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

下载积分：2000

内容提示： OceanDynamics(2019)69:123–144https://doi.org/10.1007/s10236-018-1235-8Measuringtheoceanwavedirectionalspectrum‘FirstFive’withHFradarLucy R. Wyatt 1,2Received:25February2018/Accepted:31October2018/Publishedonline:17November2018©TheAuthor(s)2018AbstractAn ability to reliably measure the first five Fourier coefficients of the directional distribution of ocean wave energy isbecoming an international requirement for any directional wave measurement device. HF radar systems are now commonlyused for surface c...

文档格式：PDF|
浏览次数：1|
上传日期：2019-11-18 07:43:42|
文档星级：

**********
OceanDynamics(2019)69:123–144https://doi.org/10.1007/s10236-018-1235-8Measuringtheoceanwavedirectionalspectrum‘FirstFive’withHFradarLucy R. Wyatt 1,2Received:25February2018/Accepted:31October2018/Publishedonline:17November2018©TheAuthor(s)2018AbstractAn ability to reliably measure the first five Fourier coefficients of the directional distribution of ocean wave energy isbecoming an international requirement for any directional wave measurement device. HF radar systems are now commonlyused for surface current measurement in the coastal ocean but robust wave measurements are more difficult to achieve. Anumber of HF radar deployments have demonstrated an ability to measure the directional spectrum, and in this paper, anevaluation 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 issuesthat do arise are presented.Keywords HF radar · Ocean wave directional spectrum · Fourier coefficient · First five · WERA · Pisces1 IntroductionOcean waves can sink ships and small boats, move sand andsediments, erode beaches and coastal defences, increase coastalflooding, and damage inshore, offshore and land-basedstructures. They can also provide power, help to break up oiland pollution slicks, and support marine activities such assurfing and fishing. In many of these cases, a measurementof waveheight alone is not sufficient; the directionaland frequency (or equivalently period or wavelength)distribution of wave energy, known as the ocean wavedirectional spectrum, is important. For example, offshoreThis article is part of the Topical Collection on the 15th InternationalWorkshop on Wave Hindcasting and Forecasting in Liverpool, UK,September 10–15, 2017Responsible Editor: Oyvind Breivik? Lucy R. Wyattl.wyatt@sheffield.ac.uk1School of Mathematics and Statistics, University of Sheffield,Sheffield S10 2TN, UK2Seaview Sensing Ltd., Sheffield, UKstructures may have dangerous resonances at particularperiods; beach erosion impacts will depend on the dominantwave directions during storms; marine renewable devicesmay have limited directional responses. As a result, manywave measuring devices now have spectral and directionalmeasurement capabilities. In coastal regions, there are anumber of factors, e.g. current shear, bottom and coastaltopography, and sea breeze, that lead to spatial variationsin wave properties. To capture this variability would requirea big investment in buoys which in turn would provideincreased hazards for shipping. Remote sensing from thecoast using HF radars provides an opportunity to measurethis spatial variability without any physical interferencewith offshore activities.The ‘First Five’ refers to parameters of the ocean wavedirectional 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 wavefrequency. These data are routinely provided by directionalwave buoys and can also be used to provide measurementsof directional spreading, skewness and kurtosis. Swail et al.(2010), in their comprehensive overview of wave measure-ments, conclude that “It is strongly recommended that alldirectional wave measuring devices should reliably esti-mate ‘First 5’ standard parameters and ‘First-5’ compliant
124 OceanDynamics(2019)69:123–144is a priority both for operational and climate assessmentrequirements”. This recommendation is also referred to inthe IOOS wave observation plan (USACE 2009) and can befound on the JCOMM website so it would appear to havewidespread international support. None of these sourcesprovide specific guidance on what constitutes a reliable firstfive measurement. Accuracy requirements are usually givenfor just a few key parameters of the spectrum, e.g. signif-icant waveheight, peak period and direction. Standards forfirst five measurement need to be developed and perhapsthis paper will play a role in stimulating that work.The measurement of waves with HF radar dates backto the 1970s; however, the development and success of theCODAR SeaSonde radar system focussed attention muchmore on the current measurement capabilities of HF radar.This is because it is much more difficult to get robust wavemeasurements from compact radars of this type although,in suitable circumstances, some wave parameters can beobtained (e.g. Long et al. 2011; Lipa et al. 2014). Phasedarray radars such as Pisces and WERA are much moresuitable for directional spectrum measurements and theresults from a number of trials demonstrating this capa-bility have been published (e.g. Wyatt et al. 2003, 2006,2011). This paper looks in particular at the accuracy ofthe ‘ First-5’ obtained from HF radar measured direc-tional spectra compared with those from directional wavebuoys.HF radar systems are normally located on the coast inpairs or, in some parts of the world, in interconnectednetworks, and measure backscatter from ocean waves ofradio waves with a frequency in the HF band (3–30 MHz).The backscatter can be measured to ranges from the coastof up to 300 km when low HF frequencies are used, or upto 50 or so km at the higher HF frequencies. Maps of wave,current and wind measurements can be made with spatialresolutions from 250 m to 5 km or more again dependingon the operating frequency, on antenna configuration and onavailable radio bandwidth.The main scattering mechanism is Bragg scattering fromlinear ocean waves with half the radio wavelength travellingtowards and away from the radar. These ocean wavespropagate with speeds determined by the linear dispersionrelationship and thus can be easily identified in the powerspectrum (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 peaksat a frequency given by, in deep water,√ 2gkrrad/swhere g is gravitational acceleration and k r is the radiowavenumber. These peaks are shifted in frequency if thereis a surface current by the component of that current inthe radar look direction and this additional shift is used todetermine that current component. Non-linear wave-waveinteractions can also generate ocean waves with the Braggscattering wavelength but these travel with different phasespeeds and are thus separated from the scatter from linearwaves because they have different frequency signatures.Double electromagnetic scattering from waves on the seasurface has a similar effect but in general is lower inamplitude in the Doppler spectrum than the hydrodynamiccontribution.The first theoretical formulation of the relationshipbetween the backscattered power spectrum and the oceanwave directional spectrum was published by Barrick(1972a, b) and Barrick and Weber (1977). This took theform of an integral equation which can be broken downinto first (linear waves)- and second (non-linear wavesand double electromagnetic)-order terms. To obtain wavemeasurements, the second-order integral equation needs tobe inverted and several attempts have been made to dothat (e.g. Lipa 1977; Lipa and Barrick 1986; Wyatt 1990,2000; Howell and Walsh 1993; Hisaki 1996; Hashimotoand Tokuda 1999; 2000; Green and Wyatt 2006). Anotherapproach has been to develop empirical relationshipsbetween the Doppler spectrum or its integral and the oceanwave frequency spectrum or its parameters, e.g. signficantwaveheight. However, these empirical methods do notprovide measurements of the Fourier coefficients so will notbe discussed further here.The nature of the integral equation puts some limitson the waveheight range that can be measured at aparticular ocean wave frequency. This is illustrated in Fig. 1.The inversion process can only provide measurements atfrequencies lower than the Bragg frequency. Taking the10 MHz case, it can be seen that its Bragg frequency istoo low to measure any waves at a waveheight of 0.2 mFig. 1 Pierson-Moskowitz spectra for different significant wave-heights (in metres, colour coded). Vertical dashed lines indicate theBragg frequencies for the radio frequencies (in MHz) shown
OceanDynamics(2019)69:123–144 125Fig.2 Significant waveheightand mean direction with WERAin Norway on 20/02/2000 @21:00 (above) and (below)significant waveheight and peakdirection with Pisces in CelticSea on 13/02/2005 @ 16:00.Radar sites shown with ? . Thebuoy image marks position ofthe buoy4˚00'4˚00'4˚10'4˚10'4˚20'4˚20'4˚30'4˚30'4˚40'4˚40'60˚30' 60˚30'60˚40' 60˚40'60˚50' 60˚50'10 kmFedjeLyngoy024msignificant waveheight (full) and wave (band) direction (full): 21:00 20/02/2000 UTC−7˚00'−7˚00'−6˚30'−6˚30'−6˚00'−6˚00'−5˚30'−5˚30'−5˚00'−5˚00'−4˚30'−4˚30'50˚30' 50˚30'51˚00' 51˚00'51˚30' 51˚30'52˚00' 52˚00'50 kmNPCM02468msignificant waveheight (all) and wave direction (peak): 16:00 13/02/2005 UTCaband is too close to the peak frequency at 0.5 m to get anaccurate inversion. At a waveheight of 1.0 m, inversion isjust about feasible. At 30 MHz on the other hand whenthe waveheight is large, the linearisation approximation
126 OceanDynamics(2019)69:123–144Fig.3 Significant waveheight comparisons for Norwegain (left) and the Celtic Sea (right) deployments. Radar measurement in blue, buoy in redused in the development of the integral equation becomesincreasingly unreliable, also it becomes much moredifficult to separate first- from second-order parts of theDoppler spectrum (the Bragg waves are much lower inamplitude than the energy containing waves) and inversionfails.The inversion method used to obtain the data presentedin this paper (Wyatt 1990; Green and Wyatt 2006)provides the ocean wavenumber directional spectrum ateach measurement location with sufficient second-ordersignal to noise. It is an iterative method, initialised witha Pierson-Moskowitz spectrum (Pierson and Moskowitz1964) and a uni-modal sech 2 directional model (Donelanet al. 1985) using an empirical model for the Pierson-Moskowitz waveheight (Wyatt 2002) and a short wavedirection determined from the two first-order peaks (Wyatt2012). The directional spectrum is modified, at eachvector wavenumber and at each iteration, according to thedifference between the radar measurement and a simulationusing the directional spectrum from the previous iteration,modified by the kernel of the integral equation. Thespectrum 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. Afurther quality control is provided by a metric measuring theconvergence of the inversion.Depending on the deployment configuration there couldbe 10 to 100 s of directional spectra measurements acrossthe field of view every 20 min to 1 h. Using standardtechniques (see Section 3.1), this spectrum can be convertedto a directional frequency spectrum (from which Fouriercoefficients are obtained) and to derived parameters suchas significant waveheight, peak period and direction, andwave power. A mean depth at each measurement location isneeded for both the inversion and the conversion processesand best available bathymetry is used for this purpose. Itis also possible to include a dynamic depth by linking theinversion to a tidal model but that has not been used in thispaper.In Section 2, the data sets are described. Section 3.1presents the methods used, Section 3.2 the radar and buoycomparisons, and Section 4 the discussion and conclusions.2 Data setsIn this paper, data from two deployments are used. TwoWERA (Gurgel et al. (1999) systems were deployed onTable1 Statistics of basicwave magnitude parametersParameter Unit Deployment Buoy mean Radar mean cc rms BiasHs m Norway 2.45 2.41 0.95 0.32 0.04Celtic Sea 2.08 2.08 0.92 0.40 0.04T E s Norway 8.67 8.63 0.90 0.66 0.04Celtic Sea 8.37 9.33 0.72 1.58 − 0.95
OceanDynamics(2019)69:123–144 127Table2 Statistics of basicwave direction parametersParameter Unit Deployment Vector correlation magnitude PhaseMean deg Norway 0.92 1.05Celtic Sea 0.90 −4.40Peak deg Norway 0.64 2.38Celtic Sea 0.87 −4.83islands off the Norwegian coast for a period of just overa month as a demonstration of HF radar capabilities forport management during the EuroROSE project (Wyatt et al.2003). Two radars, separated by 10 km to up to 100 kmdepending on radio frequency, are needed to accuratelymeasure both surface waves and currents. The Norwegianradars operated at a radio frequency of 27 MHz and thus hada maximum range for wave measurement of about 20 kmand a maximum measurable waveheight of about 6 m. ADatawell directional waverider was installed at a locationroughly 10 km offshore and some comparisons of radar bulkand spectral wave parameters with this buoy were presentedin Wyatt et al. (2003). An example of a wave map fromthis system is shown in Fig. 2. Over most of the regionmean wave direction reflects swell from the north-west. Tothe south, the wind waves are more dominant with windsacross the region being from the south-east. The seconddeployment involved a Pisces radar (Wyatt et al. 2006)which was deployed at sites on the North Coast of Devonand the South Coast of Wales in the UK looking out over theCeltic Sea. This was operational over about 18 months todemonstrate the wave measurement capability. This systemoperates over a range of frequencies in the lower half ofthe HF band giving longer range and flexibility in theevent of interference or to adapt to different environmentalconditions. However, there are limitations in this casein low waveheights particularly for the measurement ofdirectional characteristics (Wyatt et al. 2011). A Datawelldirectional waverider was deployed at 60 km from bothcoasts. Demonstrating the accuracy of wave measurementsat this range was the main requirement of the project;high spatial resolution was not needed. Figure 2 shows anexample of a wave map from this deployment during astorm. Comparisons of radar bulk and spectral parameterswere carried out (Wyatt et al. 2006).Figure 3 shows the significant waveheight comparisonsfor these two deployments. The Celtic Sea buoy unfor-tunately lost its mooring in Dec 2004 and could not beredeployed until the short break in storm conditions in mid-Jan 2005. UK Met Office model data do confirm the highsignificant waveheights measured by the radar in early Jan.Statistics of the comparisons for some of the main waveparameters are presented in Tables 1 and 2. Energy period,T E =?f −1 E(f)df?E(f)df, where E(f) is the energy spec-trum in m 2 /Hz, is a better period comparator for the radarmeasurements because these have a limited upper frequencydependent on operating frequency. This formulation is dom-inated by the lower, energy containing frequencies and iswidely used in the wave power sector. Higher ocean wavefrequencies dominate in the more standard mean, or first-moment, period, T 1 =?E(f)df?f E(f)dfso, unless the buoyfrequency range is limited to the same range as the radar, theradar will normally measure a higher mean period than thebuoy. The low waveheight limit for the Celtic Sea data setleads to lower accuracy in period and direction unless thedata are filtered to take account of this (Wyatt et al. 2011).For the data shown in the tables, periods and directionsare only included if the Bragg scattering wave frequencyis at least twice that of the peak frequency of a Pierson-Moskowitz spectrum (see Fig. 1), T pPM ? 5 √ H s , whereH s is the radar measured wavelength. During this deploy-ment the flexible frequency was used to deal with externalinterference and not to account for waveheight variationswhich would have avoided this filtering. Note that the fil-tering has only been applied in these tables and not to theFourier coefficients presented later in this paper. This pro-vides the opportunity to explore whether some parts of thespectrum are more sensitive to this limit than others. Thehigh waveheight limit for the Norwegian data is picked upas a quality issue during the inversion process so createsgaps in the data rather than errors. Peak direction is thedirection of the wave component at the peak of E(f). Meandirection is determined from the directional spectrum usingθ m = tan −1? ?S(f,θ)sinθ dθ df? ?S(f,θ)cosθ dθ dfor equivalently in terms ofthe Fourier coefficients using θ m = tan −1?E(f)b 1 (f)df?E(f)a 1 (f)df.Directions are compared here using vector correlation andphase difference as suggested by Kundu (1976). The phasedifference is the same as the mean difference between thedirection measurements.
128 OceanDynamics(2019)69:123–144Fig.4 Spectral data forNorwegian deployment on20/02/2000 at 07:30. a:frequency (with a logarithmicamplitude scale) and meandirection spectra on the left,radar in black, buoy in red,directional spectra on the rightusing a logarithmic colour scaleas shown, radar above, buoy(estimated from Fouriercoefficients as discussed in thetext) below. b: Fouriercoefficients (middle two panels)and derived parameters: upperpanel directional spreading fromfirst two coefficients on left,from 2nd two coefficients on theright; lower panel skewness onthe left, kurtosis on the right. c:spectral shape analysis asdescribed in the text(Section 3.1) radar in shades ofgrey, buoy in shades of cyan.Frequencies near the peak(larger square) are shown indarker shades In the upperframe; three standard directionaldistributions are shown withdashed/dotted grey lines
OceanDynamics(2019)69:123–144 129Fig.5 Spectral data forNorwegian deployment on21/02/2000 at 23:30. Notation asin Fig. 4
130 OceanDynamics(2019)69:123–144Fig.6 Spectral data for CelticSea deployment on 13/02/2005at 02:10. Notation as in Fig. 4.Cases where the value ofkurtosis falls outside the rangeon the yaxis are shown at the topof the plot as empty symbolsand their values
OceanDynamics(2019)69:123–144 131Fig.7 Spectral data for CelticSea deployment on 25/02/2005at 03:10. Notation as in Fig. 4
132 OceanDynamics(2019)69:123–1443 Methods and comparisons3.1MethodsThe output from the inversion process is an ocean wavedirectional spectrum, S(k) on a wavenumber, k, grid. Thegrid is uniform in√ k (where k = |k|) a convenient variablein the inversion process and is thus uniform in frequencyin deep water. In this work, where depths are variable,the√ k grid has been selected with intervals correspondingto 0.005 Hz in deep water frequency. Since all the buoydata used are provided as functions of frequency ratherthan wavenumber, the radar spectra have been converted todirectional frequency spectra, S(f,θ), taking into accountwater depth, in the standard way, i.e. S(f,θ) =dkdfkS(k)(Tucker 1991). Fourier coefficients have been determinedfrom the directional frequency spectra again using standardmethods (Tucker 1991). For example, writing S(f,θ) =E(f)G(θ,f), a n (f) =?π−πG(θ,f)cosnθ dθ.Directional wave data from Datawell buoys are providedeither as Fourier coefficients (estimated from the co- andquad spectra of the buoy measured time series of heaveandlateraldisplacement)or,equivalently,asmeandirection,directional spreading, skewness and kurtosis from whichthe Fourier coefficients can be calculated using standardmethods (e.g. Kuik et al. 1998). Both forms were providedfrom the Norwegian buoy (allowing the conversion fromone to the other to be checked) and the latter formwas provided from the Celtic Sea buoy. The data usedin this paper were provided with a frequency resolutionof 0.005 Hz below 0.1 Hz and 0.01 Hz above. Themain purpose of this paper is to compare the Fouriercoefficients from the radar and buoy but a few examples offull directional spectral comparisons are also included. Anumberofmethodshavebeensuggestedforestimatingbuoydirectional frequency spectra from the Fourier coefficients.In this paper, the Capon (1967) method, as applied byBenoit et al. (1997), has been used to estimate the buoyspectra shown in Figs. 4, 5, 6, and 7a because this providesa smoother, less peaky spectrum, more like that from theradar. It has been found (Waters 2010) that such a modelalso allows for easier and more reliable partitioning of thebuoy data.In the absence of the full directional spectrum, theFourier coefficients can be used to indicate spectralshape and the presence of bi-modality. Defining r i (f) =?a i (f) 2 + b i (f) 2 , a plot of√ r2 (f) against r 1 (f) can beused 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 potentialbimodality in the spectrum plots kurtosis against theabsolute value of skewness both of which can be determinedfrom the Fourier coefficients (Kuik et al. 1998). An analysisof this kind is included below in Figs. 4–7c and providefurther insights into the differences between radar and buoymeasurements. The relationship between√ r2 (f) and r 1 (f)for three standard directional models are shown in thefigures.3.2Radar/buoycomparisonsIndividual measurements of the directional spectrum and itsassociated Fourier coefficients are shown in Figs. 4, 5, 6and 7a,b. Also shown are the frequency spectrum, E(f),the mean direction, and the directional spreading at eachfrequency. In all cases, the radar Fourier coefficients aresmoother but in reasonable agreement with those of thebuoy. Small differences are amplified in the skewness andkurtosis calculations where, in general, the buoy skewnessis more variable and the kurtosis is significantly higher atthe spectral peak, also seen in the shape analysis plots.The inversion process requires some smoothing in bothfrequency and direction to ensure stability in the solutionwhich probably accounts for this (see Green and Wyatt(2006) for a discussion about the need for, and parametersused for, the smoothing). The shape analysis in both plotsin Fig. 4c shows evidence of bimodality in the radar datanear the spectral peak. One explanation is that the frequencysmoothing referred to above is also responsible for thisevidence of directional bimodality, i.e. the individual wavecomponents (wind-sea and swell, as seen in Fig. 4a) havemore well-defined narrower frequency ranges in the buoydata than in the radar data. That is, spectra that are bimodalin frequency but not in direction at a particular frequencyin the buoy data appear bimodal in direction in the radardata because of the frequency smoothing. Some must alsobe attributed to the evidence in both directional spectraplots, albeit clearer in the radar spectrum, of a secondswell contribution well separated from the main swell andwind-sea contributions. The buoy measurements suggestbimodality at frequencies well away from the peak bothabove (squares) and below (circles). This is not seen in theradar data and could be indicating noise in the buoy data atthese frequencies.The directional spectra in Fig. 5a appears to show 4different wave components although two are more mergedin the buoy spectrum. The kurtosis in the buoy data is higherat all these peaks. The upper plot in the shape analysis,
OceanDynamics(2019)69:123–144 133Fig.8 Time series of energyspectra (first Fourier coefficient)for Norwegian (above) and theCeltic Sea (below) deployments.Radar measurement above, buoybelow. Data gaps of 6 hours ormore are shown in white; theplotting program uses the pythonpseudocolor routine pcolormeshand shorter gaps than 6 hoursare thus colour-coded with thevalues at the end of the gap. Theeffect is most noticeable in theCeltic Sea data on about 20/1Fig. 5c, shows no bimodality in the radar data although thelower plot does indicate some multi-modality or perhapsnon-symmetry near the peak. The buoy data appears to bebimodal at very low frequencies but here the amplitude islow so this again could be noise in the data. There is noconformity to standard directional shapes in either case.The radar directional spectra in Figs. 6 and 7 whilstshowing general agreement with the buoy include an extraswell component at about 0.06 Hz. These are likely to berelated to ships, to antenna sidelobe signals associated withvariable surface currents across the measurement region(Wyatt et al. 2005) or to local current shear. Where onecontribution to the spectrum is dominant, e.g. Fig. 6 thereis some evidence in the shape analysis plots of a particulardirectional shape over a range of frequencies near the peak.This is particularly clear for the radar data in this case whichappears to align well with a sech 2 form near the peak, notingthat this is indistinguishable from the cos 2s form very closeto the peak. In general though the data are more scatteredfor both types of measurement and do not conform to a
134 OceanDynamics(2019)69:123–144Fig.9 Time series of directionspectra for Norwegian (above)and the Celtic Sea (below)deployments. Radarmeasurement above, buoy belowparticular form. In the lower plot in Fig. 6c, the buoy isshowing evidence of multi-modality or non-symmetry awayfrom the peak whereas the upper plot shows very littleevidence of bimodality. Non-symmetry is therefore likelyto be the explanation and is of course consistent with theskewness shown in Fig. 6b.TheshapeanalysisinFig.7showssomeevidencethattheradar data is consistent with the sech distribution. However,in this case, this may be biased by the initialisation sincethe peak frequency is quite high relative to the measurementrange. There is no indication of bimodality in the radardata but a slight indication of non-symmetry near the peakin the lower plot. The buoy data looks more like a cos 2sshape near the peak with some evidence of bimodality atlow frequencies where amplitude is low so again possiblynoise in the buoy data. There is also some evidence of lackof symmetry near the peak.For the remaining comparisons, the radar and buoydata at frequency increments of 0.01 Hz from 0.05 to0.2 Hz are used. The first Fourier coefficient is the Energyspectrum, E(f) =? π−πS(f,θ)dθ. This is plotted inFig. 8 at all times when both radar and buoy provide this
OceanDynamics(2019)69:123–144 135Fig.10 Time series of the a 1 (f)Fourier coefficient forNorwegian deployment. Radarmeasurement above, buoy belowmeasurement. Temporal gaps are shown as vertical whitelines. The amplitudes are colour coded according to alogarithmic scale to ensure both high and low amplitudescan be compared. The temporal variation in amplitude anddistribution with frequency seen in the buoy data is wellcaptured by the radar data although, particularly for theNorwegian data the radar amplitudes are a little lowermost likely due to the high operating frequency with aconsequent high waveheight limit. The Celtic sea radarspectra are a little noisier at low frequencies where shipsignals and antenna sidelobes can contaminate the seasignal.The spectra can also be integrated in frequency, E(θ) =? π−πS(f,θ)df to give a mean amplitude in each directionand hence some indication of the directional characteristicsof the wave field. This is plotted in Fig. 9 and again showsgood agreement with very similar temporal variations inamplitude and distribution with direction. There are somedifferences in the Celtic Sea plot during periods of lowwaves (e.g. late Jan). This is consistent with previous workFig.11 Time series of b 1 (f)Fourier coefficient forNorwegian deployment. Radarmeasurement above, buoy below
136 OceanDynamics(2019)69:123–144Fig.12 Time series of the a 2 (f)Fourier coefficient forNorwegian deployment. Radarmeasurement above, buoy below(Wyatt et al. 2011) which has shown that directions (andperiods) have a higher waveheight threshold (dependent onoperating frequency) for accuracy than waveheight itself.During this trial the flexibility in operating frequency thatPisces was used to avoid interference and not to adjust towaveheight conditions which would have minimised thisparticular problem. This has an impact on higher orderFourier coefficient comparisons. Thresholding is neededto remove the low waveheight cases and this has notyet been done for the data shown here. Note thoughtthat there is some indication that the differences aremostly confined to low frequencies so perhaps frequency-dependent thresholding would be more appropriate.A comparison of the four directional Fourier coefficientsfor the Norwegian data set are shown in Figs. 10, 11,12, and 13. The a 1 and b 2 measurements are in goodFig.13 Time series of b 2 (f)Fourier coefficient forNorwegian deployment. Radarmeasurement above, buoy below
OceanDynamics(2019)69:123–144 137Fig.14 Scatter plots and statistics of the a 1 (f) Fourier coefficient forNorwegian deployment. The frequency is shown in the lower righthand corner of each plot. x—buoy, y—radar. cc is the correlation coef-ficient; si is the scatter index but note that this is not very useful forthese data which range between − 1 and 1; N is the number of datapairs in the comparisonagreement although somewhat noisy at low frequencies inboth measurements particularly when amplitudes are low(as seen in Fig. 8). Both radar and buoy b 1 measurementsshow less variation with time. Similar features can beseen in the a 2 measurements although the larger negative,and in some cases larger positive values in the buoy dataare not seen in the radar data. These observations areconfirmed in the scatter plots shown in Figs. 14, 15, 16,and 17. Correlation coefficients of over 0.9 are seen inthe a 1 comparison over a range of frequencies. Aboveabout 0.1 Hz, the standard deviations in the radar andbuoy time series (shown in brackets after the means) aresimilar, and in each case, the rms of the comparison is lowerthan the individual standard deviations. The b 1 coefficientvaries over a smaller range and correlation coefficients arelower. Agreement is qualitatively better above about 0.1 Hzalthough rms differences are now similar in magnitudeto the individual instrument standard deviations which isa concern. The a 2 scatter plots confirm that the buoymeasurements vary over a wider range than those of the
138 OceanDynamics(2019)69:123–144Fig.15 Scatter plots and statistics of b 1 (f) Fourier coefficient for Norwegian deployment. Same notation as Fig. 14radar although the correlation coefficient of over 0.6 athigher frequencies shows reasonable agreement. However,the rms in this case is higher than the standard deviation inthe radar measurements although lower than that of the buoymeasurements. It is possible that this Fourier coefficient ismore sensitive to the inversion smoothing than the others.The correlation and rms compared to instrument standarddeviations, again above about 0.1 Hz, are better for the b 2coefficient than for a 2 .Figures 18 and 19 show the directional parameters,direction and spread, derived from the first-order Fouriercoefficients, i.e. mean direction = tan −1b 1 (f)a 1 (f) , andspread =?2(1 − (a 21 (f) + b21 (f))12 ) both expressedin degrees. Three statistical methods are used for thedirection comparisons: (a) the mean difference, its 95%confidence interval and concentration (Bowers et al. 2000);(b) the circular correlation coefficient (Fisher and Lee1983; Fisher 1993); (c) the vector correlation and phasedifference (Kundu 1976) noting that the phase differenceand mean differences are equal. The statistics improve withincreasing frequency above about 0.1 Hz with increasingconcentrations (high values occur when scatter is low) andcorrelation coefficients and decreasing direction differencesand their confidence intervals. The Kuik et al. (1998)
OceanDynamics(2019)69:123–144 139Fig.16 Scatter plots and statistics of the a 2 (f) Fourier coefficient for Norwegian deployment. Same notation as Fig. 14method has been used to calculate the standard deviationsassociated with sampling variability for the buoy directionand spread data giving mean values over the frequencyrange of 0.1–0.2 Hz of 4.1deg for direction and 8.4deg forspread. In the direction comparison, the mean differencewith its confidence interval is of a similar order. Thestandard deviation between the spread measurements is 10–11deg which is slightly higher than the value calculated forthe buoy. This is to be expected since the radar measurementalso have their own sampling variability and, in addition,there is a positive bias most likely attributable to thesmoothing in the radar measurement already discussed. Aprocedure for estimating the sampling variability of HFradar direction measurements was presented by Sova (1995)but these depend on radio frequency, directional spreadand complexity of the directional spectrum and are notcurrently being used because they are difficult to apply tonew deployments.4 Discussion and conclusionsAlthough ...