# Measuring the ocean wave directional spectrum ‘First Five’ with HF radar

**Part of the following topical collections:**

## Abstract

An ability to reliably measure the first five Fourier coefficients of the directional distribution of ocean wave energy is becoming an international requirement for any directional wave measurement device. HF radar systems are now commonly used for surface current measurement in the coastal ocean but robust wave measurements are more difficult to achieve. A number of HF radar deployments have demonstrated an ability to measure the directional spectrum, and in this paper, an evaluation of the Fourier coefficients derived from these spectra is presented. It is shown that, when data quality is good, good quality spectra and Fourier coefficients result. Recommendations for addressing some of the radar data quality issues that do arise are presented.

## Keywords

HF radar Ocean wave directional spectrum Fourier coefficient First five WERA Pisces## 1 Introduction

Ocean waves can sink ships and small boats, move sand and sediments, erode beaches and coastal defences, increase coastal flooding, and damage inshore, offshore and land-based structures. They can also provide power, help to break up oil and pollution slicks, and support marine activities such as surfing and fishing. In many of these cases, a measurement of waveheight alone is not sufficient; the directional and frequency (or equivalently period or wavelength) distribution of wave energy, known as the ocean wave directional spectrum, is important. For example, offshore structures may have dangerous resonances at particular periods; beach erosion impacts will depend on the dominant wave directions during storms; marine renewable devices may have limited directional responses. As a result, many wave measuring devices now have spectral and directional measurement capabilities. In coastal regions, there are a number of factors, e.g. current shear, bottom and coastal topography, and sea breeze, that lead to spatial variations in wave properties. To capture this variability would require a big investment in buoys which in turn would provide increased hazards for shipping. Remote sensing from the coast using HF radars provides an opportunity to measure this spatial variability without any physical interference with offshore activities.

The ‘First Five’ refers to parameters of the ocean wave directional spectrum which include the energy spectrum, *E*(*f*), and the first four Fourier coefficients, *a*_{1}, *b*_{1}, *a*_{2}, *b*_{2}, of the directional distribution of ocean waves at each wave frequency. These data are routinely provided by directional wave buoys and can also be used to provide measurements of directional spreading, skewness and kurtosis. Swail et al. (2010), in their comprehensive overview of wave measurements, conclude that “It is strongly recommended that all directional wave measuring devices should reliably estimate ‘First 5’ standard parameters and ‘First-5’ compliant is a priority both for operational and climate assessment requirements”. This recommendation is also referred to in the IOOS wave observation plan (USACE 2009) and can be found on the JCOMM website so it would appear to have widespread international support. None of these sources provide specific guidance on what constitutes a reliable first five measurement. Accuracy requirements are usually given for just a few key parameters of the spectrum, e.g. significant waveheight, peak period and direction. Standards for first five measurement need to be developed and perhaps this paper will play a role in stimulating that work.

The measurement of waves with HF radar dates back to the 1970s; however, the development and success of the CODAR SeaSonde radar system focussed attention much more on the current measurement capabilities of HF radar. This is because it is much more difficult to get robust wave measurements from compact radars of this type although, in suitable circumstances, some wave parameters can be obtained (e.g. Long et al. 2011; Lipa et al. 2014). Phased array radars such as Pisces and WERA are much more suitable for directional spectrum measurements and the results from a number of trials demonstrating this capability have been published (e.g. Wyatt et al. 2003, 2006, 2011). This paper looks in particular at the accuracy of the ‘ First-5’ obtained from HF radar measured directional spectra compared with those from directional wave buoys.

HF radar systems are normally located on the coast in pairs or, in some parts of the world, in interconnected networks, and measure backscatter from ocean waves of radio waves with a frequency in the HF band (3–30 MHz). The backscatter can be measured to ranges from the coast of up to 300 km when low HF frequencies are used, or up to 50 or so km at the higher HF frequencies. Maps of wave, current and wind measurements can be made with spatial resolutions from 250 m to 5 km or more again depending on the operating frequency, on antenna configuration and on available radio bandwidth.

The main scattering mechanism is Bragg scattering from linear ocean waves with half the radio wavelength travelling towards and away from the radar. These ocean waves propagate with speeds determined by the linear dispersion relationship and thus can be easily identified in the power spectrum (commonly referred to as the Doppler spectrum) of the backscattered signal from their frequency signature, i.e. they appear in the spectrum as high amplitude peaks at a frequency given by, in deep water, \(\sqrt {2 g k_{r}}\) rad/s where *g* is gravitational acceleration and *k*_{r} is the radio wavenumber. These peaks are shifted in frequency if there is a surface current by the component of that current in the radar look direction and this additional shift is used to determine that current component. Non-linear wave-wave interactions can also generate ocean waves with the Bragg scattering wavelength but these travel with different phase speeds and are thus separated from the scatter from linear waves because they have different frequency signatures. Double electromagnetic scattering from waves on the sea surface has a similar effect but in general is lower in amplitude in the Doppler spectrum than the hydrodynamic contribution.

The first theoretical formulation of the relationship between the backscattered power spectrum and the ocean wave directional spectrum was published by Barrick (1972a, b) and Barrick and Weber (1977). This took the form of an integral equation which can be broken down into first (linear waves)- and second (non-linear waves and double electromagnetic)-order terms. To obtain wave measurements, the second-order integral equation needs to be inverted and several attempts have been made to do that (e.g. Lipa 1977; Lipa and Barrick 1986; Wyatt1990, 2000; Howell and Walsh 1993; Hisaki 1996; Hashimoto and Tokuda 1999; 2000; Green and Wyatt2006). Another approach has been to develop empirical relationships between the Doppler spectrum or its integral and the ocean wave frequency spectrum or its parameters, e.g. signficant waveheight. However, these empirical methods do not provide measurements of the Fourier coefficients so will not be discussed further here.

The inversion method used to obtain the data presented in this paper (Wyatt 1990; Green and Wyatt 2006) provides the ocean wavenumber directional spectrum at each measurement location with sufficient second-order signal to noise. It is an iterative method, initialised with a Pierson-Moskowitz spectrum (Pierson and Moskowitz 1964) and a uni-modal sech^{2} directional model (Donelan et al. 1985) using an empirical model for the Pierson-Moskowitz waveheight (Wyatt 2002) and a short wave direction determined from the two first-order peaks (Wyatt 2012). The directional spectrum is modified, at each vector wavenumber and at each iteration, according to the difference between the radar measurement and a simulation using the directional spectrum from the previous iteration, modified by the kernel of the integral equation. The spectrum at convergence is usually very different in shape, both in frequency and direction, from the initial guess and, as will be seen, bi- and multi-modal spectra can emerge. A further quality control is provided by a metric measuring the convergence of the inversion.

Depending on the deployment configuration there could be 10 to 100 s of directional spectra measurements across the field of view every 20 min to 1 h. Using standard techniques (see Section 3.1), this spectrum can be converted to a directional frequency spectrum (from which Fourier coefficients are obtained) and to derived parameters such as significant waveheight, peak period and direction, and wave power. A mean depth at each measurement location is needed for both the inversion and the conversion processes and best available bathymetry is used for this purpose. It is also possible to include a dynamic depth by linking the inversion to a tidal model but that has not been used in this paper.

In Section 2, the data sets are described. Section 3.1 presents the methods used, Section 3.2 the radar and buoy comparisons, and Section 4 the discussion and conclusions.

## 2 Data sets

*E*(

*f*) is the energy spectrum in m

^{2}/Hz, is a better period comparator for the radar measurements because these have a limited upper frequency dependent on operating frequency. This formulation is dominated by the lower, energy containing frequencies and is widely used in the wave power sector. Higher ocean wave frequencies dominate in the more standard mean, or first-moment, period, \(\displaystyle T_{1} = \frac {\int E(f) df}{\int f E(f) df}\) so, unless the buoy frequency range is limited to the same range as the radar, the radar will normally measure a higher mean period than the buoy. The low waveheight limit for the Celtic Sea data set leads to lower accuracy in period and direction unless the data are filtered to take account of this (Wyatt et al. 2011). For the data shown in the tables, periods and directions are only included if the Bragg scattering wave frequency is at least twice that of the peak frequency of a Pierson-Moskowitz spectrum (see Fig. 1), \(T_{pPM} \simeq 5 \sqrt {H_{s}}\), where

*H*

_{s}is the radar measured wavelength. During this deployment the flexible frequency was used to deal with external interference and not to account for waveheight variations which would have avoided this filtering. Note that the filtering has only been applied in these tables and not to the Fourier coefficients presented later in this paper. This provides the opportunity to explore whether some parts of the spectrum are more sensitive to this limit than others. The high waveheight limit for the Norwegian data is picked up as a quality issue during the inversion process so creates gaps in the data rather than errors. Peak direction is the direction of the wave component at the peak of

*E*(

*f*). Mean direction is determined from the directional spectrum using \(\theta _{m} =\tan ^{-1} \frac {\int \int S(f,\theta ) \sin \theta d \theta df }{\int \int S(f,\theta ) \cos \theta d \theta df}\) or equivalently in terms of the Fourier coefficients using \(\theta _{m} =\tan ^{-1} \frac {\int E(f ) b_{1}(f) df }{\int E(f) a_{1}(f) df}\). Directions are compared here using vector correlation and phase difference as suggested by Kundu (1976). The phase difference is the same as the mean difference between the direction measurements.

Statistics of basic wave magnitude parameters

Parameter | Unit | Deployment | Buoy mean | Radar mean | cc | rms | Bias |
---|---|---|---|---|---|---|---|

Hs | m | Norway | 2.45 | 2.41 | 0.95 | 0.32 | 0.04 |

Celtic Sea | 2.08 | 2.08 | 0.92 | 0.40 | 0.04 | ||

T | s | Norway | 8.67 | 8.63 | 0.90 | 0.66 | 0.04 |

Celtic Sea | 8.37 | 9.33 | 0.72 | 1.58 | − 0.95 |

Statistics of basic wave direction parameters

Parameter | Unit | Deployment | Vector correlation magnitude | Phase |
---|---|---|---|---|

Mean | \(\deg \) | Norway | 0.92 | 1.05 |

Celtic Sea | 0.90 | −4.40 | ||

Peak | \(\deg \) | Norway | 0.64 | 2.38 |

Celtic Sea | 0.87 | −4.83 |

## 3 Methods and comparisons

### 3.1 Methods

The output from the inversion process is an ocean wave directional spectrum, *S*(**k**) on a wavenumber, **k**, grid. The grid is uniform in \(\sqrt {k}\) (where *k* = |**k**|) a convenient variable in the inversion process and is thus uniform in frequency in deep water. In this work, where depths are variable, the \(\sqrt {k}\) grid has been selected with intervals corresponding to 0.005 Hz in deep water frequency. Since all the buoy data used are provided as functions of frequency rather than wavenumber, the radar spectra have been converted to directional frequency spectra, *S*(*f*, *θ*), taking into account water depth, in the standard way, i.e. \(\displaystyle S(f,\theta ) = \frac {dk}{df} k S(\textbf {k})\) (Tucker 1991). Fourier coefficients have been determined from the directional frequency spectra again using standard methods (Tucker 1991). For example, writing *S*(*f*, *θ*) = *E*(*f*)*G*(*θ*, *f*), \(\displaystyle a_{n}(f) = {\int }_{-\pi }^{\pi } G(\theta ,f) \cos n \theta d\theta \).

In the absence of the full directional spectrum, the Fourier coefficients can be used to indicate spectral shape and the presence of bi-modality. Defining \(r_{i}(f) = \sqrt {a_{i}(f)^{2} + b_{i}(f)^{2}}\), a plot of \(\sqrt {r_{2}(f)}\) against *r*_{1}(*f*) can be used to compare data against standard directional models, e.g. \(\cos ^{2s}\) or sech^{2} and to identify potential bimodality (Hauser et al. 2005). Another approach to identify potential bimodality in the spectrum plots kurtosis against the absolute value of skewness both of which can be determined from the Fourier coefficients (Kuik et al. 1998). An analysis of this kind is included below in Figs. 4–7c and provide further insights into the differences between radar and buoy measurements. The relationship between \(\sqrt {r_{2}(f)}\) and *r*_{1}(*f*) for three standard directional models are shown in the figures.

### 3.2 Radar/buoy comparisons

Individual measurements of the directional spectrum and its associated Fourier coefficients are shown in Figs. 4, 5, 6 and 7a,b. Also shown are the frequency spectrum, *E*(*f*), the mean direction, and the directional spreading at each frequency. In all cases, the radar Fourier coefficients are smoother but in reasonable agreement with those of the buoy. Small differences are amplified in the skewness and kurtosis calculations where, in general, the buoy skewness is more variable and the kurtosis is significantly higher at the spectral peak, also seen in the shape analysis plots. The inversion process requires some smoothing in both frequency and direction to ensure stability in the solution which probably accounts for this (see Green and Wyatt (2006) for a discussion about the need for, and parameters used for, the smoothing). The shape analysis in both plots in Fig. 4c shows evidence of bimodality in the radar data near the spectral peak. One explanation is that the frequency smoothing referred to above is also responsible for this evidence of directional bimodality, i.e. the individual wave components (wind-sea and swell, as seen in Fig. 4a) have more well-defined narrower frequency ranges in the buoy data than in the radar data. That is, spectra that are bimodal in frequency but not in direction at a particular frequency in the buoy data appear bimodal in direction in the radar data because of the frequency smoothing. Some must also be attributed to the evidence in both directional spectra plots, albeit clearer in the radar spectrum, of a second swell contribution well separated from the main swell and wind-sea contributions. The buoy measurements suggest bimodality at frequencies well away from the peak both above (squares) and below (circles). This is not seen in the radar data and could be indicating noise in the buoy data at these frequencies.

The directional spectra in Fig. 5a appears to show 4 different wave components although two are more merged in the buoy spectrum. The kurtosis in the buoy data is higher at all these peaks. The upper plot in the shape analysis, Fig. 5c, shows no bimodality in the radar data although the lower plot does indicate some multi-modality or perhaps non-symmetry near the peak. The buoy data appears to be bimodal at very low frequencies but here the amplitude is low so this again could be noise in the data. There is no conformity to standard directional shapes in either case.

The radar directional spectra in Figs. 6 and 7 whilst showing general agreement with the buoy include an extra swell component at about 0.06 Hz. These are likely to be related to ships, to antenna sidelobe signals associated with variable surface currents across the measurement region (Wyatt et al. 2005) or to local current shear. Where one contribution to the spectrum is dominant, e.g. Fig. 6 there is some evidence in the shape analysis plots of a particular directional shape over a range of frequencies near the peak. This is particularly clear for the radar data in this case which appears to align well with a sech^{2} form near the peak, noting that this is indistinguishable from the cos^{2s} form very close to the peak. In general though the data are more scattered for both types of measurement and do not conform to a particular form. In the lower plot in Fig. 6c, the buoy is showing evidence of multi-modality or non-symmetry away from the peak whereas the upper plot shows very little evidence of bimodality. Non-symmetry is therefore likely to be the explanation and is of course consistent with the skewness shown in Fig. 6b.

The shape analysis in Fig. 7 shows some evidence that the radar data is consistent with the sech distribution. However, in this case, this may be biased by the initialisation since the peak frequency is quite high relative to the measurement range. There is no indication of bimodality in the radar data but a slight indication of non-symmetry near the peak in the lower plot. The buoy data looks more like a \(\cos ^{2s}\) shape near the peak with some evidence of bimodality at low frequencies where amplitude is low so again possibly noise in the buoy data. There is also some evidence of lack of symmetry near the peak.

*a*

_{1}and

*b*

_{2}measurements are in good agreement although somewhat noisy at low frequencies in both measurements particularly when amplitudes are low (as seen in Fig. 8). Both radar and buoy

*b*

_{1}measurements show less variation with time. Similar features can be seen in the

*a*

_{2}measurements although the larger negative, and in some cases larger positive values in the buoy data are not seen in the radar data. These observations are confirmed in the scatter plots shown in Figs. 14, 15, 16, and 17. Correlation coefficients of over 0.9 are seen in the

*a*

_{1}comparison over a range of frequencies. Above about 0.1 Hz, the standard deviations in the radar and buoy time series (shown in brackets after the means) are similar, and in each case, the rms of the comparison is lower than the individual standard deviations. The

*b*

_{1}coefficient varies over a smaller range and correlation coefficients are lower. Agreement is qualitatively better above about 0.1 Hz although rms differences are now similar in magnitude to the individual instrument standard deviations which is a concern. The

*a*

_{2}scatter plots confirm that the buoy measurements vary over a wider range than those of the radar although the correlation coefficient of over 0.6 at higher frequencies shows reasonable agreement. However, the rms in this case is higher than the standard deviation in the radar measurements although lower than that of the buoy measurements. It is possible that this Fourier coefficient is more sensitive to the inversion smoothing than the others. The correlation and rms compared to instrument standard deviations, again above about 0.1 Hz, are better for the

*b*

_{2}coefficient than for

*a*

_{2}.

## 4 Discussion and conclusions

Although there have been a number of studies involving buoy intercomparisons which have looked at directional parameters (e.g. Allender et al. 1989) this author has been unable to find any publications which look specifically at the Fourier coefficients although of course there are many studies looking at derived parameters such as mean direction and directional spreading. Given the stated international requirement for these coefficients perhaps such a study is needed in order to establish a benchmark for the accuracy of these parameters. In making comparisons with a directional waverider buoy and drawing conclusions about the radar data therefrom, we are therefore making the assumption that the buoy measures the true Fourier coefficients of the directional distribution. With this assumption, the results here show that the radar tends to measure a smoother distribution of the parameters with frequency and this is attributed to the smoothing that is necessary in the inversion to stabilise the numerical solution. The temporal variation in the coefficients seen in the buoy data is well represented in the radar data although the radar *a*_{2} coefficient varies over a narrower range. The comparisons have focussed on correlation coefficients and rms differences the latter having being compared with the standard deviations in the individual buoy and radar measurements. High values of correlation coefficient and low values of rms relative to, in particular, the buoy standard deviations would imply good agreement and this has been found for the *a*_{1} and *b*_{2} coefficients. The other two coefficients appear to have been measured less reliably, *a*_{2} in particular has a much wider variance in the buoy than the radar data. However mean direction comparisons are good so perhaps the apparent lower agreement for *b*_{1} is reflecting the smaller range of values of this coefficient in both measurements rather than indicating significant radar errors.

There are significant differences in all coefficients and the associated mean direction and spread at low frequencies below about 0.1 Hz. In part, these are associated with the misinterpretation of ship signals or first-order signals coming in on the antenna sidebands as swell contributions. There are three possible solutions to this. One is to remove the ship signals before inversion. A number of methods have been proposed for identifying ship signals in the radar data in order to provide a ship-tracking application but these are not yet routinely applied and probably not yet sufficiently robust. A second is to ensure careful calibration of the receive antenna array to minimise sidelobes although it is difficult to remove these altogether. Perhaps a more promising approach is to partition the radar spectra and use the temporal and spatial continuity of the radar data to identify and remove partitions that are unlikely to be either wind-sea or swell. Partitioning methods have been applied to HF radar data (see, e.g. Isaac and Wyatt 1997, Waters et al. 2013) but have not yet been used in this way although some progress is being made towards this goal. The other main factor limiting the availability of good quality directional information, in regions which experience a wide range of waveheight conditions, is the low frequency limit in low sea-states and the high frequency limit in high sea-states. Having a radar that can measure over a range of frequencies responding automatically to changing conditions is the answer here although these require wideband antenna and radar hardware systems. Another explanation for the low frequency differences is that the buoy directional measurements are also noisy in this range. There is certainly less averaging in the buoy data at these frequencies. However, there is enough evidence of problems in the radar measurements at low frequencies which need to be addressed before attributing errors to the buoys.

The shape analysis comparisons are intriguing but more work is needed to really understand the differences. There are cases showing consistency between radar and buoy and others with significant differences. In general, the radar measurements show less variation with frequency in part probably due to the smoothing in the inversion. There is some suggestions that the buoy data is noisy at frequencies well away from the peak. The two different methods appear to be consistent near the spectral peak. The more empirical (Kuik et al. 1998) method does not distinguish between multi-modality and non-symmetry, but by comparison with the other method, is likely to be indicating non-symmetry in most cases.

In both deployments discussed here, the buoy was located at a position where the angle between the look directions from the two radars is roughly \(90 \deg \). It has been shown (Wyatt and Holden 1994) that the accuracy of the radar measurements does depend on this angle with \(90 \deg \) being optimum. A more extensive validation using in situ measurements at more locations across the radar field of view would therefore be useful to clearly establish the range and azimuthal extent of accurate data.

In conclusion, this paper has demonstrated that Fourier coefficients can be obtained from HF radar data and that they agree reasonably well with those measured with a buoy at frequencies greater than about 0.1 Hz up to 0.2 Hz, the maximum frequency analysed here. The agreement is not perfect for reasons outlined but approaches to improve the quality have been identified. Of course the radar is making these measurements over wide areas of the coastal ocean so can measure spatial as well as temporal variations in these quantities. Waves vary in the coastal environment due to changes in depth and coastal topography with associated variations in current and in wind and a spatial picture with good but possibly lower accuracy may be more useful for some applications.

## Notes

### Acknowledgements

The Norwegian data were obtained during the EU-funded EuroROSE project and we thank Klaus-Werner Gurgel and all of the EuroROSE team for their contributions. The Pisces data was provided by Neptune Radar and collected during a project funded by DEFRA and the UK Met Office. The buoy data for this experiment have been provided by CEFAS. Some of these data sets were processed and provided by Seaview Sensing Ltd. Referees made a number of very useful suggestions that have improved the paper.

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