *μ*_{n} and directions *θ*_{n} were carried out. To investigate the importance of steepness, three additional experiments were carried out for each following sea state at a reduced input significant wave height *H*_{s} = 0.175 m, using the same random phases and directions as realizations carried out at *H*_{s} = 0.225 m.

Test matrix.

### c. Estimation of sea-state parameters

Measurements made by buoys are often used to compute spectral parameters, such as significant wave height *H*_{s} and peak period *T*_{p}. These summary statistics are commonly used to calibrate phase-averaged ocean forecast and hindcast models (Janssen et al. 1997; Bidlot et al. 2002) and to characterize environmental conditions across the oceans (Hasselmann et al. 1980; Pontes 1998; Mork et al. 2010; Fedele and Arena 2010; Lenee-Bluhm et al. 2011). To understand actual conditions achieved in the tank, we estimate the spectral parameters of the free surface elevation measured by the gauges. We also carry this out for our buoy measurements to examine if any difference exist between the two measurements types. We use the superscript

*m*

_{0}is the zeroth moment of the measured spectrum. The peak period

The time-domain parameters skewness *λ*_{3} and kurtosis *λ*_{4} are often used to quantify the nonlinearity of a given sea state and the likelihood of extreme waves. We also estimate these two parameters using both our Eulerian and buoy measurements to establish if any systematic differences exist between the two types of measurement.

## 3. Results

### a. Filtering

The signal processing that is carried out when buoys convert acceleration and velocity measurements to displacements, results in the high-pass filtering of the free surface elevation that they measure. For example, the resulting operational frequency band of a Datawell Waverider buoy is 0.037–0.64 Hz. To replicate this with our experimental results, we filter our measured free surface elevations over the band 0.37*f*_{p} < *f* < 6*f*_{p}. This is based on the assumption that the average peak frequency of extreme ocean waves will be approximately 0.1 Hz. Storm waves will often have frequencies lower than 0.1 Hz; for such waves the effects of filtering will be even more pronounced, making this choice of frequency conservative.

### b. Sea-state parameters

In Fig. 4, we compare the sea-state parameters measured by our model buoys to those measured by the array of Eulerian wave gauges. For each individual experiment, we take the mean across all of the available measurements (7 buoys and 8 gauges). In this figure, the black markers show parameters estimated using the unprocessed buoy data, and the red markers show the same parameters estimated using the buoy measurements after filtering. Error bars in these figures represent two standard deviations of the estimates made from the different measurements (7 buoys and 8 gauges) used to generate the mean values (see appendix C for a full discussion of sources of measurement error). We only provide error bars for the parameters estimated for the unfiltered data (black markers) and not the filtered data (red markers) to aid clarity.

- Significant wave height
*H*_{s}: The significant wave height measured by our model buoys (Fig. 4a) is consistently higher than the true value measured by the wave gauges. On average, the values estimated using the buoy measurements are 2.8% larger (≈5 mm at laboratory scale). This difference reduces slightly as the degree of spreading is increased; therefore, it may be a result of the forward transport of buoys induced by large waves, resulting in the observation of larger crests as the waves focus. Filtering the free surface elevation reduces this difference, as filtering the measurements essentially removes energy and reduces*m*_{0}. - Peak period
*T*_{p}: The peak period measured by our model buoys (Fig. 4b) is slightly larger than the true period measured by the wave gauges. However, the differences we observe are very small, on average 1%, and not always significant. An increase in period is expected between Lagrangian and Eulerian frames of reference (Longuet-Higgins 1986). Stokes drift causes a Doppler shift in the period measured by a Lagrangian observer, which is proportional to the steepness of the waves measured and reduces with increasing directional spreading. There is no observable trend between the degree of spreading and the measured Lagrangian period. - JONSWAP parameters
*n*and*γ*: The results of fitting a JONSWAP spectrum to our measurements, namely the parameters${n}^{\mathrm{\star}}$ (Fig. 4c) and${\gamma}^{\mathrm{\star}}$ (Fig. 4d) vary quite significantly across our buoys and gauges, as illustrated by the large error bars. Thus, for both parameters, it is hard to infer any trends with confidence. Ideally, to fit the exponent*n*to the tail of the spectrum, several decades of data would be used, whereas we use the relatively short frequency range$2{f}_{p}^{\mathrm{\star}}<f<6{f}_{p}$ . We fit over this band, as it represents the band of frequencies measured by a Waverider buoy, assuming a full-scale peak frequency of approximately 0.1 Hz. The slope of the high-frequency tail of the spectrum is steeper for the buoy measurements; this may be caused by the attenuation of high-frequency waves. At around 6*f*_{p}, the ratio of wavenumber to buoy radius*Dk*/2 ≈ 1. At values of*Dk*/2 greater than unity, motion will be attenuated by the buoy. As the parameter${n}^{\mathrm{\star}}$ describes the tail of the spectrum, it is not affected by high-pass filtering. - Skewness
*λ*_{3}: The skewness of the free surface elevation measured by our model buoys and wave gauges (Fig. 4e) agree reasonably well. For low degrees of spreading, the free surface elevation measured by the buoys exhibits slightly increased skewness, which is consistent with the hypothesis in item i, that forward transport by large waves may cause a buoy to spend more time at the crest of waves. Filtering the buoy measurements significantly reduces the skewness of the free surface elevation, bringing it close to zero, as subharmonic bound waves are removed. - Kurtosis
*λ*_{4}: The kurtosis of the free surface elevation measured by our model buoys (Fig. 4f) is slightly larger than that measured by the wave gauges. This difference is larger at low degrees of spreading. Filtering the buoy measurements also slightly reduces their kurtosis.

### c. Crest height distributions

Ultimately, it is wave crests and their likelihood that inform the design of fixed offshore structures. We examine the distribution of wave crests measured during our experiments in Fig. 5. Here, a wave crest is defined as the maximum value of free surface elevation between successive up- and downcrossings. In Fig. 5, we present crest heights measured by buoy and gauge pair 3, located at the center of the tank (*x* = 0, *y* = 0). In each subfigure, data measured over three realizations of the same input spreading conditions are presented. The same input significant wave height and peak period were used for all four sets of experiments with different spreading conditions. The distribution of crest heights we measure for both buoys and gauges is slightly under predicted by the second-order Forristall distribution (2). The waves we create are very steep and at times breaking. Similar deviations from second-order theory are also observed for steep sea states in Onorato et al. (2009) and Latheef and Swan (2013). Moving from left to right in Fig. 5, as the degree of spreading is increased, the measured crest heights tend toward the second-order distribution. Between the crest heights measured by the buoy and the gauge, there are some differences. For the gauge–buoy pair we present, the crests measured by the wave gauge appear to be slightly larger. However, the difference is not significant. In Fig. 6, we combine crest heights measured across all 7 buoy and gauge pairs; increasing the number of observations removes any noticeable difference between the two sets of measurements. It should be noted that combining concurrent measurements will affect the likelihood of observing a given crest height, as although the observations are made at different locations, they are correlated. Thus, we only include theoretical distributions in Fig. 6 for reference.

### d. Wave height distributions

It is well established that crest-to-trough wave height is not affected by second-order nonlinearity (Longuet-Higgins 1980). Therefore, in principle, wave heights measured by buoys and gauges should be the same even after filtering. We examine the distribution of wave heights measured during our experiments in Fig. 7. Here, wave height is defined using the maximum and minimum values of free surface elevation between successive upcrossings. In Fig. 7, we present wave heights measured by buoy and gauge pair number 3, located at the center of the tank (*x* = 0, *y* = 0). In each subfigure, data measured over three realizations of the same input spreading conditions are presented. The same input significant wave height and peak period were used for all four sets of experiments.

### e. Average wave form

The theory of quasi-determinism that underpins nonlinear crest and wave height statistics implies that the average wave form of extreme waves within given sea state follows the form of the autocorrelation function of the underlying spectrum (Lindgren 1970; Boccotti 1983). In other words, the average form of extreme waves occurring in a random sea state may be approximated by a focused wave group based on the underlying spectrum. To assess the validity of this assumption for buoy measurements, we examine the average wave form, and corresponding second-order bound components, of the largest waves examined during our experiments.

The average linear wave form (Figs. 8a–f), subharmonics (Figs. 8g–l), and superharmonics (Figs. 8m–r) extracted from our measurements are shown by the black lines in Fig. 8. The blue and red lines in Figs. 8a–f show the scaled autocorrelation function of each sea state, and the sub- and superharmonics based on the scaled autocorrelation function calculated using Dalzell (1999) (gauges) and Herbers and Janssen (2016) (buoys) are shown in Figs. 8g–l and 8m–r, respectively. We introduce the notation *η*_{T}, to denote theoretical predictions based upon the autocorrelation function of surface elevation.

For both buoy and gauge measurements, the scaled autocorrelation function provides a good approximation to the average linear wave form of extreme events. Both sets of measurements exhibit slight horizontal asymmetry, where the wave preceding the maximum crest is slightly smaller than the one proceeding it. The period and crest-front steepness of the average wave form measured by the buoys is slightly reduced in comparison to the gauge measurements. The averaged second-order bound waves follow the theoretical predictions well, considering the approximate extraction method we use. For the buoy measurements, we observe the formation of a large setup of the subharmonics and cancellation of the superharmonics as predicted for Lagrangian motion. For our Eulerian gauges, we observe a small setdown of the subharmonics and large superharmonics. Hence, the effects illustrated deterministically in McAllister and van den Bremer (2019), can be readily applied to extreme waves measured within random sea states by both buoys and Eulerian devices.

## 4. Buoy filtering reconstruction method

The experiments presented herein have illustrated that a wave-following buoy’s motion itself should not lead to the systematic underestimation of wave crest heights that is often observed for in situ measurements (see Fig. 1). We do not consider how measurements may be affected by a mooring, which may play a significant role and will be investigated in future work. We note that, if the effects of a realistic mooring were to influence the measurement of extreme crest, one might expect this to affect the distribution wave heights too. In light of our observations, we believe that the most likely cause of this systematic underestimation is filtering, as also suggested in Marthinsen and Winterstein (1992), Seymour and Castel (1998), Prevosto et al. (2000), and Forristall (2000).

Seymour and Castel (1998) present a simple method by which the true maximum crest height measured by a buoy may be estimated. This method involves a process of reverse filtering, then performing an approximate conversion to a Eulerian frame of reference. After applying this approximate correction method to in situ buoy measurements, they suggest that for sea states of steepness *k*_{p}*H*_{s} > 0.1, a 20% correction should be applied to maximum crest height *η*_{c}/*H*_{s}.

We have illustrated that the second-order contribution to crest height, or to the skewness of the free surface elevation, is the same for Lagrangian and Eulerian observers, as highlighted by Srokosz and Longuet-Higgins (1986). Therefore, it is unnecessary to convert buoy records into an Eulerian frame of reference to correct crest heights, as suggested by Seymour and Castel (1998). Moreover, reverse filtering in the frequency domain may not be possible for surface elevation records at frequencies where no phase or amplitude information exists.

*a*

_{0}and

*k*

_{0}as the characteristic amplitude and wavenumber of the linear wave group at focus. In deep water, the Eulerian subharmonic setdown contribution to crest height

*k*

_{0}

*d*= 3 in unidirectional conditions and will be smaller for large directional spreading). It should be noted that

*σ*

_{θ}which we use to define the spreading width of our experiments in (6) corresponds to width of the energy spectrum, whereas

*σ*

_{θ}in (10) corresponds to the width of the amplitude distribution. Under the framework of quasi-determinism extreme waves will have an amplitude distribution based on the underlying energy spectrum. Hence, in this case, the definitions of

*σ*

_{θ}in both equations are consistent.

*η*

_{c}are approximately linear (

*η*

_{c}and the wavenumber

*k*

_{p}, effectively making the assumption each crest is a local maximum (cf. theory of quasi-determinism). The parameter

*ε*

_{hpf}is an ad hoc correction factor that accounts the for relative frequency at which the measurements are filtered, as this affects the amount of second-order contribution to crest height that is removed. For extratropical storms, the majority of subharmonic energy lies between 0 and 0.5

*f*

_{p}[see McAllister et al. (2019) for an example where this is not the case]. However, the total second-order subharmonic contribution to crest height extends to frequencies higher than 0.5

*f*

_{p}. For extreme waves based on the JONSWAP spectra we examine, this frequency band ranges from 0 to approximately 2

*f*

_{p}. Therefore, the exact amount of second-order subharmonic energy removed by filtering and the required correction depends on the relative frequency cutoff frequency

*f*

_{c}/

*f*

_{p}. We define the parameter

*ε*

_{hpf}empirically using exact second-order theory for a Lagrangian observer, where

*ε*

_{hpf}represents the ratio energy being removed from the data to the total second-order subharmonic contribution, and estimate the relationship,

*γ*≈ 3.3) (see appendix D for further details).

Figure 9 shows filtered *η*_{Filt,c} (Figs. 9a,c), and then corrected *η*_{Corr,c} (Figs. 9b,d) buoy measurements from our experiments using the proposed filtering reconstruction method. In this figure, measured wave crests are taken from all seven buoys and all three realizations of each sea state. The approximate correction works well, and the corrected measurements match the original unfiltered data for both following (Figs. 9a,b) and crossing (Figs. 9c,d) sea states.

## 5. Conclusions

Wave buoys are commonly considered to underestimate and effectively linearize extreme crest heights. We examine the nature of this phenomenon. Through stochastic calculations based on exact second-order theory (Dalzell 1999; Herbers and Janssen 2016), we show that crest heights measured by Eulerian and Lagrangian (buoy) sensors should not differ significantly. To confirm this is also the case for a partially submerged wave-following buoy in steep directionally spread irregular waves, we have carried out a new series of experiments.

In general, the wave-averaged sea-state parameters measured by our buoys do not vary significantly from the true values measured by the Eulerian wave gauges. The significant wave height *H*_{s} and peak period *T*_{p} measured by the buoys are approximately 2.8% and 1% larger, respectively. Filtering the buoy measurements, in a manner representative of the process carried out by buoys when converting measured accelerations to displacements, slightly reduces the measured significant wave height and kurtosis *λ*_{4}, and quite significantly affects the skewness *λ*_{3} of the measurements.

The crest and wave height distributions measured by both our buoys and gauges follow our theoretical predictions and do not exhibit significant differences. The average linear wave form of the extreme waves in our measurements is well approximated by the autocorrelation function of their underlying spectra. The corresponding second-order sub- and superharmonics are also well approximated using second-order calculations based on these autocorrelation functions.

We propose an approximate means of correcting crest heights measured by buoys that have been filtered. We apply this approximate correction to our experiments and show it is capable of retrospectively accounting for the effects of filtering and correcting in situ buoy measurements, both in directionally spread and crossing sea states.

This project was funded by the EPSRC and Wave Energy Scotland (grant no. 24840078 and 24841554). TSvdB was supported by a Royal Academy of Engineering Research Fellowship. The authors thank Dr. S. Draycott and Mr. E. Nixon, Dr. R. Gabl, and Dr. T. Davey at the FloWave Ocean Energy Research Facility for their assistance in planning and carrying out the experiments.

# APPENDIX A

## Buoy Data

The data in Fig. 1 were obtained from a single buoy within CDIP’s network of buoys. The buoy’s mooring (33°13′29″N, 119°52′54″W) lies to the west of San Nicholas Island, which is located 61 miles from the Californian Coast. This particular buoy was chosen, as it is situated in deep water (*d* = 274 m and *k*_{p}*d* ≈ 10 with *k*_{p} as the peak wavenumber) far from the coast, in an area exposed to large waves. Of the available data, five 30-min duration measurements were selected (see Table A1), giving a sample of *N* = 1066 waves and hence similar probability levels to the waves created in our experiments. Measurements were first selected based on their steepness, then their spectral characteristics in both frequency and direction (peak period *T*_{p} and direction *θ*_{p}) to ensure they individual wave crests were samples from similar sea states.

CDIP data from station 67 (San Nicholas Island), 30-min duration measurements recorded by a Waverider buoy on a depth of 274 m.

# APPENDIX B

## Second-Order Exceedance Probability

*α*and

*β*in (2) are related to the steepness parameter,

# APPENDIX C

## Measurement Error

To quantify the sources of error affecting our comparison between Eulerian and buoy measurements, we examine the role of the various sources of error and uncertainty involved in our experiments.

For Eulerian measurements, the leading source of error is associated with wave gauge calibration and is approximately 0.4 mm (two standard deviations). Of the eight cameras, each possible pairing is used to measure the buoys’ location; the residual values measure the time varying error in the position of the buoys and ranged from ±0.5 to ±1.5 mm.

Additional to sources of measurement error, spatial inhomogeneity and small variations in the construction of the buoys may also introduce error. To estimate these sources of error, we compute the energy measured at each wave gauge and for each buoy. We then estimate the variation in energy across the tank as the normalized standard deviation of energy across all gauges or buoys for a single experiment. The variation in energy, averaged over all of our experiments, measured across the tank by the wave gauges is 3.4%, and between the model buoys is 5.4% (two standard deviations). To draw comparison between gauge and buoy pairs, it is important that they are located at the same position, and any misalignment may also cause error. However, as the buoys have a degree of freedom to move horizontally some misalignment during experiments is unavoidable. The error between the mean position of our model buoys and wave gauges is given in Table 1 and ranges from 0.11 to 0.1 m.

# APPENDIX D

## Relative Filtering Frequency Parameter *ε*_{hpf}

Depending on the peak frequency *f*_{p} at which a given sea state occurs, the relative high-pass-filter frequency cutoff *f*_{c}/*f*_{p} changes. This relative cutoff frequency determines how much of the second-order subharmonic contribution to crest height is removed by filtering; that is, if *f*_{c} ≪ *f*_{p} the effect of filtering will be less significant. Hence, the retrospective correction we make to measured crest heights must account for this. Therefore, we seek a correction factor to account for the amount of second-order subharmonic contribution to crest height that is filtered out for a given ratio of *f*_{c}/*f*_{p}. To do so we use exact Lagrangian second-order theory (Herbers and Janssen 2016).

*γ*values of 1, 3.3, and 6.6 (dotted, solid, and dashed lines, respectively). We also illustrate the effects of directional spreading, for following-sea (black and gray lines) and crossing-sea conditions (red lines). When normalized directional spreading does not have a strong effect on the shape of the amplitude distribution. When plotted on a logarithmic scale, it becomes clear that the subharmonic amplitude distributions can be approximated by the form

*y*= exp(

*ax*+

*b*). This simple approximate relation is less accurate for very narrow-banded conditions (large

*γ*). For broadbanded conditions (

*γ*≈ 3.3), this approximate relation can be used to estimate the ratio of subharmonic contribution to crest height filtered out to the total subharmonic contribution to crest height,

*a*is derived from the data in Fig. D1a:

*a*≈ −2.0. Figure D1b illustrates this process schematically.

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