Estimating ocean wave directional spreading from an Eulerian surface elevation time history

T. A. A. Adcock, P. H. Taylor

Abstract

The directional spreading of sea states is an important design parameter in offshore engineering. Wave directionality affects the resulting wave kinematics, which affects the forces exerted on offshore structures. In this paper, we develop a method for estimating the amount of spreading, when the only information available is the time history of free surface elevation at a single point in space.

We do this by predicting the second-order bound waves that occur at the difference in frequency of two freely propagating waves. The magnitude of these second-order bound waves is a function of the angle between the interacting waves. Thus, it is possible to infer some information about spreading from a single-point time history.

We demonstrate that this approach works for wave groups in a fully nonlinear numerical wave tank. We create a synthetic random sea state and introduce noise into the analysis and thus show that our approach is robust and insensitive to noise, even with a signal-to-noise ratio of unity in the difference waves. This approach is also applied to random waves in a physical wave tank where spreading was directly measured and also to a storm recorded in the North Sea. In all cases, we find our estimate of spreading is in good agreement with other measurements.

1. Introduction

Directional spreading is an important parameter in the accurate description of a sea state. The degree of directional spreading affects the forces on offshore structures (Bea et al. 1991) and the height of extreme waves (Jonathan & Taylor 1997; Johannessen & Swan 2001) and as such is of importance to offshore engineers.

Unfortunately, measuring directionality is more complicated than merely measuring the elevation of the free surface motion with time. Much field data consist of point measurements of the free surface without any information about directionality recorded. One instance of this is the much analysed Draupner wave where the nearest measurement of spreading known to the authors was 180 km away.  Walker et al. (2005) proposed a method of estimating directional spreading by using the ‘sum’ term of Stokes-type second-order bound waves. Our work builds on this, but uses the low-frequency ‘difference’ terms to infer the directional spreading.

First, the theoretical magnitude of the difference waves is shown to be dependent on the directional spreading of the freely propagating waves. We demonstrate that this can be used to estimate spreading in a fully nonlinear numerical wave tank.

To demonstrate the robustness of our approach, synthetic random data containing noise are created. We also apply our method to random wave data from a physical wave basin. Finally, we show that our analysis is consistent with other spreading measurements for severe storm waves recorded at the Draupner platform in the North Sea.

It is worth noting that, while the amount of directional spreading may be inferred from the nonlinear wave–wave interactions, it is, of course, not possible to determine the mean wave direction.

2. Directional spreading

A directional sea state may be described by a frequency-directional spectrum S(f,μ). We assume that this is the product of a power spectrum and a directional distribution, i.e.Embedded Image 2.1

In this work, for simplicity, we make the assumption that the directional distribution is independent of frequency. However, more detailed analyses such as Mitsuyasu et al. (1975) and Ewans (1998) show that spreading is a function of frequency. Thus, our estimates are of the weighted average spreading.

Here, we model the directional distribution using a wrapped normal spreading functionEmbedded Image 2.2 where μ is the angle relative to the mean wave direction. This spreading function is theoretically not bounded in angle and will ‘wrap around’, as noted by Tucker & Pitt (2001). In this paper, the largest value of spreading we consider is σ=35° and for computational reasons we curtail this spreading at μ=±100°.

In this work, there is no attempt to estimate directional spreading on a wave-by-wave basis. Instead, we aim to estimate the degree of directional spreading occurring in a sea state (typically taken to be 3 hours in duration), and comparable to what would be recorded from a directional wave buoy, to be captured as the value of a single parameter. We choose, for convenience, a Gaussian spreading model but other forms such as Embedded Image, as used for the experiments discussed in §6, could be used to equal effect. In terms of a single parameter to describe spreading, all such forms are equivalent. As a consequence of this ‘broad-brush’ approach, the method breaks down for crossing seas such as might occur as the combination of a local wind–sea and swell. However, a study of worldwide field measurements of directional spreading in extra-tropical storms by Forristall & Ewans (1998) shows that the representation of real storms as unimodal sea states with relatively small directional spreading is a reasonable assumption for engineering designs.

3. Second-order bound waves

(a) Theory

Freely propagating water waves will interact to produce ‘bound’ components that do not move with their own dynamics, but are merely functions of the freely propagating waves. To second-order, these waves will occur at the sum and difference of the frequencies of the freely propagating waves. A typical spectrum based on 20 min of data recorded at the Draupner platform in the North Sea (see §7) is shown in figure 1. Here the largest components, predominately linear terms, occur at frequencies above 0.04 Hz. In this paper, we concentrate on the second-order difference terms, here below 0.04 Hz. These are small but separable by simple filtering in frequency. The freely propagating waves may be written asEmbedded Image 3.1 where an is the Fourier coefficient, N the number of Fourier components used, andEmbedded Image 3.2 where ξ gives the relative phase of the component.

Figure 1.

The spectrum of a typical 20 min time history recorded at the Draupner platform. The data are slightly smoothed for clarity.

The waves in equation (3.1) will interact to give a second-order record given byEmbedded Image 3.3 whereEmbedded Image 3.4 where κ+ and κ are the interaction kernels given in equation (3.5). The kernels for deep water were given by Longuet-Higgins (1962) and for finite depth by Dean & Sharma (1981) (see also Dalzell (1999) and Forristall (2000), who both give corrections to the finite depth case)Embedded Image 3.5

The wavenumber |k| and natural frequency ω are related by the linear dispersion relation. The angle between the interacting components is θ. The relative variation in the kernels with angle and frequency is shown in figures 2 and 3.

Figure 2.

The variation in κ+(θ)/κ+(0) for kmd=2 and ωm=1 rad s−1.

Figure 3.

The variation in κ(θ)/κ(0) for kmd=2 and ωm=1 rad s−1.

It should be noted that, as the ratio of ωn/ωm increases, the magnitude of the kernels also increases. The kernels were derived on the assumption that the second-order term is small compared with the linear wave and that the ratio between the components is small; this assumption becomes invalid at higher frequency ratios. This work uses only interactions where the frequency ratio is less than 3. The interaction kernels are functions of frequency, water depth and the angle between the free components. Thus, the second-order terms are functions of the directional spreading of the free components.

This paper uses the difference terms to estimate spreading. To show how the difference term varies under directionally spread waves, we use a linear NewWave group (see Tromans et al. 1991). The wavegroup has unit amplitude when all components are at focus and is based on a JONSWAP spectrum with Tz=12 s and a peak enhancement of 3.3. The free waves are shown in figure 4.

Figure 4.

A linear NewWave group with unit amplitude.

The difference terms may be calculated from this for different water depths and spreading angles. The wavenumber, k, used to characterize depth, corresponds to the mean period. The variation with spreading angle and depth is shown in figure 5. It can be seen in figure 5 that these waves are small compared with the free waves. The magnitude of the bound waves, of course, varies with the square of the linear wave.

Figure 5.

The difference term under a NewWave wavegroup. (a) Variation with spreading (kd=2). Dashed line, σ=0°; thick line, σ=15°; thin line, σ=30°. (b) Variation with depth (σ=15°). Dashed line, kd=1; thick line, kd=2; thin line, Embedded Image.

(b) Fully nonlinear simulations

 Gibbs & Taylor (2005) carried out fully nonlinear simulations of the evolution of directionally spread NewWave groups using the numerical scheme developed by Bateman et al. (2001). The wavegroups were based on a narrow-banded Gaussian spectrum in deep water and a mean period of 12 s.

The odd and even harmonics can be separated by running simulations that are π out of phase (equivalent to replacing an by −an in equation (3.1)) but have the same linear envelope, producing ‘crest’ and ‘trough’ focused events. The odd and even harmonics are then given byEmbedded Image 3.6

The inversion of the wave group permits the straightforward extraction of the spectral contributions from different orders, because the nonlinear contributions to the surface profile scale as powers of the linear components an. As an example, the third-order (cubic) terms are inverted by inverting the linear term (Embedded Image), whereas the second-order (quadratic) terms remain the same (Embedded Image). The free waves can then be extracted by filtering in the frequency domain.  Figure 6 shows the free waves for a run where the linear focused steepness was ak=0.1. Figure 7 shows the second-order difference waves for this run, along with the bound waves calculated from the the linearized data using equation (3.4) assuming a spreading of 15°.

Figure 6.

Free waves extracted from fully nonlinear simulation for the ak=0.1 case.

Figure 7.

Comparison of fully nonlinear bound waves from fully nonlinear simulation for the ak=0.1 case, with those predicted from the free waves assuming σ=15°. Dotted line represents fully nonlinear simulation, whereas the solid line represents calculation using interaction kernels.

On a linear basis, this group would have had a 15° spread at focus. Because of the nonlinear changes in group structure as the group converges, Gibbs & Taylor (2005) estimated the local spreading to 14.2° at focus, based on the analysis of the spatial shape of the wave group. It can be seen that the time history of the second-order difference term using 15° is very slightly too small, as would be expected from the rather weak nonlinear dynamics occurring for this group.

Gibbs & Taylor (2005) carried out other simulations using steeper wavegroups with the same initial directional spreading. These fully nonlinear simulations examined the complete spatio-temporal evolution of nonlinear groups. As the steeper groups focused, there were substantial changes in the group structure, such as a reduction of directional spreading below that of the initial conditions. There was also a contraction of the steeper wave groups along the mean wave direction substantially greater than that predicted by linear focusing. Each simulation started from a linear wave group with a Gaussian spectrum and directional spreading of 15° with the phasing set so that the group would focus after 20 periods, assuming linear behaviour. The actual spreading for various cases, based on the group structure at focus, is given in table 1. We also estimate the second-order difference term for these data for various spreading angles. Table 1 also shows the estimated spreading angle at which the theoretical difference time history matches that in the numerical simulation.

View this table:
Table 1.

Spreading estimates for different input steepness of wave group.

The disagreement for the more nonlinear cases is partly due to inaccuracy in extracting the linear data. However, the two spreading parameters compared here are also significantly different. Gibbs & Taylor (2005) calculated spreading by considering the spatial structure of the whole group over the entire free surface at the instant of temporal focus, whereas analysis of the difference term, as here, uses the complete time history at the spatial focus point—here an estimate over approximately eight wave periods. Thus, some discrepancy between the small spreading angles is to be expected for the most nonlinear and fast evolving cases. However, the reduction of the directional spreading over time as the groups focus is identified by both methods.

4. Method for analysing non-deterministic data

(a) Linearizing data

This work relies on being able to separate accurately a wave record into the different components in equation (3.3). Walker et al. (2005) introduced a method of linearizing datasets so as to extract the freely propagating, or linear, waves.

First the data are filtered to remove all low-frequency waves and very high-frequency waves. The assumption is then made that the remaining waves, ηfiltered, are dominated by free waves with a small contribution from the second-order sum waves. The next step uses the Hilbert transform, which introduces a π/2 phase shift into a signal. For a single-frequency signal given byEmbedded Image 4.1 its Hilbert transform isEmbedded Image 4.2 The ‘sum’ term for the signal given in equation (4.1) will occur at twice the original frequency, the location in frequency of which can also be found from the Hilbert transform,Embedded Image 4.3

For irregular seas, which are not too broadbanded, the shape of the sum term is approximatelyEmbedded Image 4.4 where K is a scalar. Thus the free waves are given byEmbedded Image 4.5 The value of K is found by searching for the value of K that sets the skewness of the free waves, ηfree, to zero. This approach works equally well for unidirectional or directional spread waves. Taylor et al. (2005) showed that this Hilbert-based approach accurately reproduces the second-order sum waves for relatively broadbanded wave groups measured in a physical wave tank.

If the second-order sum components are a sensitive function of directional spreading (figure 2), why analyse the smaller difference terms? The reason is the frequency range of the various terms: the upper tail of the linear components overlaps in frequency with the sum term making accurate separation rather difficult. In contrast, the difference terms are small but measurable at frequencies well below any significant linear term and so may be separated by simple filtering, as is clear from figure 1.  

(b) Analysis of results

Once the free waves have been extracted from the data, the difference terms may be calculated using equation (3.4) for a given model of spreading. Initially, a value is assumed for the root mean square spreading parameter. Each frequency component in the linearized signal is assumed to be the sum of components all with the same frequency but spread around a single mean direction with the assumed wrapped normal distribution (we discretize the wrapped normal distribution into 30 components). Then a double sum over direction and frequency is performed (equation (3.4)) to predict the long-wave component. Various different spreading values are tried and the results compared with the low-frequency waves in the original signal.

The comparison is carried out by identical low-pass filtering of both the calculated difference terms and the original data, so that no free waves from the linear range remain in the latter (see figure 1). Filtering is also done to remove any very long-period waves from the data (typically periods over 4 min at field scale). The two datasets are then subtracted. The Euclidian norm is then taken of this vector before it is non-dimensionalized as shown in equation (4.6) to find the long-wave discrepancy,Embedded Image 4.6

In the absence of free long waves, the linear wave field can be assumed to consist of a sum of many linear components of different frequency, with different phase and different direction of motion. Our analysis accounts for the random amplitudes and phases of these components by Fourier decomposition. However, it cannot include the directional information for each component because this is unavailable in a single-point time history of surface elevation. We simply assume that each component has exactly the same spreading distribution around the same mean direction and estimate what second-order difference signal would be expected.

We seek the best possible correspondence between the measured long waves and those predicted using second-order theory based on the measured linear components. As no wave-by-wave estimation of spreading is performed, we seek only an optimum in the goodness of fit based on a single averaged value of spreading over the entire record. Various other measures of this fit were tried. While many of these produced numerically lower values for the discrepancy overall, all gave more erratic results for the position of the minima of the discrepancy curves when trialled with noisy synthetic data and wave-basin data.

The matching process is also complicated by noise in the data. Both in laboratory experiments with random waves and in the open ocean, the wave fields are relatively very noisy at low frequency. Bound long waves arising from second-order difference interactions with wave trains are small—typically smaller than approximately 5 per cent of the overall wave height. On the open sea, the measured low-frequency waves will be a combination of the bound waves used in this analysis, and free waves uncorrelated with the local linear structure of the wave group. This noise will arise from any process that disrupts wave groups. As an example, consider wave groups incident on a distant coastline. These have bound long waves that are amplified as the groups shoal and then released as free waves to propagate offshore when the groups break. More local and incremental release of long-wave components would occur from intermittent breaking on deeper water close to the measurement point. Thus, every breaking wave event in the entire sea state is likely to be a weak source of free long waves. These processes, both local and far away, would lead to a relatively high background level of free long components, within which the bound structure we seek is embedded.

In laboratory basins and flumes, the absorption of the intended linear wave components is relatively straightforward, either by active feedback control on the paddles or passively. In contrast, long-wave components (including those related to mass transport—Stokes drift) are much harder to control or absorb and will be inevitably present to a significant degree in random waves. In work on the distribution of crest heights in random seas, Forristall (2000) encountered such large uncorrelated long waves in tank data that he used high-pass filtering to remove all long waves from his analysis.

In the open sea, we do not know the main source of the discrepancy between the measured and modelled long-wave components. Both the effects discussed above may be important. The analysis of deliberately injected noise given in §5 shows that even high levels of noise in the long-wave components do not significantly degrade our ability to estimate the mean spreading.

5. Synthetic test data

(a) Method for generating data

As a test of this approach, synthetic second-order data may be analysed. Simulated random data are generated using the approach set out in Forristall (2000).

First, a free wave directional spectrum is created. We use a JONSWAP spectrum with Hs=6 m, mean period Tz=12 s and γ=3.3. This is discretized with Δω=0.0052 rad s−1 and an angular resolution of 3°. The spectrum is curtailed at three times the mean frequency and at two standard deviations from the mean wave direction. Free waves may then be generated by assigning a random phase to each component. A typical time series is shown in figure 8.

Figure 8.

A typical time series of synthetic linear waves from a JONSWAP spectrum.

Once the free waves have been calculated, the bound waves are found using equation (3.4) for a depth of 60.1 m. For the linear time series in figure 8, the sum and difference terms are shown in figure 9. The difference terms have been filtered to remove components with frequencies higher than the lowest frequency linear waves.

Figure 9.

Typical sum and difference terms for synthetic data. The thin line represents the sum term and the thick line represents the difference term.

(b) Analysis of noise-free data

Random second-order data may be analysed using the method set out in §4. Various values of spreading parameter σ may be trialled, and the results compared using equation (4.6).

Results are shown in figure 10. Data were created with a mean directional spread of 15°. The small inaccuracy in the analysis is due to two effects. First, the removal of the sum bound waves is not exact for a finite bandwidth record. Second, the mean wave direction will vary slightly over time. The waves were created assuming a spreading around the mean wave direction over the duration of the record, whereas the analysis measures the spreading around the instantaneous mean wave direction.

Figure 10.

The long-wave discrepancy between the synthetic data with σ=15° and estimates. Fifteen data points are used.

(c) Analysis of noisy data

In all recorded data, noise will be present. It is important to establish the sensitivity of an analysis technique to noise. We assume in this analysis that all the noise is uncorrelated to the original data. For the purposes of this study, there are three types of noise that we will consider:

  1. sensor noise in recording long waves,

  2. spurious long waves which are not bound to the free waves; for instance, freely propagating long waves could be released by short-wave wave groups breaking at a coastline and propagating offshore, and

  3. sensor noise in the linear record.

The effect of noise in the ‘free wave’ range of frequencies will lead to errors in the estimates of the bound waves. The effect of this can be examined by comparing the signal-to-noise ratio (SNR) in the ‘free waves’ with that introduced from this into the estimate of the second-order signal. We define SNR asEmbedded Image 5.1

If we consider the interaction of two wave trains which have a small error in amplitude ε, then the linear wave will be given byEmbedded Image 5.2 and the SNR in the linear range will beEmbedded Image 5.3 The interaction of two waves will then be approximatelyEmbedded Image 5.4 If we ignore very small terms, the SNR of the second-order difference estimate is thenEmbedded Image 5.5 If the noise is Gaussian, these will sum geometrically to give a general resultEmbedded Image 5.6

This can be checked numerically by injecting Gaussian noise into the linear range and by calculating the SNR ratio of the resulting second-order difference waves, using equation (3.4), as shown in figure 11. Thus noise in the linear part of the spectrum will cause an inaccuracy in the estimated second-order difference term. This will have the same order SNR as that in the linear range.

Figure 11.

The signal-to-noise ratio in the second-order difference term estimate, introduced by noise in the free waves.

We now make the simplifying assumption that the noise from types (1) and (2) will be more important than the noise in the linear range to the estimate of discrepancy. This assumption is based on the physical waves being far larger in the linear range, and so absolute inaccuracies in the measurement will lead to smaller SNRs. This is obviously dependent on the sensor type, analogue to digital conversion, etc. The amount of disturbance from type (2) is specific to the location and sea state. A more complex analysis than the one presented here would involve several varying noise parameters, but would lead to very similar results, so we present the simplified case.

We will lump the noise from types (1) and (2) together and introduce this into the filtered data term in equation (4.6). The amount of noise is given by the SNR (the signal being the filtered data). This can then be systematically varied.

In the following examples, the data were created with a spreading value of 15° and are analysed with various levels of noise added using the method set out in §4. In all cases, 20 min of data was used for the analysis. Figure 12 shows a plot of the discrepancy for several SNRs. It can be seen that the discrepancy increases with noise and that the minimum is much flatter than in the noiseless case (figure 10). It can also be seen that the location of the minimum does not occur exactly at 15°. To examine this, figure 13 shows the variability of the estimate in spreading as the noise level increases. While the average value of predicted spreading remains close to 15°, the standard deviation of estimates based on 20 min of data (roughly 100 waves) increases.

Figure 12.

Typical long-wave discrepancy for various levels of noise, based on 20 min of data. The data points used are shown with circles and these results spline fitted. Dashed line, SNR=0.5; dotted line, SNR=1; dash-dotted line, SNR=2; solid line, SNR=3.

Figure 13.

The average estimate of spreading when noise is added to the calculation. The results are based on 250 runs with 20 min of data in each. The standard deviation of these estimates is shown. The true spreading is 15°.

Thus, we can make an estimate of spreading, even when the level of noise in the low-frequency waves is comparable to the actual signal. However, as noise increases, we will need to analyse more data to make an accurate estimate of the spreading.

6. Wave basin data

(a) Data parameters

To assist with the design of a pair of offshore structures for the Sea of Okhotsk, tests were commissioned by Shell and performed in a wave basin at the Canadian Hydraulics Centre. These tests included irregular, unidirectional and multi-directional sea states, in the absence of any structure. The tests were carried out with length scaling of 1:45. All surface elevation data presented here are scaled up to full scale, using Froude number scaling as appropriate. Further details may be found in Cornett et al. (2002).

The spectrum used was a JONSWAP spectrum with γ=2. The undisturbed water depth was 53.13 m, the significant wave height was 9.9 m and the peak period was 14.3 s.

The directional spreading model used in the experiments wasEmbedded Image 6.1 where Γ is the gamma function, θ0 the mean wave direction, s the spreading index and |θθ0|<90°. At each frequency component, only one plane wave component was synthesized, so equation (6.1) was only fulfilled over a range of frequencies. A spreading index of 7 was used, which equates to a theoretical standard deviation of 15°. Although different in form, the two models for the spreading function (equations (2.2) and (6.1)) with suitably matched values of the constants are visually indistinguishable when plotted. Thus fitting a wrapped normal angular distribution to Embedded Image data presents no difficulty or ambiguity in interpretation unless there is significant wave energy propagating at angles >90° to the mean wave direction. The actual wave spreading in the basin was measured using two-axis electromagnetic current meters. These found a mean spreading angle of 17° and it was observed that measured spreading varied with frequency, with the minimum spreading under the spectral peak. For nominally unidirectional waves, spreading of 4° was measured in the basin.

(b) Analysis

Point surface elevation data were available for 186 min for both spread and unidirectional seas. For convenience, this was divided into 20 min sections, which were then analysed using the approach set out in §4. A comparison of low-pass-filtered data with best estimate for the bound difference waves is given in figure 14 for a typical 20 min section of the spread sea case. The long-wave discrepancy over the entire record is shown for various assumed spreading angles in figure 15.

Figure 14.

A typical comparison of low-pass-filtered data with the best estimate of the second-order difference components after identical filtering, based on Canadian Hydraulics Centre data. Thin line, filtered data; thick line, best estimate.

Figure 15.

The long-wave discrepancy between the low-pass-filtered data and the estimate of second-order difference components for various spreading angles. Eight data points were used, shown with multiplication symbols, and the results spline fitted. Solid line, the unidirectional case; dotted line, the spread sea case.

Over the nine sections of spread sea data analysed, the mean estimated spreading was found to be 17° with a standard deviation in the results of 3°. The smallest estimate for a 20 min period was 13° and the largest 22°, with be compared with the intended spreading angle of 15° and 17° achieved in the wave basin.

For the unidirectional case, we estimate mean spreading to be 7° with a standard deviation of 0.7°. It should be noted that the long-wave discrepancy was much larger for these data than for the directional spread data.

Thus, the mean spreading is accurately predicted when the whole record is considered. However, for these data, there is some error when shorter time intervals are considered. In wave tanks, it is very difficult to absorb low-frequency waves, so it is likely that spurious low-frequency waves will be present, possibly at a higher level than in field data and these are likely to be significantly larger in unidirectional as opposed to spread sea states.

7. Field data

(a) Draupner field data

The Draupner platform is located in the Norwegian sector of the North Sea in a water depth of 70 m. The surface elevation was measured using a downward-pointing laser. The platform structure is rather sparse and the effect of the structure on the waves is assumed to have been minimal. For more information on the structure, the instrumentation, the meteorological conditions and the wave record, see Haver (2004).

The data used in this paper were recorded during a storm on 1 January 1995, between 14:00 and 19:00. The sea-state properties remained fairly constant over the storm’s 5 hour duration. Data were recorded for 20 min in every hour. Significant wave height was around 12 m and average wave period was around 12.5 s. The data have been much analysed (see, for instance, Jensen 2005; Walker et al. 2005) because of the ‘New Year wave’ that occurred during the storm. This had a crest elevation of 18.5 m. This is shown in figure 16.

Figure 16.

The surface elevation time history recorded at the Draupner platform that includes the New Year wave. For convenience, the time is set to zero under the big wave.

No directional information was recorded at the Draupner location. The only relevant directional spreading measurement known to the authors is from a directional buoy close to the Auk platform, 180 km away (K. C. Ewans 2005, personal communication). Gibson (2004) based his analysis of this record on the same directional measurement. The data recorded here had a similar frequency spectrum to those recorded at Draupner and are assumed to be part of the same storm. The spreading measured at Auk was around 20° although there was considerable variation with frequency. Using the ECMWF WAM type hindcast model P. A. E. M. Janssen (2009, personal communication) estimated the spreading to be 24° at 18:00.

(b) Analysis

Six 20 min records are available at Draupner, with the start of each recording being on the hour. The record containing the New Year wave, shown in figure 16, was the second of these.

We observe that something unusual happens to the low-frequency waves around the time of the giant wave. Forristall (2000) pointed out that the kernels for the second-order difference terms usually predict a set-down under a large wave. This is supported by figure 5, which shows a set-down under that wave group. However, Walker et al. (2005) observed that there is a set-up under the Draupner wave as shown in figure 17. This also shows our prediction for the second-order difference terms based on 20° spreading. It can be seen that these are dramatically out of phase around the large wave. We will consider this inconsistency in greater detail in a subsequent paper, in which we propose that the giant wave arose in a locally crossing sea (Adcock & Taylor 2009). The present method assumes that sea states are unimodal in direction with relatively modest directional spreading. This is consistent with most of the data analysed in the worldwide study of wave directional spreading in extra-tropical storms by Forristall & Ewans (1998).

Figure 17.

The low-pass-filtered data for the time history containing the New Year wave, together with the estimate of the long waves based on a wrapped normal spreading distribution with σ=20°. Thin line, low-pass-filtered data; thick line, predicted waves for σ=20°.

A more typical result is shown in figure 18 for the 20 min period an hour after the one containing the giant wave. The low-pass-filtered signal is shown as in the identically filtered best estimate.

Figure 18.

The low-pass-filtered waves for a typical 20 min section of data. The best estimate is also shown. Thin line, low-pass-filtered data; thick line, predicted waves for σ=20°.

Figure 19 shows the long-wave discrepancy, as defined by equation (4.6), for the storm (with a small section around the big wave excluded). It can be seen that the minimum discrepancy over all the available data is at 20°, which is consistent with a previous, cruder, estimate by Walker et al. (2005) based on the second-order sum terms. The lowest estimate of spreading is 12° in the first 20 min section. The highest is 25° in the final 20 min section. Without accurate local spreading measurements, we cannot say whether these variations are due to inaccuracy in our method or natural variation of the sea-state properties with time.

Figure 19.

Long-wave discrepancies for the Draupner data. (a) The long-wave discrepancy of each 20 min section of the Draupner time histories. (b) The average long-wave discrepancy for Draupner time histories. The results are spline fitted.

8. Conclusions

In this paper, we have demonstrated that it is possible to infer the degree of spreading without direct measurement. This may be done using the low-frequency, second-order difference waves, whose magnitude is dependent on the directional spreading of the linear waves.

The technique is shown to work in a numerical wave tank, for mildly nonlinear waves that are representative for waves in a typical storm.

Analysis of synthetic random data shows that our approach is robust, producing reasonable estimates of spreading even if the SNR is of the order of 1. This analysis highlights that the length of data available, and over which the sea-state parameters are approximately constant, is important in determining the accuracy of the spreading estimate.

We apply our approach to random wave data recorded in a physical wave basin. While the analysis is not perfect when only 20 min of data are considered, when longer sections of data are analysed, the results are excellent.

We also show that this approach can be applied to field data recorded in the North Sea.

The approach in this paper has the potential to be a useful tool in the analysis of wave records where spreading has not been directly measured. It is applicable to analysis of rough seas, when the properties of the storm are stationary for several hours.

Acknowledgements

The authors are grateful for the data provided by Dr Richard Gibbs, Dr Kevin Ewans (Shell) and Dr Sverre Haver (Statoil). T. A. A. A. was supported by an EPSRC studentship.

Footnotes

    • Received January 20, 2009.
    • Accepted July 16, 2009.

References

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