Ocean Dynamics (2016) 66:1025–1035DOI 10.1007/s10236-016-0967-6
Spatial characteristics of ocean surface waves
Johannes Gemmrich1 · Jim Thomson2 ·W. Erick Rogers3 ·Andrey Pleskachevsky4 ·Susanne Lehner4
Received: 28 January 2016 / Accepted: 16 June 2016 / Published online: 2 July 2016© Springer-Verlag Berlin Heidelberg 2016
Abstract The spatial variability of open ocean wave fieldson scales of O (10km) is assessed from four different datasources: TerraSAR-X SAR imagery, four drifting SWIFTbuoys, a moored waverider buoy, and WAVEWATCH III� model runs. Two examples from the open north-eastPacific, comprising of a pure wind sea and a mixed seawith swell, are given. Wave parameters attained from obser-vations have a natural variability, which decreases withincreasing record length or acquisition area. The retrieval ofdominant wave scales from point observations and modeloutput are inherently different to dominant scales retrievedfrom spatial observations. This can lead to significant dif-ferences in the dominant steepness associated with a givenwave field. These uncertainties have to be taken into accountwhen models are assessed against observations or whennew wave retrieval algorithms from spatial or temporal dataare tested. However, there is evidence of abrupt changes in
This article is part of the Topical Collection on the 14th Inter-national Workshop on Wave Hindcasting and Forecasting in KeyWest, Florida, USA, November 8-13, 2015
Responsible Editor: Oyvind Breivik
� Johannes [email protected]
1 University of Victoria, Physics and Astronomy, PO Box 1700,Victoria, BC, V8W 2Y2, Canada
2 Applied Physics Laboratory, Seattle, WA, USA
3 Naval Research Laboratory, Stennis Space Center,MS, USA
4 German Aerospace Centre - DLR, Bremen, Germany
wave field characteristics that are larger than the expectedmethodological uncertainties.
Keywords Spatial wave observations · Wave retrievalfrom SAR · Wavewatch III model · Model-data comparison
1 Introduction
The random, multi-scale characteristics of wave fields atthe surface of oceans and lakes are commonly describedby a small set of wave parameters. These parameters arebased on statistical properties of the wave field, like signif-icant wave height Hs , which is defined as the mean of theone third highest waves or spectral properties of the mostenergetic wave scales, like dominant wave length λp, speedcpk or period Tp, and dominant direction θp. These scalesare the scale of the maximum of the energy spectrum and,therefore, are also termed the ‘peak’ wave scales. The vastmajority of wave observations are obtained as time series ofsurface elevation, pressure, or orbital velocities at a singlelocation. Most commonly, wave observations are obtainedfrom surface following floats equipped with accelerometersor global positioning devices (GPS). However, a variety ofmeasuring devices has been developed over the last 60years, or so, ranging from bottom-mounted and sub-surfacemoorings, like pressure sensors and acoustic Doppler cur-rent profilers (ADCPs), to acoustic or optical range-gaugingsensors mounted on platforms or ships tens of metres abovethe surface. The underlying basic principle to all thesedevices is that the wave field parameters are defined asquantities obtained during a finite measuring period, and thewave field is assumed to be stationary over this period. Thisacquisition period is not universally agreed on, but usuallylies in the range of 20 to 40 min. Operational networks of
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wave buoy observations are concentrated in a band of about500 km width along the coast with typical spacing of O(100km), (see e.g., National Data Buoy Center, www.ndbc.noaa.gov).
Alternatively, wave field characteristics may also bederived from a snapshot of the surface elevation over a largeenough spatial extent that allows reliable statistics of allrelevant wave scales. Scanning the water surface with therequired spatial coverage and resolution can be achievedwith marine X-band radars, coastal HF radars, and airborneor space-borne synthetic aperture radar (SAR). As datafrom this type of observations become more prevalent, newinsight on the variability of wave fields over spatial scales ofa few tens of kilometers will be gained. This will have impli-cations ranging from the comparison of sea state parametersderived in frequency or wavenumber space to the apparentoccurrence rate of extreme waves (Heller 2006; Gemmrichand Garrett 2011).
Here, we compare the spatial variability of sea stateparameters in the open ocean obtained from three differentsources: space-borne SAR images, a set of closely spacedbuoys, and a widely used wave model, WAVEWATCH III� (Tolman and the WAVEWATCH III Development Group2014) (in the following: TWDG2014).
2 Methods
We are interested in the spatial stationarity of wave fieldsaway from coastal or heterogeneous, strong atmosphericinfluences. Wave, wind, and surface layer turbulence datawere collected in the vicinity of OceanWeather Station P at50 N, 145 W during a mooring turnaround cruise in Jan-uary 2015 aboard the R/V T. G. Thompson. There were twoTerraSAR-X overpasses coinciding with the in situ obser-vations at Station P. Wind observations were made with a3-axis sonic anemometer mounted to the jackstaff at the bowof the ship at 16 m height above the surface. More infor-mation about additional measurements, and in particular theenergy input into and dissipation of the wave field can befound in Thomson et al. (2016).
2.1 Wave retrieval from synthetic aperture radar SAR
Over the ocean, synthetic aperture radar is capable of pro-viding wind and wave information by measuring the rough-ness of the sea surface. In particular, TerraSAR-X data havebeen used to investigate the highly variable wave climate incoastal areas (Bruck and Lehner 2013; Lehner et al. 2014).In addition, TerraSAR-X data provide accurate estimates ofthe wind field over the ocean.
TerraSAR-X operates in the X-band with 31-mm wave-length from 15 sun-synchronous orbits per day at 514
km, yielding a repeat-cycle of 11 days. However, by vary-ing the incidence angle repeat coverage at mid-latitudelocations can be achieved approximately every 3 days,and more frequently in polar regions. The low altitude,short wave length, and high resolution of TerraSAR-X areadvantageous for the retrieval of meteo-marine parameters,compared to previous satellite-based SAR systems.
The TerraSAR-X data used for this study are Multi-Look Ground Range Detected (MGD) standard products.We use VV-polarization in the so-called StripMap mode,which covers a 30-km wide swath at a pixel spacing of 1.25m. Details on the conversion of image intensity to normal-ized radar cross section σ0 can be found in Pleskachevskyet al. (2016).
The retrieval of wind speeds from TerraSAR-X data isbased on the XMOD algorithm (Ren et al. 2012). The nor-malized radar cross section σ0 is a function of the incidenceangle α, the wind direction φ relative to the SAR flightdirection, and the wind speed U .
σo = ao + a1U + a2 sin(α) + (a3 + a4U) cos(2φ), (1)
where a0...a4 are empirical constants. Calibration of theobserved radar backscatter, i.e., the image brightness, yieldsσ0 (in decibel), and the wind speed U is calculated solving(1) iteratively. The wind direction can be retrieved directlyfrom the SAR image (e.g, Koch (2004)) or taken fromnearby in situ observations or other independent sources.Here, we take the results from the atmospheric modelproviding the wind input for the wave model.
At the same time, the corresponding sea state can be esti-mated from the same image. The empirical X-WAVE modelfunction for obtaining integrated wave parameters from X-band data has been developed for deep water (Bruck andLehner 2013; Bruck 2015). Here, we use a reduced ver-sion of the original model function X-WAVE to estimate thesignificant wave height
Hs = A0(β) + A1
√E tan(α), (2)
where A0, A1 are empirical coefficients. The varyingdegrees of non-linear imaging distortion inherent in SAR-observations are accounted for by the dependence on β, theangle between the dominant wave direction, and the satel-lite azimuth, i.e. flight direction (0o ≤ β ≤ 90o). For adetailed discussion of TerraSAR-X imaging artifacts, seePleskachevsky et al. (2016).
The parameter E is an integral measure of the scaledradar backscatter
σ̃ = σo
σ̄o
− 1 : (3)
E =∫ k2
k1
∫ 2π
0P(k, θ)dkdθ, (4)
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where P is the 2-d power spectrum of σ̃ . The wave num-ber integration limits k1, k2 correspond to wave lengthsbetween 500m and 30m. TerraSAR-X flies at an orbit of514 km, which is lower than most space-borne SAR sensors,resulting in relatively weak SAR-specific imaging artefacts.Nevertheless, to remove the energy associated with poten-tial image distortions, the 2-D power spectrum is scaled in away that the 1-d wave number spectrum
S(k) =∫ 2π
0P(k, θ)dθ ∝ k−3, (5)
for 1.5kp < k < k2, where kp is the wave numberassociated with the dominant wave scale, in case of a single-peaked wave spectrum. For double-peaked spectra kp isassociated with the dominant wind sea (Fig. 1), definedas the spectral peak in the actively forced spectral rangek < g/U2, corresponding to spectral wave ages cp/U < 1,with phase speed cp.
The constants A0, A1 have been tuned to match Hs
observations over NDBC wave buoys. The dominant wavelengths and wave direction are given by the maximum
Fig. 1 Spectra of TerraSAR-X sub-scene centred at 50.03oN,145.12oW on 5 January 2015. a 2-D wavenumber spectrum. Circlescorrespond to wave lengths of 30, 50, 100, 200m (decreasing radius).b 1-D wavenumber spectrum (line) and dominant wave scale (circle).High wavenumbers are scaled to obey theoretical dependence (black).Raw spectrum is shown in gray. c 1-D frequency spectrum. The dashedline indicates a ω−4 dependence, the circle the dominant frequency ofthe wind-sea
value of the 2-D image wavenumber spectrum. Utilizing thedeep-water wave dispersion relation without currents
ω2 = gk (6)
with wave frequency ω = 2πf and gravitational accel-eration g, the spectrum can be converted into frequencyspace.
Wave parameters are obtained from sub-scenes of theSAR image. Typically, we downsample the TerraSAR-X stripmap image to a 2.5 m pixel resolution, and basethe wave and wind retrieval on consecutive 1024 × 1024pixel scenes. To increase spectral resolution, and to reducealiasing effects, the sub-scenes are centered within a zero-padded 2048 × 2048 array. Thus, wave parameters arederived at a spatial resolution of approximately 2.5 km.The sensitivity of the results on the choice of the sub-scenesize is discussed in Section 4. The 180o ambiguity of theSAR spectra (Fig. 1a) can be resolved with complex data orestimated from the image directly (Koch 2004). Here, wesimply chose the peak closest to the wind direction.
2.2 In situ wave buoy observations
A suite of instruments called SWIFT (Surface Wave Instru-ment Floats with Tracking) has been developed to observewave breaking and turbulence close to the surface in a wave-following frame of reference (Thomson 2012). SWIFT’s arefreely floating and are designed to follow the free surface.GPS sensors in the hull collect velocities and heave accel-erations at 4 Hz, from which surface elevation time seriesand directional information are obtained (Herbers et al.2012). Data segments of 30 min are used to estimate signif-icant wave heights and 1-d frequency spectra. Here, up to 4SWIFTs have been deployed within the TerraSAR-X imagecoverage, or close-by.
In the open ocean, the SWIFT drift velocity vS is acombination of wind drift, wave-induced Stokes drift, andinertial currents. Typical drift speeds are |vS | � 1m/s atangles relative to the dominant wave propagation θS < 30o.For a better comparison with the observations at fixed loca-tion, we correct the dominant wave frequency observed bySWIFTs ω̃p for the effect of the drift
ωp = ω̃p + kp |vS | cos(θS) (7)
prior to calculating the dominant wave length via the disper-sion relation (6).
In addition, a 0.9m Datawell DWR MKIII directionalwaverider moored at Station P (CDIP166, and NDBC46246) yields Hs as well as directional wave spectra basedon 20-min acquisition records. The waverider collects buoypitch, roll, and heave displacements at 1.28 Hz on a 30-minduty cycle, following the Datawell standards. Both types ofbuoys process data onboard at the end of each duty cycle
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and transmit the spectral moments to servers on shore viaIridium satellites.
2.3 Wave model
The numerical wave model used here is WAVEWATCHIII � (TWDG2014), with the physics package of Ardhuinet al. (2010). The geographic grid is global, at 0.5 degreeresolution, and the spectral grid includes 36 directionalbins and 31 frequency bins (0.0418 to 0.72 Hz, logarith-mically spaced). An obstruction grid is used to representunresolved islands (TWDG2014). Winds and ice concentra-tions are taken from the Navy Global Environmental Model(NAVGEM) (Hogan and Coauthors 2014), with the formerinput at 3-h intervals and the latter at 12-h intervals. Thesimulation is initialized 0000 UTC 1 December 2014 andends 0000 UTC 1 February 2015. With invalid spin-up timeremoved, the valid portion of the simulation is 0000 UTC 15December 2014 to 0000 UTC 1 February 2015. The physicspackage of Ardhuin et al. (2010) requires specification of aparameter, βmax which is used to compensate for the meanbias of the input wind fields, or lack thereof; βmax = 1.2 isused for this hindcast. The model version number employedis development version 5.04. In context of the present appli-cation, this is not substantially different from public releaseversion 4.18.
Hs is calculated as 4 times the square root of the total seasurface variance, which is the integral of the variance den-sity spectrum, which is the prognostic variable of the model.The dominant wave length λp is calculated using the lineardispersion relation for deep water using peak frequency fp,which is calculated within the model using a parabolic fitaround the discrete peak of the one-dimensional frequencyspectrum (for details, see TWDG2014). The peak direc-tion’ is the mean direction of the peak frequency. The mean
direction is arctan(b1/a1), where a1 and b1 are a1(fp) andb1(fp), the Fourier coefficients of the directional spreadingfunction at the peak frequency (see e.g., Longuet-Higginset al. (1963)) . It is important to note that this ‘peak direc-tion’ is calculated in the same way as for the waverider andthe SWIFTs. However, ‘peak directions’ from the modeland the buoys are therefore neither the peak of the 2-Ddirectional spectrum nor the peak of the 1-D directionalspectrum.
3 Results
The first TerraSAR-X overpass occurred on January 1, 2015at 16:04 UTC, during the passing of a storm north of thesite. Winds had shifted from southerly winds prior to 9:00UTC to north-westerly direction starting around 12:00 UTC.The analysis of the SAR images shows a gradual increaseof wave height from Hs ≈ 2.3m at the southern end of theswath to Hs ≈ 3.3m about 100 km further north (Fig. 2).The increase in wave height coincides with a similar gradualincrease of wind speed from u ≈ 10.5m/s in the southernregion to u ≈ 13m/s in the north, indicating that the wavefield is predominantly a local wind-generated sea.
The SAR-retrieved wave heights are in good agreementwith the in situ observations obtained from three SWIFTbuoys and the waverider. However, the in situ observationswere spaced too closely to make any inference about thespatial evolution of the wave field. Wind speeds retrievedfrom the SAR image and wind speeds utilized for the wavemodel are consistent and are about 20 − 25 % higher thanreported by the SWIFTs. The SWIFT anemometer is only1 m above the surface, and wind speeds have not beenadjusted to the standard 10 m reporting height. Wave heightsobtained from the wave model capture the gradual south - to
Fig. 2 Fields of significantwave height Hs (left) and windspeed u (right) on 1 January2015, retrieved fromTerraSAR-X images. Individualsymbols give location andvalues as obtained from SWIFTbuoys (red triangle), waverider(green x), and results from thewave model (black,o)
Ocean Dynamics (2016) 66:1025–1035 1029
Fig. 3 Fields of dominant wavelength λp (left) and dominantwave direction θp (right) on 1January 2015, retrieved fromTerraSAR-X images. Individualsymbols give location andvalues as obtained from SWIFTbuoys (red tringle), waverider(green x), and results from thewave model (black,o)
- north increase, but with a smaller gradient and generallyhigher values: 3.4m ≤ Hs ≤ 3.8m .
The dominant wave length λp and dominant wave direc-tion θp are indirect parameters related to a measure of thespectral maximum. Open ocean wave fields commonly con-sist of the superposition of several wave fields of differentspectral strength, which can result in multi-peaked spectra.The SAR-retrieved dominant wave parameters as given bythe maximum of the 2-D image spectrum shows rather shortwave lengths increasing from λp ≈ 60m in the south toλp ≈ 75m in the north, at a direction of about 285o (Fig. 3).Contrary to the trend obtained from the SAR-image, thewave model predicts a much longer dominant wave lengthof λp ≈ 115m, slightly decreasing from south to north. Thebuoy observations yield intermediate wave lengths of λp ≈80m from the SWIFT buoys and λp = 108m from thewaverider buoy, with no clear spatial trend. Thus, dependingon the data source, vastly different dominant wave lengthsand reversed trends can be obtained. Furthermore, as will bediscussed in more detail below, the definition of the dom-inant wave scale depends on the spectral analysis method,
and results will differ for different methods. The differencesbetween the various observations are even more pronouncedfor the dominant wave direction. According to the SARanalysis and the SWIFT observations, the dominant wavedirection is from WNW, θp ≈ 290o, whereas the wave riderand the model show waves from S, θp ≈ 170o to 190o
(Fig. 3).The second TerraSAR-X overpass, 5 January 2015, 03:30
UTC, extents about 80 km along-track. Easterly winds atabout 14m/s prevailed, with little variation over the imagedregion (Fig. 4). The SAR image indicatesHs values of about3.5m south of 50oN sharply increasing to Hs ≈ 4.4min the northern section of the image. This division of thewave field is also seen in the dominant wave length, jump-ing from λp ≈ 80m in the south to λp ≈ 200m in thenorth, and a shift in dominant wave direction from θ ≈ 90o
to θ ≈ 50o (Fig. 5). There was one SWIFT within thesouthern section of the SAR image and another 3 SWIFTsabout 25 km west of the central image section. ReportedHs values vary from 3.8m in the south to 4.1m outsidethe central section, roughly consistent with the values and
Fig. 4 Fields of significantwave height Hs (left) and windspeed u (right) on 5 January2015, retrieved fromTerraSAR-X images. Individualsymbols give location andvalues as obtained from SWIFTbuoys (red triangle), waverider(green x), and results from thewave model (black,o)
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Fig. 5 Fields of dominant wavelength λp (left) and dominantwave direction θp (right) on 5January 2015, retrieved fromTerraSAR-X images. Individualsymbols give location andvalues as obtained from SWIFTbuoys (red triangle), waverider(green x), and results from thewave model (black,o)
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spatial pattern obtained from the SAR images. Results fromdifferent SWIFTs less than 1 km apart vary by 0.1m, indi-cating Hs ≤ 0.1 m is the uncertainty inherent in these30-min wave observations. Modeled wave heights increasefrom Hs = 3.8m in the south to Hs = 4.0m in the north,supporting the increase, but not the relatively sharp jump,in wave height retrieved from the SAR image. In general,wave models do have an inherent tendency of smoothingsharp gradients. The modeled wave spectra and, thus, waveheights, are a result of spatial and temporal integration ofthe effect of the wind field on the waves, together withadvection. The integration naturally produces smoothing,and the winds themselves are already smooth, as they do nothave the resolution to predict sub-mesoscale features in theatmosphere.
Only the SAR-retrieved wave field suggests the divisionof easterly, wind-sea dominated waves in the south, andnorth-easterly swell in the north, due to a relative shift ofspectral energy (Fig. 6). All SWIFTs report wind seas withλp ≈ 90m, but a change in dominant wave direction fromθp = 73o in the southern section to θp = 47o and 345o westand northwest of the SAR image. The wave model predictsa complex sea state consistent of multiple systems. Reduc-ing this multi-modal system to single dominant parameterscan give misleading values. Here, the resulting dominantwave lengths fall between the wind sea and swell with λp ≈120m in the south, increasing to λp ≈ 130m at the northernedge of the SAR image, and θp ≈ 45o (Fig. 5). Overall, inboth examples, the significant wave heights obtained fromthe four data sources agree relatively well. Reducing multi-peaked directional spectra to “dominant parameters” leadsto noticeable differences in dominant wave direction and inpeak wave length. As discussed below, this also leads tosignificant differences in wave steepness.
4 Discussion
The comparison of the SAR-retrieved wave parameters tothe in situ and the modeled wave parameters revealed some
significant differences, in particular when dealing withdominant scales. Historically, most wave measurementsare obtained as time series at a fixed location. Therefore,best practices are defined in the time domain, and wavescales are based on frequency, rather than wave numberspace. For spatial wave observations at a fixed time, Hs esti-mates might be expected to depend on the survey area, aswell as the directional spreading and crest length of the wavefield.
Based on the dispersion relation, wave parameters ofindividual waves may readily be expressed in temporal or
Fig. 6 Image spectra on 5 January 2015; a and b for southernsection at 49.64oN, 144.88oW, and c, d for northern section 50.39oN,145.07oW
Ocean Dynamics (2016) 66:1025–1035 1031
spatial description. However, for a spectrum of wave scales,the dominant parameters do not readily convert (Gebhardtet al. 2015). Thus, the question arises what are best practicesfor obtaining wave field parameters from spatial obser-vations, and what are the limitations of assessing spatialalgorithms with the help of temporal observations, or viceversa.
4.1 Observational record size
Significant wave height is defined as the average of thehighest one third waves within a record. For a random seawith a Gaussian distribution of surface elevations, this iswithin 1 % error the same asHs = 4ση, where ση is the stan-dard deviation of the surface elevation time series. However,even for stationary wave records, the result will fluctuatedepending on the the record length. We simulate a 10-h-longstationary surface elevation record for a fully developed sea(Gemmrich and Garrett 2011), and calculate Hs values atthe beginning of each hour, but with varying acquisitionlengths from 10 min to 1 h, each yielding 10 estimates forHs (Fig. 7). The average value of Hs is in all cases within1 % of the correct value specified in the simulation. How-ever, for the shortest acquisition (10 minutes), the estimatesspan a range of > 20 %, more than twice the uncertainty
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Fig. 7 Variability of significant wave height Hs as function of acqui-sition period T. Black errobars give range of Hs calculated from 10sections with same starting time but different length.Gray circles showthe range for all consecutive records. All Hs estimates are based on thesame simulated, stationary surface elevation time series
obtained for acquisition lengths T ≥ 40min. Conversely,one could consider calculating Hs from all consecutive datasections for the entire data record, thus increasing the num-ber of independent estimates for acquisition times T <
1 hour. For all acquisition lengths, the average Hs value is 4m. However, for the shortest acquisition, the estimates spana range of 3.56m ≤ Hs ≤ 4.57m. Uncertainties seem tostabilize at ≈ 10 % for acquisition length T > 30min.
A recent study examined the statistical uncertainty ofwave height and wave period estimated from waveriderobservations in the North Sea (Bitner-Gregersen andMagnusson 2014). They found the variability of waveheight estimates to be greater than the uncertainty in waveperiods. They also point out that “sampling variability of Hs
may have a significant impact on (...) validation of differentdata sources and wave models”.
Taking Tp = 10 s as a typical dominant wave period, thestandard operational acquisition length of 38 min (Environ-ment Canada) yields about Nt = 225 dominant waves asthe basis for estimating wave field parameters. Therefore,it would be desirable to base the analysis of spatial recordson a similar number of waves Ns ≈ Nt . For unidirectional,long-crested waves, an equivalent spatial acquisition size oflength La = Nsλp and arbitrary width could be defined.However, for directional, short-crested wave fields com-monly observed at mid-latitude open ocean locations, it isgenerally not known how many independent dominant wavecrests are contained within an acquisition area La × La .Here, we compare Hs , and λp estimates obtained from ourstandard acquisition area with La = 2560 m to to estimatesbased on acquisition areas with La = 5120 m. For con-venience, we assume Ns = (La/λp)2 as an upper boundfor the number of independent dominant waves within theacquisition area.
On 1 January 2015, the dominant wave length was about70 m, yielding Ns � Nt for both sizes of the acquisi-tion area. Thus, there is no systematic difference of the Hs
estimates between the two acquisition sizes (Fig. 8a), andfor the majority of the estimates the difference is |Hs | ≤0.2 m (Fig. 8c). On 5 January 2015, the influence of longerswell prevailed in the northern section of the observationalarea, with λp ≈ 220 m, yielding Ns � Nt only for thelarger acquisition area, but Ns � Nt for La = 2560 m. Forthe northern section, where Hs > 4 m, estimates obtainedfrom the smaller (i.e. standard) acquisition are clearly biasedtoward lower wave heights (Fig. 8b). Individual differencesreach |Hs | > 1 m, but again the majority of the estimatesare in the range |Hs | ≤ 0.2 m (Fig. 8d).
For both datasets, dominant wave length and dominantdirection do not show a systematic dependence on recordsize (not shown). However, in the 5 January 2015 data, thepeak jumps occasionally from the dominant wind sea to thedominant swell.
1032 Ocean Dynamics (2016) 66:1025–1035
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Fig. 8 Top Variability of significant wave height Hs over entire imagearea, for acquisition areas of 2560m × 2560 m (gray) and 5120m ×5120m (black). Bottom:Difference of Hs estimates between 5120m×5120m and 2560m×2560 m acquisitions. For 1 January 2015 (a), (c)and 5 January 2015 (b), (d)
4.2 Dominant wave scale
Viewing TerraSAR-X imagery of the open ocean scenes onecan often readily determine by eye a dominant wave lengthand a dominant wave direction. The equivalent data analysis
method is to determine the location kp, θp of the peak of thetwo-dimensional wave number spectrum S(k, θ) (Fig. 1a).Integrating the 2-D spectrum over all directions gives the 1-D wave number spectrum (Fig. 1b). For most wave fields,the dominant wave length λp = 2π/kp from the 2-D and 1-D wavenumber spectra match. However, there are situationswhere the integration can lead to a different dominant scale.For example, assume two moderate swell systems with sim-ilar wave lengths but different directions and a wind sea thatis stronger than either of the swell systems; the sum of theswell systems can have a higher 1-D peak than the wind seaand the dominant wave scale would jump from the wind seaas obtained from the 2-D spectrum to the dominant scale ofthe swell. The opposite case, where the peak of the longerwaves dominates the 2-D spectra, but shorter waves are themost energetic directionally-integrated scales occurred on 5January 2015 (Fig. 9a,b). In the southern part of the obser-vations, both methods are in good agreement, whereas in thenorthern section, there is a clear division of scales.
For temporal wave observations, the dominant scale isdefined as the peak of the frequency spectrum (Fig. 1c). Thissituation can be emulated from a 2-D wavenumber spec-trum via the dispersion relation and the Jacobian (k ∂k/∂ω).Due to this conversion, the peak in frequency space cor-responds to slightly reduced peak wave lengths (Gebhardtet al. 2015). This effect can clearly be seen in the dataset of 5January 2015, where frequency-based wave lengths are sys-tematically shorter than the dominant wave scale obtainedin 1-D and 2-D wave number space (Fig. 9c).
Dominant wave parameters in the wave model are calcu-lated from various moments and weights of spectral energy(e.g., TWDG2014), ensuring a smooth evolution of waveparameters in time and space. For comparison with the
Fig. 9 Comparison of dominantwave length λp on 5 January2015, derived from TerraSAR-Ximages based on 2-D wavenumber spectra (a), 1-D wavenumber spectra (b), and 1-Dfrequency spectra (c). Graycolour scale depicts swell,ranging from 190 to 240m (darkto light)
Ocean Dynamics (2016) 66:1025–1035 1033
SAR-retrieved parameters, more direct estimates might bedesirable. Therefore, we calculate different estimates ofdominant wave parameters directly from the model outputspectra, along a transect through the SAR images at 145W(Fig. 10). The dominant wave length is obtained via the dis-persion relation from estimates of the peak wave period Tp.Here, we include the standard wave model output Tmm10, aswell as the peak of the 1-d frequency and the 2-D frequency-direction spectra. In terms of wave length, the results canvary by up to λp = 80m, but all estimates are larger thanthe SAR analysis, and the SWIFT buoys, suggest. The dom-inant wave direction in the model is commonly calculatedas the mean direction of the wave components of the dom-inant frequency. In addition, we extract the direction fromthe peak of the modeled 2-D wave spectra. Both methodsagree within 30o with each other, but can be vastly different
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100
[o]
49.6 49.8 50 50.2 50.4
Latitude [N]
2.5
3
3.5
4
4.5
Hs [
m]
Fig. 10 Comparison of dominant wave parameters based on differentdefinitions, from model results (symbols) and from SAR image analy-sis (line), along a cross section at 145W. Left column 1 January 2015,right column:5 January 2015. Top row Dominant wave length λp fromstandard model output (black circles), from 2-D spectra (red circles),from 1-D spectra (blue circles), from 1-D k-spectra (black line), from2-D k-spectra (red line), and from 1-D ω-spectra (blue line). Middlerow Dominant wave direction θp from 2-D frequency spectra (red tri-angles), from 1-D spectra (black triangles), and from 2-D k-spectra(red line). Bottom row Significant wave height Hs
from the SAR and SWIFT results. This is likely due to thesmoothing and relative coarse directional resolution inher-ent in the model result. The significant wave height hascontributions from all spectral components, rather than onlya small subsection like dominant period or direction. Dif-ferences in Hs between the model and the observations aretherefore mainly due to the coarse spatial resolution of themodel.
Depending on the data source and the processing methodthe results for the dominant wave lengths can vary byalmost a factor of 2. In addition, for multi-peaked wavesystems, the dominant scale can jump between peaks ofsimilar strength. This has huge implications on the domi-nant steepness Hs/λp. For moderate sea states, it is oftenthe steepness of a wave field rather than the absolute waveheight that determines marine risks. Furthermore, dominantsteepness is used as first-order predictor of breaking ratesof dominant waves (Banner et al. 2000) and, thus, air-seaexchange processes. However, the overall wave breakingrate and the associated turbulence levels also depend onthe steepness in the saturation range of the spectrum, andmean-square slope or the related wave saturation, are bet-ter indicators of breaking than dominant steepness (Banneret al. 2002; Schwendeman et al. 2014; Schwendeman andThomson 2015).
For pure wind seas, the “3/2 power law” predicts Hs ∝T3/2p (Toba 1972), whereTp is the peakwaveperiod.Utilizing
the deep water dispersion relation (6) this is equivalent to
Hs ∝ λ3/4p . (8)
Although our two datasets include swell components, theyare mainly dominated by developing wind sea, and Eq. 8 canprovide a consistency check. For 1 January 2015 waves inthe northern section, where Hs ≥ 3.0m, agree well with the“3/2 power law”, and SAR-retrieved wave parameters andSWIFT results are consistent. Model-based wave steepnessare significantly less, and do not indicate a coherent spatialwave evolution (Fig. 11a). Surprisingly, the waverider givesan intermediate wave steepness, lower than the estimatesfrom SWIFT and SAR. The waverider internal processingincludes a coarse low-pass filter of the 1-d frequency spec-trum. It is likely that the peak at an intermediate frequencyis the result of the smoothing and coarse resolution of thespectrum, rather than being caused by difference in theobservational technique.
The second dataset, 5 January 2015, is characterized bywind sea in the southern section and swell-dominated seasin the northern part. Wave lengths retrieved in the 2-D wavenumber space in the wind sea section closely follow the evo-lution curve. Wave parameters from SWIFT are on the samesteepness curve but at greater wave development (Fig. 11b).The waverider is located in the transition between wind seaand swell-dominated region and the energy at the lower
1034 Ocean Dynamics (2016) 66:1025–1035
Fig. 11 Comparison ofsignificant wave height Hs asfunction of dominant wavelength λp for different datasources and methods: SWIFTbuoys (red �), waverider (green×), wave model (black,◦),TerraSAR-X images based on2-D wave number spectra (blue),1-D wave number spectra (gray×), and 1-D frequency spectra(orange ). Dashed lines givewave height evolutionHs ∝ λ
3/4p for normalized initial
steepness 1, 0.5 and 0.33 (left toright) (Toba 1972). a 1 January2015, b 5 January 2015
101
102
[m]
2
2.5
3
3.5
4
4.5
5
Hs [
m]
a
101
102
[m]
2
2.5
3
3.5
4
4.5
5
b
frequency is slightly higher than the energy in the frequencyrange corresponding to the wind sea (not shown). Thus, thewaverider, and the SAR results from the northern sectionwith Hs > 3.5m, have the lowest steepness of all datasources. These data are swell dominated and do not show aspatial wave evolution. Peak wave lengths from the SAR 1-D spectra are in the wind sea range over the entire image.However, they are roughly constant, and do not indicatewave evolution, even for the southern section. The wavemodel predicts an evolving wave field, but at about 60 % ofthe steepness measured by SWIFTs or TerraSAR-X. Basedon these results, the retrieval of dominant wave scales fromthe 2-D wave number space is recommended.
5 Conclusions
Dominant wave field parameters are estimated based on acertain observations record size. Due to the random natureof oceanic wave fields, these parameters possess an inherentuncertainty which decreases with increasing record length.For spatial observations and pure wind seas, a standardacquisition area of 2.5 km × 2.5 km seems to be appro-priate. However, for mixed wind seas and swell, a largerarea is recommended. For point observations of a station-ary wave field, estimates of significant wave heights basedon 30-min time series have about half the uncertainty com-pared to estimates from a 10-min record length. Spectraldifferences in wave number and frequency space lead toslightly different dominant wave scales from time series orspatial wave observations. The reduction of complex multi-system sea states to single dominant parameters can yieldhuge uncertainties in dominant wave length and direction.These natural and introduced fluctuations should be kept in
mind in comparison of different observational methods orin model validation studies.
Nevertheless, wave fields in the open ocean can includetrue wave height variations that exceed the measurementuncertainty at spatial scales of O (10 km), stressing the needfor co-located observations for the assessment of modelresults or empirical wave retrieval algorithm. Satellite-borneSAR imagery is an ideal tool to retrieve such meso-scalewave field changes, which tend to be smeared out in opera-tional wave predictions.
The next step will be to explain the physics of theserelatively abrupt wave field fluctuations. They seem notassociated with spatial pattern in wind speed. We spec-ulate, wave modulations by inertial currents might be acontributing factor (Gemmrich and Garrett 2012).
Acknowledgements The TerraSAR-X data were provided by DLRunder AO OCE1837 (JG). The in situ data collection was fundedby the US-National Sciences Foundations (JT). We thank Joe Talbertand Alex de Klerk for SWIFT engineering and the crew of the R/VT.G. Thompson for deployments and recoveries of the buoys. Thecontribution by ER for this work was supported by the Office ofNaval Research through an NRL Core Project, Program Element#0602435N. This work was partially supported by the US Officeof Naval Research through the DRI “Sea State and Boundary LayerPhysics in the Emerging Arctic”. We thank the two anonymousreviewers for their comments.
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