FEATURE

THE OCEAN WAVE DIRECTIONAL

SPECTRUM

By Lucy R. W y a t t

T H E DIRECTIONALSPECTRUMS(k) [or S(f,0)] mea- that is marketed by Marconi Radar Co. as a sur-
sures the distribution of wave energy in wave face current measurement radar and has a maxi-
number, k, (or frequency, f) and direction. Differ- mum range of -20 km for wave measurement. The
ent contributions to local wave energy, e.g., swell method has also been used with two different
from distant storms and locally wind-generated radar systems: Pisces, a system developed by Nep-
waves, can be identified in a measurement of the tune Radar Ltd. from a University of Birmingham,
directional spectrum (see Fig. 1). The direction of U.K., prototype that has a maximum range of
propagation of wave energy and the period (l/f) of - 1 5 0 km for wave measurement (Wyatt, 199l),
the most energetic waves are important for many and WERA (WEllen RAdar), a system under de-
practical applications, e.g., the design and opera- velopment at the Institut ftir Meereskunde, Univer-
tion of coastal and offshore structures and storm sity of Hamburg, Germany, with a maximum
surge forecasts. range similar to OSCR.
The use of HF radar to make measurements of
The Method
the spectrum is based on equations developed by
The concept of the method is simple. Signifi-
Barrick (1972a,b) that relate the power spectrum
of the backscattered signal to the ocean wave
cant waveheight and wind direction are estimated T hmeasurement
e
from the radar data using methods referred to else-
spectrum through a nonlinear integral equation. of the directional
where in this issue (Fernandez et al.. 1997; Graber
The backscattered signal is dominated by Brags
and Heron, 1997; see also Wyatt 1988: Wyatt et spectrum is only
scattering, i.e., the ocean waves responsible for the
al., 1997). These are used to derive a model direc-
scatter have a wavelength of one-half the radio possible because
tional spectrum using both the waveheight to de-
wavelength. The measurement of the directional
termine a Pierson-Moskowitz wave number spec- ocean waves are not
spectrum is only possible because ocean waves are
trum and also the wind direction in a directional
not simple sinusoidal forms satisfying the linear simple sinusoidal
distribution. This model spectrum is fed into the
dispersion relationship. If they were linear there
integral equation that is integrated directly to de- forms • . ,
would be only the two first-order peaks in the
termine the corresponding Doppler spectrum. Dif-
backscattered power (Doppler) spectrum. The non-
ferences between this Doppler spectrum and the
linear properties of ocean waves give rise to waves
measured spectrum are used to modify the direc-
with the correct wavelength for Brags scatter but
tional spectrum at each wave number (within a
with different frequencies and hence different
limited range) and direction. This process is re-
Doppler shifts. This is the second-order part of the
peated until the differences become sufficiently
spectrum described by Barrick's integral equation.
small. The way in which the modification is car-
To make the measurement, the integral equa-
ried out is the key to the success of the method.
tion must be solved. This is not straightforward
The details are too complicated to be included
and a number of methods have been proposed
here, but it is important to realize that although the
(Lipa, 1978: Wyatt, 1990: Howell and Walsh,
method starts with a simple directional spectrum
1993: Hisaki, 1996). No method as yet has gained
with a single wind-wave mode, the solution (when
widespread acceptance and there are no opera-
the procedure has converged) can be very differ-
tional HF radars making routine wave measure-
ent; for example, it has no problem detecting the
ments. Here we concentrate on the Wyatt method
presence of swell propagating in a very different
and show some measurements from the Ocean
direction.
Surface Current Radar (OSCR) HE radar system
The method does not provide a measurement of
the directional spectrum at all wave frequencies.
Lucy R. Wyatt, SheffieldCentre for Earth Observation Sci- High-frequency waves are not adjusted in the pro-
ence Schoolof Mathematics and Statistics (AM), Universityof cedure and retain the memory of the initializing
Sheffield, Hounsfield Road, Sheffield $3 7RH, UK. spectrum. The cutoff, at OSCR frequencies, is

OCEANOGRAPHY'VoI.10, No 2"1997 85
 06 28/12/95 054 22 21/12/95 055 07 22/12/95 055 17 17/12/95 074
Radar 0.5 Radar 1.9 Radar 2.0 Radar 0.7

200

(D x 1.0 m 2/H#rad
"{~ 100 ] above 1.~
t-
O L_ i ] 0.90 - 1.00

¢,~
0 --
0.0
-
0,1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.1 0.2 0.3 o,'--] 0.80 - 0.90

waverider 1.0 waverider 1.9 waverider 3.4 waverider 1.3 / 0.70 - 0.80
C3 0.60 - 0.70
~3o0
> 0.50 - 0.60
0.40 - 0.50
0.30 - 0.40
0.20 - 0.30
100
! 0.10 - 0.20
below 0.10
0 !--
0.0 0.1 0.2 0.3 0.4 0,1 0.2 0.3 0.4 0.1 0.2 0.3 0,4 0.1 0.2 0.3 0.4

Frequency (Hz)
Fig. 1: Directional spectra measured by the radar (top row) and estimated from the directional waverider measurement (bottom
row) with frequency in Hz on the horizontal axis and direction toward which the waves are propagating on the vertical. Four differ-
ent types of wavefield are shown: (left) swell-dominated sea with a peak frequency of 0.09 Hz; (mid-left) swell (0.09 Hz) with some
wind sea with a peak frequency ~0.18 Hz; (mid-right) wind-sea (0.16 Hz) with some swell (0.09 Hz); (right) wind-sea peaking at
0.14 Hz. The hour and date of each measurement is shown together with the cell location of the radar measurement (all close to the
wavebuoy site). The number above each plot is the peak in the directional spectrum for that plot. Arrow heads, short wave direction.
The color coding is indicated on the right: dark blue, <0.1; yellow, >1.0.

~0.38 Hz, the exact value depending on the look components in the spectrum that are similar in fre-
directions of the two radars at the measurement quency but different in direction. This, of course,
O n e advantage of
point. The Lipa, Howell, and Walsh methods have applies both to the wavebuoy and the radar analy-
this method is that similar limitations, the exact cutoff in each case sis. In the Wyatt method the directional spectrum is
depends on the model assumed for high frequen- discretized in vector wave number and the solution
wave components in
cies and on the range of Doppler frequencies in- is determined at each discrete vector wave number.
the spectrum that are cluded in their analyses. Hisaki's method is a non- One advantage of this method is that wave compo-
linear optimization method and in principle nents in the spectrum that are similar in frequency
similar in frequency extends the range of frequencies, although the but different in direction emerge naturally during
but different in method has not been exhaustively verified. For the inversion (Wyatt and Holden, 1994). Validating
most applications it is the longer, lower-frequency, the detailed structure in the radar-measured spec-
direction emerge energy-containing waves that are of interest. Wind trum is not straightforward because wavebuoys
speeds would have to be very low or fetch very only measure limited detail. One approach is to
naturally during the
short for the peak in the spectrum to be at frequen- partition the wave spectra into swell and wind-
inversion. cies close to the cutoff. wave modes and compare integrated parameters for
An important difference between the Wyatt these. Such an approach is under development
method and those of Lipa, Howell and Walsh, and (Isaac and Wyatt, 1997).
Hisaki is the form of the solution. In the other three
methods the directional spectrum is discretized in Holderness Measurements
scalar wave number or frequency and is expressed The data set used here to demonstrate the suc-
as a truncated Fourier series in direction, which al- cess of the method was collected using the OSCR
lows the problem to be expressed as a matrix equa- HF radar during a deployment on the U.K. coast at
tion for the Fourier coefficients. The Fourier coeffi- Holderness. Directional waveriders were deployed
cients used are the same five used in the analysis of at two offshore locations about midway between
directional wavebuoys; methods employed in the the radar sites, one in -10 and the other in -20 m
interpretation of buoy data can be applied directly. water depth. Depth contours are roughly parallel to
One limitation is difficulty in identifying wave the coast deepening to 10 m -1 km offshore. The

86 OCEANOGRAPHY"Vol. 10, No. 2"1997

 4| . radar x
uo,_

},
0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

3oo-
200-
I--

100 = l l l l l = l l l J l l l l l l l l l l l J l l l r l l l i
.-,10-
u}

;2
"E

0 1
17 19 21 23 25 27 29 31 2 4 6 8 10 12 14 16
December 1995 January 1996
Fig. 2: Time series of amplitude (significant waveheight), mean direction, and mean period obtained from One big advantage
the directional spectra measured by the radar ( X) and from the directional parameters measured by the
HF radar has.., is
waverider ( ) in both cases integrated over the frequency band of 0.05-0.4 Hz.
the ability to monitor

OSCR system used was provided by Wimpey En- in the early afternoon, a relatively calm period wave development in
vironmental (now GEOS Ltd.). Data were col- with high pressure over eastern Europe, low pres-
space as well as
lected over a period of 1 mo from 17 December sure over Scandinavia and to the west of the
1995 to 16 January 1996 with short breaks over British Isles, driving a weak south-easterly wind over time.
the Christmas-New Year period (the data require-
ments for wave measurement require an operator
53.90
to be on site). The radar measurement period in- peak direction ~"
21 Dec 95
cluded two high sea-state events, high for this re-
gion at least, with significant waveheights reaching 14:00-15:0( Peak

>3 m at the buoy positions (Fig. 1). Surprisingly Period (sec)

for this region, there were very few examples of [-~ AeOVE ~zo
fetch-limited development. Figure 2 shows time ~M
~_.l ~1.o- 12.o
series comparisons of significant waveheight, _ ~. 10.0"11.0

mean direction, and mean period obtained from 53.80 " m 9.0 - lO.O
the directional spectrum measured by the radar 8.0 - 9.0
and a co-located wavebuoy; very good agreement ~1 7o- e o
can be seen. Similar comparisons can be made for i 6.0 - 7.0
these parameters measured over limited frequency 5.0 - 6.0
bands; for example, waves with frequencies <0.1 4.0 - 5.0
Hz, which, during this experiment at least, are 3.0 - 4.0

swell waves. Again good agreement is found. BELOW 3.O

Quantitative measures of the accuracy of a range 53'700.1'0 0.00 0.10 0.20 0.30
of parameters describing the directional spectrum
are being determined. Fig. 3: The Holderness coastline is shown to the south-west of the map with
One big advantage HF radar has over conven- the two radar sites indicated as M and S. Directions shown with arrowheads
tional wave measurement systems is the ability to are those toward which the peak of the radar measured spectrum is propa-
monitor wave development in space as well as gating. The period of the peak is color coded as shown. This figure shows
over time. This advantage is demonstrated very swell dominating over most of the region propagating from the north and re-
clearly with data collected on 21 December 1995 fracting toward the coast. To the south, south-easterly wind waves dominate.

OCEANOGRAPHY'VoI.10, No. 2"1997 87

 -0.10
54.00
0.00 0.10 0.20 0.30 0.40 0.00 0.10 0.20 0.30 0.40 creases. Swell energy is dissipated (not shown
here), probably by bottom friction or by wave-cur-
rent interaction. Tidal current speeds were large at
the time of the measurement (>0.5 m/s), and al-
though uniform in direction themselves (in
53.90
roughly a southward direction), the direction rela-
tive to the swell varies as the swell is refracted by
the bottom topography. Wave breaking is unlikely
to be a problem at these amplitudes in these
63.80 depths. As a result of the dissipation of swell en-
ergy, the spectral maximum in the south of the re-
gion shifts to the wind-wave field with a peak pe-
riod of <5 s (blue shades on the figure).
53.70 Comparisons of amplitude, direction, and spread
64.00
as functions of frequency measured by the radar
and wave buoy confirm the swell and wind-sea di-
rections and also show good agreement in relative
amplitudes at the buoy location.
63.90 A second example showing both temporal and
spatial variability is presented in Figure 4 and
\ ttr.tttak~tttt•~P-t~P~•tt I•1
shows significant waveheight determined from the
! ¢~P~PtPt PP ~¢ • ~Pt
I directional spectra and short-wave directions
53.80 \ I P•'•P~'J'••••,I~..t~P'~_I
(Wyatt et al., 1997) at each measurement position.
\ I t t V ' ¢ ~ ' # • • • ~ ? ~ • • l Short-wave directions can be made over a wider
IN f~ PIAl'ttrttt
area because they use only the first order part of
lilt T the Doppler spectrum. The figure shows the re-
69.70 sponse of the wave field to a passing low-pressure
. ,-. l , , ~ . ~ band: full ~ ~ 45 • S o
~.00 ~lfi~t I--'-7 40.4~ system. Waveheights are initially low in response
23 Dec 95 .~ u e c ~ Waveheloht (m) i
mi
s~ .4o
so -&s to offshore winds but increase after the low has
03:00i04:00 09:00-10:00 r~ ,;.~
passed through, leaving a strong north-easterly
I" I I i ,5 .z, wind pattern and hence longer fetch.
##t 0-5 " 10
63.90 - - •
/t/•l•// •t[•••
¢ •• ~'•/•I'P~•p• • • • • l.#tta#t#* I i ~
tt¢f~-~••P'*~ I Concluding Remarks
i
. •,# • • ~ • • ,#/'~"~'•~'•~'~'•"*1 The data presented here have demonstrated that
OSCR can measure the ocean wave directional
53,80 ~lt:'--'~.'~.~ . . . . . . .
spectrum. These OSCR measurements are much
better than those previously published (Wyatt and
Ledgard, 1996) because this Holderness deploy-
ment configuration was optimized for wave mea-
53.70 I ~1 • L. . . . . . . . . [ surement. Good agreement with wave buoy pa-
0.40 0.00 0,10 0.20 0.30 0.40
-0.10 0.00 0,10 0.20 0,30 rameters was shown and the comparison is now
the subject of rigorous evaluation to provide quan-
Fig. 4: Short-wave (wind) directions are shown with arrowheads. Significant titative measures of parameter accuracy. Oceano-
waveheight is color-coded as shown and varies from <1.0 m near to the graphic consistency of the data over the region of
shore on 22 December 1995 at 1500 to >5 m offshore at 0900 on 23 De- the measurements is also being demonstrated.
cember 1995. The figure shows changes because of the passage of a low- There remain some problems that need to be
pressure system through the region. overcome before OSCR can become an opera-
tional tool for wave monitoring. Some of these are
over the region. Fronts were building up in the purely technical; for example, the use of modern
west to come through the region over the next 24 computer systems and, in particular, reliable and
T h e data presented h, but in the meantime the synoptic pattern sug- high-capacity data storage devices would signifi-
gested that the wave field would be dominated by cantly increase the temporal coverage because
here have demon- there would no longer be an onsite operator re-
swell propagating southward down the North Sea.
strated that OSCR This pattern is exactly the same as seen over most quirement. Others require further research, some
of the radar coverage region. Figure 3 shows the of which is currently underway at Sheffield. One
can measure the direction of the peak of the spectrum superim- of the problems we have identified is a reduction
ocean wave direc- posed on the contoured peak period. Of interest is in data availability during periods of varying sur-
the spatial variation in these parameters. Swell, face current. This is because the first order peaks
tional spectrum. with a period of - 1 0 s (yellow shades), can be move about in frequency during the measurement
seen propagating from the north and being re- period (responding to different current compo-
fracted toward the coast as bottom depth de- nents) and prevent a clear distinction between first

88 OCEANOGRAPHY'VoI. 10, No. 2-1997

and second order in the power spectrum. At pres- References
ent OSCR data from three separate surface current Barrick, D.E.• 1972a: First-order theory and analysis of
MF/HF/VHF scatter from the sea. IEEE Trans. Anten-
measurement periods (10 min in every 20) are av- nas Propag., AP-20• 2-10.
eraged before extracting wave information (a sta- - - • 1972b: Remote sensing of sea state by radar. In: Re-
tistical requirement). Any current variability on mote Sensing of the Troposphere. V.E. Derr. ed.
time scales of <1 h causes a problem. This could NOAA/Environmental Research Laboratories, Boulder•
be overcome by longer coherent data collections, CO. 12.1-12.6.
Fernandez, D.M., H.C. Graber• J,D. Paduan and D.E. Barrick,
thus obtaining sufficient averaging for wave mea- 1997: Mapping wind direction with HF radar. Ocean-
surement from a shorter time period, but at the ex- ography, 10. 93-95.
pense of reduced surface current sampling. Other Graber, H.C. and M.L. Heron, 1997: Waveheight measure-
approaches to this problem are under develop- ments from HF radar. Oceanography, 10, 90-92.
Hisaki• Y.• 1996: Nonlinear inversion of the integral equation
ment. Another important problem, identified many
to estimate ocean w a v e spectra form HF radar. Radio
years ago by Lipa and Barrick (1986), is a possi- Sci.. 31, 25-39.
ble upper waveheight limit beyond which the the- Howell, R. and J. Walsh• 1993: Measurement of ocean w a v e
ory used in the inversion no longer applies. So far spectra using narrow beam HF radar. IEEE J. Ocean.
we have seen no evidence of the particular prob- Eng., 18. 296-305.
Isaac, F.E. and L.R. Wyatt• 1997: Segmentation o f HF radar
lem they describe, although there are certainly dif-
measured directional w a v e spectra using the Voronoi
ferences between the second-order theory and the Diagram. J. Atmos. Ocean. Tech.. 14, 950-959.
radar measurements in higher sea-states that may Lipa, BJ., 1978: Inversion of second-order radar echoes from
be affecting the accuracy of our measurements the sea. J. Geophys. Res.. 83, 959-962.
(Wyatt, 1995). Measurements are required in sea- - - and D.E. Barrick, 1986: Extraction of sea state from
HF radar sea echo: mathematical theory and modelling.
states higher than are seen in the Holderness re- Radio Sci., 21, 81-100.
gion to investigate these issues further. Recent Wyatt, L.R., 1988: Significant waveheight measurement with
measurements with the W E R A system off the HF radar. Int. J. Remote Sens., 9. 1087-1095.
north Netherlands' coast at Petten during a very • 1990: A relaxation method for integral inversion ap-
plied to HF radar measurement of the ocean w a v e di-
stormy period in N o v e m b e r 1996 may provide
rectional spectrum. Int. J. Remote Sens., 11, 1481-
some answers. 1494.
, 1991: HF radar measurements of the ocean w a v e di-
Acknowledgments rectional spectrum. IEEE J. Ocean. Eng., 16, 163-169.
The observations off Holderness described here • 1995: High order nonlinearities in HF radar backscat-
were funded in part by the Ministry of Agriculture ter from the ocean surface. In: lEE Proceedings on
Fisheries and Food (U.K.) under its Flood Protec- Radar, Sonar and Navigation, 142, lEE. London.
293-300.
tion Commission with NERC's Proudman Ocean- - - and G.J. Holden• 1994: HF radar measurement of
ographic Laboratory. The data were collected and multi-modal directional w a v e spectra. Global Atmos.
analyzed by Louise Ledgard. The Natural Envi- Ocean Sys., 2, 265-290.
ronment Research Council also supported these - - and L.J. Ledgard. 1996: OSCR w a v e m e a s u r e m e n t - -
observations through its Land Ocean Interaction some preliminary results. IEEE J. Ocean. Eng., 21, 64-
76.
Studies project. Other support has come from the • L.J. Ledgard and C.W. Anderson. 1997: Maximum like-
EC MAST SCAWVEX project (MAS2CT940103) lihood estimation of the directional distribution of 0.53 Hz
and EPSRC (grants GR/J07341, GR/J50934). ocean waves. J. Atmos. Ocean. Tech.. 14, 591-603. O

OCEANOGRAPHY'VoI. 10, No. 2.1997 89