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Interpreting directional data from large pitch-roll-heave buoys
Ocean Engng, Vol. 16, No 2, pp. 173-192, 1989. Printed in Great Britain.
0029--8018/89 $3.00 + .00 Pergamon Press plc
INTERPRETING DIRECTIONAL PITCH-ROLL-HEAVE
DATA FROM BUOYS
LARGE
M. J. TUCKER 6 Highlands, Taunton, Somerset, U.K. Abstract--Surface-following buoys are commonly used to measure directional wave spectra. At each frequency, four directional parameters can be calculated from the auto- and cross-spectra of pitch, roll and heave. It is shown that for many buoys, particularly large data buoys, the response of the buoys must be taken into account when calculating the spread parameter s~ or related parameters such as the r.m.s, angular spread tr. The detection of noise in the records is discussed, and some likely sources identified. These include the effects of compass tilt. It is shown that the mean direction 0M~ is robust against the presence of noise, but that s~ and tr are very sensitive to it. The theoretical results are illustrated with data from the U.K. data buoys DB1 and DB2.
1. INTRODUCTION 1.1. General discussion THE DIRECTIONALproperties of a wave system have been of interest for many years, the classic fields of application being coastal engineering and naval architecture, with the design of offshore structures coming to the fore in recent years. It was in fact a major U.K. programme of research into ship motion which triggered the development of the technique of using the pitch, roll and heave of surface-following buoys to estimate the directional wave spectrum. Some results from the research are given by Canham et al. (1962), and the development of the buoys is described by Longuet-Higgins et al. (1963). A buoy which measured the curvature of the sea surface (the "Clover-leaf buoy") was also developed (Cartwright and Smith, 1964) but the use of this has been confined to occasional research projects, since it has not proved to be practicable to develop it for use as a routine wave recorder. The three components of curvature which it measures allow in principle an additional two angular harmonics to be determined (see below for the definition of these) but the curvature due to long-wavelength components of the spectrum is so small that noise limits the useful results to the higherfrequency part of the spectrum. As time progressed it became obvious that routine measurement of the directional properties of waves well offshore was desirable, and in 1972 the National Institute of Oceanography (now the IOS Deacon Laboratory of the Natural Environment Research Council) proposed to the U.K. Department of Industry that they should support the development of a large data buoy which measured and telemetered ashore a variety of metocean data, including the pitch, roll and heave of the buoys. This proposal was accepted and resulted in the development of DB1 (Rusby et al., 1978). After trials off the east coast of England, the U.K. Offshore Operators Association (UKOOA) paid for the deployment of the buoy in a depth of 180 m of water in the Western Approaches to the English Channel, where it operated successfully for three years from 1978. As 173
174
M . J . TUCKER
a result of their encouraging experience with this buoy, UKOOA funded the development and manufacture of two units of an improved version (DB2 and DB3) which have been successfully deployed to the West of the U.K. from mid-1984 through to the present time. DB2 is shown in Fig. 1. The theory given in the classic papers referred to above assumes that over the frequency range of interest, the buoy follows the slope of the sea-surface perfectly. However, this would clearly not be the case for large data-buoys, and the work reported here was started in 1985 to see what effect this response would have on the results, and whether it could be corrected. It is shown that the response has no effect of the estimates 0Mr and 0u2 of mean wave direction, nor on the spread parameter s2 determined from the second angular harmonic (for definition of these parameters see Section 2.2). It does affect sl, the spread factor determined from the first angular harmonic, and the related r.m.s, spread parameter cr, but the effect is easy to correct. The effect of noise is examined in the second part of the paper. It is shown that the mean direction parameters, in particular 0MI, are robust against the presence of noise, but that the spread parameters, in particular s~ are very sensitive to noise and must be interpreted with care. For the present paper, it is assumed that the buoy response is isotropic in direction. Barstow et al. (1986) describe the development of two large buoys, both of which have resonances at frequencies below 0.5 Hz. In both cases the cross-spectra are corrected for the responses using a best fit transfer function before the directional analysis is carried out. However, experience with DB1, DB2 and DB3 shows that the responses are not constant with time, and cannot be described by simple resonance effects. Mooring forces are also important and depend on wind and currents as well as
Satellite wPial hi
l
I //
//
Ill
l
\\
t lll
seuors
I
kk J
~;.rrt~t wter FIG. 1. T h e U . K . data buoy DB2. It is 6.0 m in diameter with a draft of 1.25 m. DB1 was a similar discus buoY but with a diameter of 7.6 m. Both were moored in the SW Approaches to the English Channel using 3-point catenary moorings. DB1 was in a depth of 168 m and DB2 in a depth of 184 m.
Measurement of directional wave spectrum
175
the wave system itself. Thus, a method of correction is required which does not involve fitting a simple resonance-type formula to the variation of response with frequency. The results will be illustrated using data from DB1 and DB2. DB1 was 7.6 m in diameter but the same draft as DB2 and otherwise rather similar in shape. In both buoys pitch, roll and heave were measured using a Datawell "Hippy" sensor. In DB1 the heave acceleration was integrated twice aboard the buoy using a Datawell analogue integrator. This displacement signal, the pitch, the roll and the compass bearing were sampled at 0.834 Hz and transmitted to shore by radio. After quality control, eleven subsections each 100.8 sec long were then Fourier-transformed and averaged to give the recorded spectral estimates at approximately 0.01 Hz intervals and with 22 degrees of freedom. Further details of DB1 are given by Freathy et al. (1981). In DB2 the heave acceleration signal was digitised directly, and all four channels were sampled at 1.28 Hz for 2000 sec. These series were divided into 400 sec segments which were tapered with a cosine function extending over the first and last 10% of the segment. The Fourier transforms of these were averaged over four adjacent harmonics and over the five subseries to give 0.01 Hz resolution with approximately 36 degrees of freedom. These computations were carried out on the buoy and the results were recorded aboard as well as being transmitted ashore via a satellite. 1.2. Convention for the coordinate system This presents a problem for the practical oceanographer, since the output of the analysis is always required in terms of geographical angles. Also, for locally-generated waves, the wind and wave directions must be the same, which means that 0 has to be the direction from which the waves have come, which is the opposite of the usual convention used in physical theory. However, it is usual to carry through the analysis in the conventional physical/mathematical system and to leave it to the user to convert the results into the practical system. This will be done here. Thus, the system is: x positive east; y positive north; z positive up; roll is tilt, east up positive; pitch is tilt, north up positive; bearing angles are positive anticlockwise from east; 0 is the direction towards which the waves are travelling. 2. THEORY The classic method of analysis was worked out by Longuet-Higgins et al. (1963) and by Cartwright (1963). The analysis which follows is basically the same theory extended to include the response of the buoy. It is assumed throughout that we are dealing with a stationary Gaussian process. 2.1. The effect o f buoy response on the determination of the angular harmonics The wave system is taken as the superposition of a large number of sinusoidal components travelling independently of one another:
~(t) = ~ a" cos (l,,x + m,,y - to,,t - d~,,) n
(1)
176
M.J. TUCKER
where a" has a dash to distinguish it from an used later as a cosine component In = m. = ton = +n = 0=
k. cos 0 (x component of wave number); kn sin 0 (y component of wave number); angular frequency; a random phase; and direction of travel.
The slopes are
Sx(t) = O~ = _ ~ a" l~ sin(/,,x + m~y -tont - ~n) Ox sy(t) - Og Oy _
~'~ an m . sin(/nx + m,0, - t o n t - ~bn).
Taking the origin at the centre of the buoy, the vertical displacements and slopes of the water surface at this point in the absence of the buoy would be ~o(t) =
a" cos(to o t + +n)
(2)
(an costo, t - b. sintod)
Sx(t) =
In (b. costo, t + a. sinto~t)
(3)
Sy(t) =
rn. (b. coston t + an sintod)
(4)
where a. = a" cos+. and b. = a" sind~.. Now this is the point at which one would normally expect to move into complex notation. However, for this particular problem it is simpler and clearer to stay in real notation. Considering two signals represented as the sum of sinusoidal components
vl(t) = ~ (al(n) costo, t + bt(n) sinto, t)
(5)
vz(t) --- ~] (az(n) coston t + b2(n) simon t).
(6)
Then the co- and quadrature spectra are given by
cizAf = 1/2 ~ (a,az + b,bz) af
(7)
q,2Af = 1/2 ~ (a,b2 - azb,). a[
(8)
The concept of these is that the signals are first filtered to leave the band Af, then c,2 is the mean of Vl(t)'v2(t) and qt2 is the mean of vt(t) times vz(t) with each component of vz advanced by 90° in phase. Equations (7) and (8) are used in algorithms to compute estimates of ctz and qt2 from simultaneous finite records, and in this case the right-hand side of Equations (5)
Measurement of directional wave spectrum
177
and (6) are Fourier series. The mean of vl(t).v2(t) works out neatly since all crossproducts between different harmonics average to zero over the length of the record. However, for the present purpose we require the expectation, or ensemble averages, of c12 and q12, and in this case the same result is obtained if non-harmonically related frequencies are used on the right-hand sides of Equations (5) and (6), and there can even be a number of components at the same frequency as long as their phases are random. The cross-products of the unrelated components will have random phases and will therefore be randomly positive and negative, so that only related components will contribute systematically to the sum as the number of components is notionally taken to infinity. For the first computation, vx(t) will be taken as the heave of the buoy and v2(t) will be taken as the roll of the buoy. Let the amplitude and phase of the heave response be given by r~a and otH respectively (a positive phase angle being a phase advance). These are functions of frequency, but will be assumed to be constant within a bandwidth Af. Then using Equation (2)
vl(t) = ~ rHa" [cos(0n + otn) costont - sin(0n + an) sintont]
(9)
similarly, if the amplitude and phase of the tilt response are rr and otr respectively v2(t) = ~, In rra" [sin(0n + err)COStOn t+ COS(0n + Otr)sintont].
(10)
From Equations (9) and (10) using Equation (7) cur Af = 1/2 ~, 1, ru rr (a') 2 [cos(0n + an) sin(0n + a t ) A[ --
sin(0 + or,v) cos(0n + otr)]
= 1/2 ~'. In rn rr (a') 2 sin(otr - ~xn). af Since rH, rr, a n and a r are assumed constant within Af, then putting In -- kn cos 0n, and assuming k,, also to be constant within Af gives
ctaR A f = r n r r k s i n ( a t - an) ~ 1/2(a') 2 cos 0n. af
(11)
Consider components travelling in a range of directions between (0 - 1/2A0) and (0 + 1/2A0), then by definition of the directional spectral density S(f,0): Expectation ~'~ 1/2(a') 2 = S(f,0) AfA0. Af.A0
Multiplying both sides by cos0n and summing over all 0 gives 1/2(a;,) 2 cos0, ---->Af ~.f
S(f,0) cos0 dO Jo
178
M.J. TUCKER
and Equation (11) becomes sin(~r - all)
CHR = rH r r k
COS 0 S(f,0) de
or putting (ar--~H) = a for convenience: sins
S(f,0) cos0 d0.
(12)
qHR = r H r r k
coso~
S(f,O) sinO dO
(13)
Clip = r H r r k
sins
S(f,0) cos0 d0
(14)
q~e = rHrr k
cosa
S(f,0) cos0 d0
(15)
CHR = r H r r k
Similar analysis gives
CHH --- ~H2
(16)
S(f,0) dO )
CRR = r r 2 k 2
S(f,0) sinZ0 dO
(17)
cos20 dO
(18)
)
Cee = r r 2 k z
S(f,O) )
CpR = r r 2 k 2
S(f,0) sin0 cos0 dO
(19)
}
(20)
qPR = 0. To interpret these it is conventional to put sff,0)
=
S(t).G(O)
where S(f) is the one-dimensional or point energy spectrum, G(0) is a function of both 0 and frequency, and .1"2,, G(0) dO = 1. G(0) is expressed as a Fourier series of angular harmonics: G(0) = ~ ~ +
[A, cos n0 + B,, sin n0] .
(21)
n=l
Putting this into Equations (12)-(20) gives CHR = r H r r
(sins) k S(f) B,
(22)
Measurement of directional wave spectrum
qnR
= rarr(coset) k S(f) B,
179
(23)
clip = rarr
(sina) k S(f) A t
(24)
qm, = rnrr
(cosct) k S(f) A1
(25) (26)
Can = r~ S(f) CRR = rr 2 k 2 S ( f )
[1/2 - 1/2 A2]
(27)
c e e = r7a k 2 S ( f )
[1/2 + 1/2 A2]
(28)
c e n = rr 2 k z S ( f ) • qen
1/2 B2
(29)
= 0.
(30)
Normalising these equations gives cap [CnH(Cvp + CRn)] v2 = A 1 sin
qHP
[CnH(Cpp + CRR)] m = A t cnn
[CHH(Cpp "F CRR)] 1/2
= B1
a
cos cx sin ~x
(31) (32)
(33)
[ c m , ( c p pq a+R cRR)] t'~ = B, cos
(34)
C p p - CRR Cpp + CRR
As
(35)
= B:.
(36)
2 cen
Cpp -F CRR
-
When a = 0, Equations (22) and (24) show that c n e and c a r = 0 so that Equations (31) and (33) disappear and the rest reduce to the classic relationships for a perfect surface-following buoy. The best way to use Equations (31) and (32) to determine Aj and (33) and (34) to determine B1 will be discussed in Section 2.2. The above analysis is in terms of heave displacement. If heave acceleration is used, a factor of -to,, z appears in Equation (1) and follows through to appear on the righthand side of Equations (22)-(25). It appears as to4 in Equation (26). Equations (31)-(34) then have the signs of the R.H.S. changed. A "check ratio" R is also usually computed and for calculations starting with heave displacement is defined by I" R = k[cep
cH---H- / I/2 + cnRj
"
For calculations starting with heave acceleration in S.I. units
(37)
180
M.J. TUCKER
e :kl
CH--H-
/ 1/2
0~2 [c,~p + CRRJ
(38)
"
Note that in practice acceleration is usually specified as a proportion of g, in which case a factor of g appears on the right-hand side of Equation (38). For deep water, gk/¢o 2 = 1 and for shallow water gk/to 2 = 1/tanh(kh) where h is the water depth. From Equations (26)-(28) (39)
R = (rnn/rrr) in.
R is therefore a measure of the ratio of the heave and pitch amplitude responses of the buoy (or of errors in calibration!). However, the dispersion relationship for a moored buoy can be changed by Doppler effects due to currents changing the apparent phase velocity of the waves (Kuik and Holthuijsen, 1981) and this also modulates the check ratio. The departure of R from 1 at middle frequencies shown in Fig. 2b is almost certainly due to mooring restraints (it is too large to be due to currents), and a time series of R at, say, 0.1 Hz shows a strong semi-diurnal variation, as well as longer-term variations. When one of the three moorings broke, the value rose to about 2. Note that apart from the equations for R, the dispersion relationship does not enter into the above calculations, which are therefore valid for shallow as well as deep water. 2.2. The effect on the mean direction and spread parameters and methods of correction The components of the first and second angular Fourier harmonics are insufficient to allow a useful reconstruction of G(0) by simple superposition. Some workers favour fitting by maximum likelihood methods, which use the additional information that G(0) cannot be negative (Jefferys et al., 1981). Long and Hasselmann (1979) have developed a variational technique to minimise a "nastiness function". Kuik (1984) favours interpretation into 0M~ and or as described below, but with skewness and kurtosis parameters derived from the second angular harmonic instead of 0M2 and s2. However, the commonest practice is to follow Cartwright (1963) and assume that the directional spectrum is of the form G(0) = K cosz' 1/2(0 - 0M)
(40)
where s and 0M vary with frequency. Then separate estimates of 0M and s at each frequency can be obtained from the first and second angular harmonics: 0MI = arctan BI/AI
(41)
0Ma = 1/2 arctan B J A z
(42)
C1 where C12 = A1E + B12 s~ - l - C 1 1 + 3Cz + (1 + 14Ca + Ca2) 1/2 s~ =
2(l-G)
(43) where C22 = A22 + B22.
(44)
Measurement of directional wave spectrum
181
A further parameter tr is sometimes used, which for narrow directional distributions is the r.m.s, spread about 0M1 tr = (2 - 2C1) '/2.
(45)
On any particular occasion and in any particular frequency band, the similarity or otherwise between sl and s2 and between 0MI and 0M2 can be used as a measure of whether Equation (40) is a reasonable description of the directional spectrum. In low energy parts of the spectrum the parameters may be corrupted by noise (see Section
3). It is immediately clear from Equations (35) and (36) that A2 and B2 would be correctly computed using the classic equations, and so the computed values of 0M2 and s2 do not depend on buoy response. The same is true for 0M~, since the classic equation can be put in the form 0M~ = arctan(q,R/qHe) and from Equations (23) and (25) this still gives the correct result when the effect of buoy response is included. Note, however, that if et approaches 90 °, the result becomes noisy and so this is not in practice the best equation to use. However, sl and tr are affected because the value of C~ computed using the classic equations is actually C~ coset if the effect of buoy response is included. Equations (43) and (44) both include factors of 1 - C~, so that when s~ is high (that is, C1 is approaching 1) this effect can become critical. This problem, including its relationship to noise, will be discussed further below. Figure 2 shows that for DB2 et is significantly non-zero at high and low frequencies. For DB1 a can be significantly different from zero over the whole frequency range. What is more, analysing many occasions for the same buoy installation shows it varying also with time, presumably as the mooring restraints vary with wind, waves and currents. Thus, it is not practicable to fit and use a fixed relationship between ot and frequency. In principle, two estimates of a can be obtained at each frequency for each record. Using Equations (22) and (23) tan OIHR = CHR/qHR.
(46)
CHP/qHp.
(47)
Similarly tan OtHt, =
In practice, it is rare to get good values for both etHR and OtH,,. Figure 2 illustrates this: when the waves are coming from the west, there is little or no correlation between heave and N-S tilt, so that aHe is largely random. Returning to the practical computation of A~ and B~, it will be seen that the righthand sides of Equations (31) and (32) are the two components of a vector whose modulus is A~ and whose argument is ct, and similarly for B; and Equations (33) and (34). It is necessary to preserve the signs of A~ and B~ in order to resolve the 180° ambiguity in 0M~ (Equation 44)), so in most cases the simplest solutions are
I CH2+qHp2 /1/2 (Cpp+ CRR)J
A1 = sign(qHp) tcHH
(48)
182
M . J . TUCKER
(a)
l o g t o Slq) S ( f ) i n m~/Hz
"--"--~x,__ -2
I
I
i
~,._.
I
2
(b)
0
I
I
Check r a t i o
R
I
j',&
~/2
-~/2
I
I
~/2 (d)
------~./~,.~
-~12
I
0
0.1
~Ew
,~-_.,.~_.__----~/-..
I
I
0.2
0.3
0.4
F r e q u e n c y Hz.
FIG. 2 a - d .
Data
f r o m D B 2 11.00 hr, 21 March 1985. H~ = 9.50 m; T~ = 11.80 s e c .
/ c"R2+ qHR2 11'2 B~ = sign(quR) [Cull(C--------p-V-+C R R ) )
"
(49)
H o w e v e r , where et approaches 90 ° this could result in some erroneous signs, so if this seems likely to be a problem it might be better to rotate the vectors A ~ - c u t , sin et - q u v cos et
[cuu(c,.,. +
c,,,0l
(50)
and similarly for B1. In this case the value of ot used should be that corresponding to the c o m p o n e n t (pitch or roll), since this reduces the effect of any asymmetry of the buoy response. That is, for A l , tan a = CurJqHP and for Bt, tan ot = CHR/qHR.
Measurement of directional wave spectrum
183
40
(e)
s,
20
0
I
I
I
40
(f)
s2
20
0
I
I
I
S
(g)
E
eN=
N W S
I
I
I
S (h)
O~2
E
N W S
I
0
O.1
I
I
O.2
0.3
O.4
F r e q u e n c y Hz.
FtG.
2e-h.
In those cases where one or other of these has a large random fluctuation as described above, this is not of importance since the corresponding Aj or BI will be low. 3. THE D E T E C T I O N OF NOISE AND ITS EFFECTS
3.1. General Examination of records when the sea is comparatively calm indicates that for the three buoys mentioned in the introduction, there is little noise not generated in some way by buoy motion. However, other tests indicate that in higher sea-states there is noise present and that its effect is significant. In Section 3.2 the methods for detecting the presence of noise are described, in Section 3.3 the probable sources are identified, and in Section 3.4 the effects on the directional parameters are examined.
184
M . J . TUCKER
"Noise" in this context does not include the statistical instability of the estimates of the parameters: that is, the random errors in these estimates arising from the uncertainties inherent in using finite samples. These have been examined by Long (1980). One can often obtain a good feel for them in any particular case from the way the estimated parameters vary from frequency to frequency along the spectrum. One cannot expect Sl and s2 to agree closely in typical circumstances since the sea at exposed sites is a complex mixture of wave energy from various sources and it is unlikely to fit the simple cos 2s 1/2(0--0M) angular distribution. Similarly, 0M1 and 0M2 are unlikely to agree for much of the time, Thus, "discrepancies" in these parameters do not necessarily indicate anything wrong with the measuring system. However, as discussed below, there should be occasions when they agree quite closely, and so if the discrepancies are always present it does indicate something wrong. Ewing and Laing (1987) using numerical simulation show that with a bimodal sea, large discrepancies between sl and s2 occur only when the directions of the two systems differ by more than 90°. 3.2. Identifying the presence of noise In this context, noise can be defined as any signal present in the pitch, roll and heave outputs which is not generated by linear response to the waves. In general, it will therefore be uncorrelated with the first-order wave signals. The only really positive indication of the presence of noise arises because it has been established that there is usually very little wave energy in the sea at frequencies below 0.04 Hz. This was demonstrated by Munk et al. (1959) for Californian waters, and some unpublished work by the author's former colleague Ian Vassie in which he analysed long records from a pressure recorder in 50 m of water west of the Outer Hebrides, showed that in a storm with a significant waveheight of 6.7 m, the spectral density in the frequency range 0.2-0.4 Hz was three orders of magnitude below the peak spectral density, which was at approx. 0.6 Hz. On calmer occasions the ratio could approach four orders of magnitude. If acceleration or tilt spectra are used, then the ratios will be an order of magnitude greater (compare Figs 2a and 4 which show the same data: but note the different frequency scales). At frequencies below about 0.01 Hz the spectral density starts rising again, but this does not concern us here. In the North Atlantic Ocean, there will be perhaps one or two severe storm events during a year which generate significant spectral energy below 0.04 Hz. Thus, the presence of significant spectral density in the 0.01 and 0.03 Hz frequency bands is a positive indication of noise at the low frequency end of the spectrum. In modern buoys, the sampling frequency is usually sufficiently high to allow the analysis to extend to well above the buoy resonances, so that these and diffraction effects reduce the spectral density to low values at the upper end of the frequency range. The primary purpose is to avoid aliassing of high frequency energy into the main part of the spectrum. If the spectrum flattens out at the upper end more than would be expected by aliassing, this can indicate the presence of noise. The third method for detecting noise from internal evidence in the data is more subtle and requires some oceanographic judgement. Tucker (1987) has shown that if the wave energy is confined to a narrow range of directions (in practice, to an r.m.s.
Measurement of directional wave spectrum
185
beam width of less than about 20°), then independent of the shape of the angular distribution:
Cl --') 1 - 12/2 (?2""> 1 - 212
where l2 = f d+~(O-OM)2 O _ G(O) which is the mean square angular spread of 0 about 0M. From which Cl = (3 + C2)/4.
(51)
The exact relationship when G(0) = K cos2s 1/2(0-0M) is C 2 = Cl(2C
1 -
1)/(2 - Ct)
which reduces to Equation (51) as C~ tends to 1. Now at the low-frequency end of the wave spectrum it is common to find swell bands from distant storms which have narrow spreads in both frequency and direction. Thus, if one plots (?2 against C1 for, say, all frequencies below 0.1 Hz and for a considerable number of records, there should be some points falling near the above relationships for high values of Ct and (?2. However, if there is noise present, then, as will be shown later, this is likely to affect C~ more than C2 and so the points will fall at lower values of Cl. These effects are shown in Figs 5 and 6 (where the graph relating C~ and C2 using Equations (40) and (41) is included for general interest: as explained above, it tends to Equation (51) for high values of s). DB2 behaves as expected, but noise is obviously present in DB1 (the source of the noise will be identified later). Note that this type of diagram was first used by Cartwright (1963) but for a different purpose. 3.3. Sources of noise It is impossible to cover all possible sources of noise, but some which have been identified as important in practice will be mentioned.
a. Frequency spreading due to analysis of short record sub-sections. This is the main source of noise in DB1. This buoy was developed in 1973 and used a dedicated minicomputer to receive and analyse the telemetered time series of pitch, roll, heave displacement and compass bearing. The limited computing power available at that time made it necessary to divide the record into 100-sec lengths for Fourier analysis without tapering. Those components in the sea spectrum which do not have an exact number of waves in 100 sec have some of their energy spread to other frequencies by this process. The effect if well known, and an assessment of it for this case shows that it has the right order of magnitude to account for the observed low frequency noise levels in DB1 (Fig. 3). For reasons to do with the stochastic nature of the data, it is not practicable to correct for it. However, the original DB1 time-series have been kept and a small number of sample records have been re-analysed using Fourier transforms of the complete records without subdivision. These demonstrate a large reduction of noise
186
M . J . TUCKER
log S ( f )
-2
i
0
i
0.1
i
0.2
0.4
0.3
Frequency
Hz
FIG. 3. A heave displacement spectrum from D B I . 06.00 hr, 9 December 1979. SOt) is in m2/Hz; H~ = 5.25 m; Tz = 8.45 sec.
and an improved agreement between s~ and s 2. However, the most practically useful parameters of the data, Hs T~ and 0M1, are not significantly affected so it is probably not worth re-analysing the whole data set. The low-frequency noise levels in DB2 (Fig. 2a) are similar in relative magnitude to those in DB1, but in fact this seems to be a coincidence, and their significance i~ very different. In DB1 the heave displacement signals were Fourier transformed and the noise due to frequency spreading would rise into the main part of the spectrum. In the case Of DB2, 400-sec lengths the heave acceleration signal were tapered and then Fourier transformed. This resulted in negligible levels of spread energy in the acceleration spectrum, indicated by the low spectral levels at low frequency in Fig. 4. i
1
--2
log Cal
-4
-/a
a
0
i
i
0.2
i
0.4
Frequency
i
O.&
Hz
FIG. 4. A heave acceleration spectrum from DB2. 11.00 hr, 21 March 1985. C,, is in g2/Hz;H,. = 9.50 m; 7", = l l . 8 0 sec.
Measurement of directional wave spectrum
187
However, multiplying by 1/o04 to get the displacement spectrum greatly magnifies the levels at 0.01-0.03 Hz. This diagnosis is supported by the evidence from plotting C] against C2 discussed in Section 3.2 (Figs 5 and 6).
BoLow
Q.ln
+~ +*~4+ :+
HI
C2
+
+
%
7L+=
+ +
+ + + + + +4"+ .4.+ +++ ÷ ++
+
4-+
÷+
+ +
+ +
++
+++ +
+
+ +++
+ .
J
,
~
,
Cl FIG. 5. Plot of C, against C2 for all spectral estimates below 0.1 Hz which have R > 0.9. Five days of DB2 data: 16-21 March 1985.
+
++ +
l /
+
/ • el+ow •,l.
+H,.U.tF.,.tl~:~++ 4~, +
Nat
/
C2
++++++++; / +
+ +++
+ +,~ +
+,** ;÷
J
I CIL
FIG. 6. Plot of CI against C 2 for all spectral estimates below 0.1 Hz which have R > 0.8. Ten days of DB1 data: 1-9 December 1979.
188
M.J. TUCKER
b. Noise due to compass tilt. It is easiest to explain this effect initially in terms of a 3-component fluxgate compass fixed to the hull of the buoy. If the buoy is tilted by an angle +R in the E-W direction, it will introduce a component of the earth's vertical magnetic field Hv into the E-W sensor given by HE = Hv sin ~bR.
If the N-S component of the earth's field is HN, then this rotates the apparent bearing of the buoy by an angle 0E given by tan 0E = HE/HN = (Hv/HN) sin +n.
(52)
(We shall not worry about sign conventions here since the analysis is only qualitative.) In UK waters Hv/HN varies from about 2 in the south to 3 in the north, so this effect is considerable. By apparently rotating the buoy, it introduces a component ~bpE of the roll angle into the N-S (pitch) channel: d~eE = d~Rsin d~E.
For small angles, using Equation (52) gives dPpE = d~n2Hv/HN.
(53)
Being a square-law error, this introduces sum and difference frequency noise components across the whole spectrum. They are detectable at low frequencies using the first test described in Section 3.2. Tilting in the N-S direction does not rotate apparent north to first order. Carrying out a full analysis shows that some noise can be generated in the E-W tilt channel, depending on the directional properties of the wave system, but that in general it is less than that in the N-S channel. This noise can generate spuriously high values of s2 at the low-frequency end of the spectrum where the true spectral density is low. In the buoys known to the author, the compass is in fact gimballed, but the sideways accelerations of the buoy cause tilts of the same order of magnitude as if the compass were fixed rigidly to the hull. A further complication is that because of these effects, the designers smooth the response of the compass: by oil damping in the case of the card compass, and by electronic or computational means in the case of both card and fluxgate compasses. These circumstances make it impracticable to correct for the effect, but an approximate calculation shows that it produces low-frequency noise comparable to that observed in the tilt channel of DB2. Thus, the high low-frequency noise level in the tilt channels of DB2 (Fig. 7) is probably due to this effect, but it falls off through the spectrum, as indicated by the low spectral density at the high-frequency end. c. Errors in the heave sensor. These can develop for a number of reasons, one of the most common being a tilt of the stabilised platform supporting the accelerometer, usually resulting from a damaged suspension. They are detectable as low-frequency noise in the heave spectrum, or as zero wandering in the displacement time-history produced by integrating the heave acceleration signal twice. They are familiar from long experience with one-dimensional wave recording buoys and will not be discussed further here.
Measurement of directional wave spectrum J
I
I
i
i
189
I
I
i
i
-2
log T~(f:
-4
_~
i
0
0.2
,.
0.4
Frequency
O.&
Hz
FIG. 7. A total tilt spectrum from DB2. 11.00 hr, 21 March 1985; Tr(f) is c~ + c33 in rad2/Hz.
3.4. The effect of noise on the directional parameters First consider the effects of noise on the cross-spectral components. Define a noise voltage e(t) by
e(t) = ~ g, cos(to, t + ~J,,) . Then adding this to the heave channel vl(t) (Equation (9) with the sine and cosine components re-combined) gives
v~(O = ~ [rHa" cos(co, t + dp, + all) + g, COS(tO, t + 0,)]From which, by similar calculations to those in Section 2.1 the new value C'Hn is given by
C'HRAf = 1/2 ~ 1, rn rr ( a ' ) 2 sin (aT -- an) + ~'~ rrg, a" sin(~b, - ~b, - an). af
(54)
af
Now by definition of the noise, ~b, - ~b, is random in the interval 0-2~r, and so the expectation of the second term is zero, in which case Equation (54) reduces to the equation preceding Equation (11). The same is true if uncorrelated noise is also added to the roll channel, and similar results are obtained for qun, cup and que. Thus, noise does not affect the expectations of these four cross-spectral components. If large, it will increase the random variability of them, but even a signal/noise ratio of 1 is not likely to produce large errors. There are possible mechanisms by which the noise in one channel could be partially correlated with the noise in another channel, but as long as the noise is a small
190
M . J . TUCKER
proportion of the total signal this is a second-order effect and unlikely to be significant, Carrying through similar analysis for the auto-spectra gives cHu = SNn(f) + run f S(f,0) dO
(55)
CRR = SNR(f) + rrrk z I S(f,O) sin20 dO
(56)
Cpp = SNe(jz) + rrr k 2 f S(f,O) cos20 dO
(57)
where suH(f), SuR(f) and Suv(f) are respectively the spectra of the noise in the heave, roll and pitch channels. To sum up the above, the cross-spectral estimates are unlikely to be significantly affected by noise in the more energetic parts of the spectrum, but the auto-spectra are all biased upwards. Turning now to the effect on the directional parameters, it is perhaps easiest to see that 0M1 is unaffected by noise by using Equations (41), (32) and (34), which together give tan 0M1 = qnR/qnv. Since qnR and qnP are unaffected by noise, then so is 0M~. To assess the effect of noise on 0M2, it is best to consider A2 and B2 as the components of a vector whose argument is 20M2 (Equation (42)). Then Equation (35) shows that although A 2 can suffer large proportional changes due to noise when it is <<1, the effect on 0M2 will only be large if B2 is also small at the same time. In this case 0~2 will be largely indeterminate due to the statistical instability in the cross-power spectral components. Thus, though noise does affect 0M2, the. effect is likely to be of rather marginal practical importance in the more energetic parts of the spectrum. The situation is very different in the case of sl. In this case from Equation (43) ds, _ s,(1 + s 0
dCl
(58)
Cl
Putting Equations (51) and (52) together gives C l 2 = CHp 2 + q n p 2 + CHR 2 -.1--q n R 2
cnn (Cep + CRR)
(59)
Considering noise in the heave channel, which always increases cnn (see Equation (55)), differentiating the above gives dC1
C1
dcuH- 2cHn"
(60)
Taking this with Equation (58) gives ds--! = 1/2 (1 + S 0 dcn____y_H S1
CHH
(61)
Measurement of directional wave spectrum
191
A similar result is obtained if noise is added to (cee + CnR), which is the total tilt spectrum. While this equation is useful in assessing the confidence in measured values of s~, it is in some ways more instructive to consider the case where the true value of s is infinity, that is, CI = 1. Then adding 2.5% of noise to both CHILl and (cm, + Cnn) in Equation (59) reduces Ct to approx. 0.975 and hence from Equation (43), st to 40. Figure 2 shows a storm for which this is the estimated value at the peak of the spectrum, so clearly, even low levels of noise can be quite critical for s~. Considering s2, the situation is more complicated. For example, consider a very narrow beam sea coming from due north. Then without noise, CRn ~ 0 and cen "-->0 and combining Equations (35) and (36) gives (72 --~ 1 and s2 ---> infinity. Adding noise to cpp makes no difference to (72, but if a small amount of noise is also added to Cnn this does make a difference. If this noise has a spectral density of v'cm,, then (72 is reduced to approx. 1-2v. Putting v = 0.025 to allow a comparison with the calculation for s~ gives C2 = 0.95 and from Equation (44), s2 = 77.5. This demonstrates that s2 is generally less sensitive to noise than st, but the situation is rather more complicated if the noise is generated by the mechanisms described in Section 3.3(a) and (b), since the noise is not then equally divided between the two tilt channels. Ewing and Laing (1987) studied the effect of noise on sl and s2 by numerical simulation and came to the conclusion that s~ is very sensitive to noise and that s2 is the more reliable parameter when noise is present. 4. CONCLUSIONS It has been shown that the amplitude response of a pitch-roll-heave buoy does not affect the determination of directional spectral parameters, but that the phase difference between the heave and tilt responses must be taken into account when calculating the spread parameters derived from the first angular harmonic. In practice, the response may depend on mooring restraints which are varied by a number of factor s including current, so a method of correction is required which does not assume a constant response. Fortunately, the information required for correction is contained within the measured spectra. Ways of detecting the presence of noise in the records are described, and some probable sources identified. In particular, it has been shown that tilting of the compass can introduce significant noise into the tilt channels. The effects of noise on the directional parameters has been analysed, and it is shown that the mean direction 0mr derived from the first angular harmonic is very robust. 0m2 is less robust, but not likely to suffer large errors except when the spread parameter is low. s~ and the related parameter cr are very sensitive to noise, which always tends to decrease sl and increase e. It seems likely that the spread parameters at the peaks of storm spectra have been underestimated because of this, but it is difficult to quantify. s2 is less sensitive, by a factor in the region of 2 for high values. Some records show large differences between 0mr and Ore2, particularly at middle frequencies (Fig. 2 shows one such case). In the author's experience, s is low when this happens, indicating a complex sea, and the difference is probably genuine. In such cases it is sometimes difficult to resolve the inherent 180 ° ambiguity in 0m2, but when s is reasonably high, 0M1 and Ore2 agree well.
192
M . J . TUCKER
Acknowledgements--This paper is based on work supported by the U.K. Offshore Operators Association, to whom the author is grateful for permission to use the DB1 and DB2 data. A report to them dated 9 January 1987 was seen by a number of scientists. In particular, helpful comments were received from M. Bradley, R. Jeffereys and A.J. Kuik. The author would also like to thank J.A. Ewing of the I.O.S. Deacon Laboratory for his interest throughout the course of this work.
REFERENCES BARSTOW, S.F., UEta~qD, G. and FossoM, B.A. 1986. The wavescan second generation directional wave buoy. Proc. Oceanology Int. Conf., Brighton, U.K., March 1986. Society for Underwater Technology, London. CANaAn, H.J.S., CARTVCgIGm, D.E., GOODRICH, G.J. and HOGBEN, N. 1962. Seakeeping trials on O.W.S. "Weather Reporter". Trans. R. Instn nav. Archit. 1114, 447-492. CARrWRIGnT, D.E. 1963. The use of directional spectra in studying the output of a wave reorder on a moving ship. In Ocean Wave Spectra, p. 203. Prentice-Hall, Englewood Cliffs, N.J. CARrWRIGnr, D.E. and SmTn, N.D. 1964. Buoy techniques for obtaining directional wave spectra. Trans. 1964 Buoy Technology Syrup., Washington, 24--25 March 1964, pp. 112-121. Marine Technology Society, Washington, D.C. EWING, J.A. and LAlrqG, A.K. 1987. Directional spectra of seas near full development. J. phys. Oceanogr. 17, 1696--1706. FREATHY, P.E., HOOPER, A.G. and MACDONALD, H.W. 1981. Wind and wave directional data obtained from DB1 in the S.W. Approaches to the United Kingdom. Proc. Int. Conf. on Wind and Wave Directionality, Paris, 1981. Editions Technip., Paris. JEFFERYS, E.R., WAREHAM,G.T., RAMSDEN, N.A. and PLAI"rs, M.J. 1981. Measuring directional spectra with the M.L.M. Proc. Conf. on Directional Wave Spectra Applications, Berkeley, California, 14-16 September 1981. Am. Soc. Civil Engrs, New York. KUIK, A.J. 1984. Proposed methods for the routine analysis of pitch-roll buoy data. Proc. Symp. on Description and Modelling of Directional Seas, Paper No. A-5. 18-20 June 1984. Technical University of Denmark, Danish Hydraulic Institute. KUIK, A.J. and HOLTnUIJSEN, L.H. 1981. Buoy observations of directional wave parameters. Proc. Conf. on Directional Wave Spectra Applications, Berkeley, California, 14-16 September 1981. Am. Soc. Civil Engrs, New York. LONG, R.B. 1980. 3~he statistical evaluation of directional spectrum estimates derived from pitch/roll buoy data. J. phys. Oceanogr. 10, 944-952. LONG, R.B. and HASSELMANN, K. 1979. A variational technique for extracting directional spectra from multicomponent wave data. J. phys. Oceanogr. 9, 373-381. LONGUET-HIGGINS, M.S., CARTWRtGHT,D.E. and SMITH, N.D. 1963. Observations of the directional spectrum of sea waves using the motions of a floating buoy. In Ocean Wave Spectra, p. Ill. Prentice-Hall, Englewood Cliffs, N.J. MUNK, W.H., SNODGRASS,F.E. and TUCKER, M.J. 1959. Spectra of low-frequency ocean waves. Bull. Scripps Instn. Oceanogr. 7, 283-362. RusBY, J.S.M., KELLY, R.F., WALL, J., HUNTER, C.A. and BUTCHER, J. 1978. The construction and offshore testing of the U.K. Data Buoy (DB1 project). Proc. Oceanology Int. 1978 Tech. Session J., pp. 64-80. Society for Underwater Technology, London. TUCKER, M.J. 1987. Directional wave data: the interpretation of the spread factors. Deep-Sea Res. 3,633-636.
0029--8018/89 $3.00 + .00 Pergamon Press plc
INTERPRETING DIRECTIONAL PITCH-ROLL-HEAVE
DATA FROM BUOYS
LARGE
M. J. TUCKER 6 Highlands, Taunton, Somerset, U.K. Abstract--Surface-following buoys are commonly used to measure directional wave spectra. At each frequency, four directional parameters can be calculated from the auto- and cross-spectra of pitch, roll and heave. It is shown that for many buoys, particularly large data buoys, the response of the buoys must be taken into account when calculating the spread parameter s~ or related parameters such as the r.m.s, angular spread tr. The detection of noise in the records is discussed, and some likely sources identified. These include the effects of compass tilt. It is shown that the mean direction 0M~ is robust against the presence of noise, but that s~ and tr are very sensitive to it. The theoretical results are illustrated with data from the U.K. data buoys DB1 and DB2.
1. INTRODUCTION 1.1. General discussion THE DIRECTIONALproperties of a wave system have been of interest for many years, the classic fields of application being coastal engineering and naval architecture, with the design of offshore structures coming to the fore in recent years. It was in fact a major U.K. programme of research into ship motion which triggered the development of the technique of using the pitch, roll and heave of surface-following buoys to estimate the directional wave spectrum. Some results from the research are given by Canham et al. (1962), and the development of the buoys is described by Longuet-Higgins et al. (1963). A buoy which measured the curvature of the sea surface (the "Clover-leaf buoy") was also developed (Cartwright and Smith, 1964) but the use of this has been confined to occasional research projects, since it has not proved to be practicable to develop it for use as a routine wave recorder. The three components of curvature which it measures allow in principle an additional two angular harmonics to be determined (see below for the definition of these) but the curvature due to long-wavelength components of the spectrum is so small that noise limits the useful results to the higherfrequency part of the spectrum. As time progressed it became obvious that routine measurement of the directional properties of waves well offshore was desirable, and in 1972 the National Institute of Oceanography (now the IOS Deacon Laboratory of the Natural Environment Research Council) proposed to the U.K. Department of Industry that they should support the development of a large data buoy which measured and telemetered ashore a variety of metocean data, including the pitch, roll and heave of the buoys. This proposal was accepted and resulted in the development of DB1 (Rusby et al., 1978). After trials off the east coast of England, the U.K. Offshore Operators Association (UKOOA) paid for the deployment of the buoy in a depth of 180 m of water in the Western Approaches to the English Channel, where it operated successfully for three years from 1978. As 173
174
M . J . TUCKER
a result of their encouraging experience with this buoy, UKOOA funded the development and manufacture of two units of an improved version (DB2 and DB3) which have been successfully deployed to the West of the U.K. from mid-1984 through to the present time. DB2 is shown in Fig. 1. The theory given in the classic papers referred to above assumes that over the frequency range of interest, the buoy follows the slope of the sea-surface perfectly. However, this would clearly not be the case for large data-buoys, and the work reported here was started in 1985 to see what effect this response would have on the results, and whether it could be corrected. It is shown that the response has no effect of the estimates 0Mr and 0u2 of mean wave direction, nor on the spread parameter s2 determined from the second angular harmonic (for definition of these parameters see Section 2.2). It does affect sl, the spread factor determined from the first angular harmonic, and the related r.m.s, spread parameter cr, but the effect is easy to correct. The effect of noise is examined in the second part of the paper. It is shown that the mean direction parameters, in particular 0MI, are robust against the presence of noise, but that the spread parameters, in particular s~ are very sensitive to noise and must be interpreted with care. For the present paper, it is assumed that the buoy response is isotropic in direction. Barstow et al. (1986) describe the development of two large buoys, both of which have resonances at frequencies below 0.5 Hz. In both cases the cross-spectra are corrected for the responses using a best fit transfer function before the directional analysis is carried out. However, experience with DB1, DB2 and DB3 shows that the responses are not constant with time, and cannot be described by simple resonance effects. Mooring forces are also important and depend on wind and currents as well as
Satellite wPial hi
l
I //
//
Ill
l
\\
t lll
seuors
I
kk J
~;.rrt~t wter FIG. 1. T h e U . K . data buoy DB2. It is 6.0 m in diameter with a draft of 1.25 m. DB1 was a similar discus buoY but with a diameter of 7.6 m. Both were moored in the SW Approaches to the English Channel using 3-point catenary moorings. DB1 was in a depth of 168 m and DB2 in a depth of 184 m.
Measurement of directional wave spectrum
175
the wave system itself. Thus, a method of correction is required which does not involve fitting a simple resonance-type formula to the variation of response with frequency. The results will be illustrated using data from DB1 and DB2. DB1 was 7.6 m in diameter but the same draft as DB2 and otherwise rather similar in shape. In both buoys pitch, roll and heave were measured using a Datawell "Hippy" sensor. In DB1 the heave acceleration was integrated twice aboard the buoy using a Datawell analogue integrator. This displacement signal, the pitch, the roll and the compass bearing were sampled at 0.834 Hz and transmitted to shore by radio. After quality control, eleven subsections each 100.8 sec long were then Fourier-transformed and averaged to give the recorded spectral estimates at approximately 0.01 Hz intervals and with 22 degrees of freedom. Further details of DB1 are given by Freathy et al. (1981). In DB2 the heave acceleration signal was digitised directly, and all four channels were sampled at 1.28 Hz for 2000 sec. These series were divided into 400 sec segments which were tapered with a cosine function extending over the first and last 10% of the segment. The Fourier transforms of these were averaged over four adjacent harmonics and over the five subseries to give 0.01 Hz resolution with approximately 36 degrees of freedom. These computations were carried out on the buoy and the results were recorded aboard as well as being transmitted ashore via a satellite. 1.2. Convention for the coordinate system This presents a problem for the practical oceanographer, since the output of the analysis is always required in terms of geographical angles. Also, for locally-generated waves, the wind and wave directions must be the same, which means that 0 has to be the direction from which the waves have come, which is the opposite of the usual convention used in physical theory. However, it is usual to carry through the analysis in the conventional physical/mathematical system and to leave it to the user to convert the results into the practical system. This will be done here. Thus, the system is: x positive east; y positive north; z positive up; roll is tilt, east up positive; pitch is tilt, north up positive; bearing angles are positive anticlockwise from east; 0 is the direction towards which the waves are travelling. 2. THEORY The classic method of analysis was worked out by Longuet-Higgins et al. (1963) and by Cartwright (1963). The analysis which follows is basically the same theory extended to include the response of the buoy. It is assumed throughout that we are dealing with a stationary Gaussian process. 2.1. The effect o f buoy response on the determination of the angular harmonics The wave system is taken as the superposition of a large number of sinusoidal components travelling independently of one another:
~(t) = ~ a" cos (l,,x + m,,y - to,,t - d~,,) n
(1)
176
M.J. TUCKER
where a" has a dash to distinguish it from an used later as a cosine component In = m. = ton = +n = 0=
k. cos 0 (x component of wave number); kn sin 0 (y component of wave number); angular frequency; a random phase; and direction of travel.
The slopes are
Sx(t) = O~ = _ ~ a" l~ sin(/,,x + m~y -tont - ~n) Ox sy(t) - Og Oy _
~'~ an m . sin(/nx + m,0, - t o n t - ~bn).
Taking the origin at the centre of the buoy, the vertical displacements and slopes of the water surface at this point in the absence of the buoy would be ~o(t) =
a" cos(to o t + +n)
(2)
(an costo, t - b. sintod)
Sx(t) =
In (b. costo, t + a. sinto~t)
(3)
Sy(t) =
rn. (b. coston t + an sintod)
(4)
where a. = a" cos+. and b. = a" sind~.. Now this is the point at which one would normally expect to move into complex notation. However, for this particular problem it is simpler and clearer to stay in real notation. Considering two signals represented as the sum of sinusoidal components
vl(t) = ~ (al(n) costo, t + bt(n) sinto, t)
(5)
vz(t) --- ~] (az(n) coston t + b2(n) simon t).
(6)
Then the co- and quadrature spectra are given by
cizAf = 1/2 ~ (a,az + b,bz) af
(7)
q,2Af = 1/2 ~ (a,b2 - azb,). a[
(8)
The concept of these is that the signals are first filtered to leave the band Af, then c,2 is the mean of Vl(t)'v2(t) and qt2 is the mean of vt(t) times vz(t) with each component of vz advanced by 90° in phase. Equations (7) and (8) are used in algorithms to compute estimates of ctz and qt2 from simultaneous finite records, and in this case the right-hand side of Equations (5)
Measurement of directional wave spectrum
177
and (6) are Fourier series. The mean of vl(t).v2(t) works out neatly since all crossproducts between different harmonics average to zero over the length of the record. However, for the present purpose we require the expectation, or ensemble averages, of c12 and q12, and in this case the same result is obtained if non-harmonically related frequencies are used on the right-hand sides of Equations (5) and (6), and there can even be a number of components at the same frequency as long as their phases are random. The cross-products of the unrelated components will have random phases and will therefore be randomly positive and negative, so that only related components will contribute systematically to the sum as the number of components is notionally taken to infinity. For the first computation, vx(t) will be taken as the heave of the buoy and v2(t) will be taken as the roll of the buoy. Let the amplitude and phase of the heave response be given by r~a and otH respectively (a positive phase angle being a phase advance). These are functions of frequency, but will be assumed to be constant within a bandwidth Af. Then using Equation (2)
vl(t) = ~ rHa" [cos(0n + otn) costont - sin(0n + an) sintont]
(9)
similarly, if the amplitude and phase of the tilt response are rr and otr respectively v2(t) = ~, In rra" [sin(0n + err)COStOn t+ COS(0n + Otr)sintont].
(10)
From Equations (9) and (10) using Equation (7) cur Af = 1/2 ~, 1, ru rr (a') 2 [cos(0n + an) sin(0n + a t ) A[ --
sin(0 + or,v) cos(0n + otr)]
= 1/2 ~'. In rn rr (a') 2 sin(otr - ~xn). af Since rH, rr, a n and a r are assumed constant within Af, then putting In -- kn cos 0n, and assuming k,, also to be constant within Af gives
ctaR A f = r n r r k s i n ( a t - an) ~ 1/2(a') 2 cos 0n. af
(11)
Consider components travelling in a range of directions between (0 - 1/2A0) and (0 + 1/2A0), then by definition of the directional spectral density S(f,0): Expectation ~'~ 1/2(a') 2 = S(f,0) AfA0. Af.A0
Multiplying both sides by cos0n and summing over all 0 gives 1/2(a;,) 2 cos0, ---->Af ~.f
S(f,0) cos0 dO Jo
178
M.J. TUCKER
and Equation (11) becomes sin(~r - all)
CHR = rH r r k
COS 0 S(f,0) de
or putting (ar--~H) = a for convenience: sins
S(f,0) cos0 d0.
(12)
qHR = r H r r k
coso~
S(f,O) sinO dO
(13)
Clip = r H r r k
sins
S(f,0) cos0 d0
(14)
q~e = rHrr k
cosa
S(f,0) cos0 d0
(15)
CHR = r H r r k
Similar analysis gives
CHH --- ~H2
(16)
S(f,0) dO )
CRR = r r 2 k 2
S(f,0) sinZ0 dO
(17)
cos20 dO
(18)
)
Cee = r r 2 k z
S(f,O) )
CpR = r r 2 k 2
S(f,0) sin0 cos0 dO
(19)
}
(20)
qPR = 0. To interpret these it is conventional to put sff,0)
=
S(t).G(O)
where S(f) is the one-dimensional or point energy spectrum, G(0) is a function of both 0 and frequency, and .1"2,, G(0) dO = 1. G(0) is expressed as a Fourier series of angular harmonics: G(0) = ~ ~ +
[A, cos n0 + B,, sin n0] .
(21)
n=l
Putting this into Equations (12)-(20) gives CHR = r H r r
(sins) k S(f) B,
(22)
Measurement of directional wave spectrum
qnR
= rarr(coset) k S(f) B,
179
(23)
clip = rarr
(sina) k S(f) A t
(24)
qm, = rnrr
(cosct) k S(f) A1
(25) (26)
Can = r~ S(f) CRR = rr 2 k 2 S ( f )
[1/2 - 1/2 A2]
(27)
c e e = r7a k 2 S ( f )
[1/2 + 1/2 A2]
(28)
c e n = rr 2 k z S ( f ) • qen
1/2 B2
(29)
= 0.
(30)
Normalising these equations gives cap [CnH(Cvp + CRn)] v2 = A 1 sin
qHP
[CnH(Cpp + CRR)] m = A t cnn
[CHH(Cpp "F CRR)] 1/2
= B1
a
cos cx sin ~x
(31) (32)
(33)
[ c m , ( c p pq a+R cRR)] t'~ = B, cos
(34)
C p p - CRR Cpp + CRR
As
(35)
= B:.
(36)
2 cen
Cpp -F CRR
-
When a = 0, Equations (22) and (24) show that c n e and c a r = 0 so that Equations (31) and (33) disappear and the rest reduce to the classic relationships for a perfect surface-following buoy. The best way to use Equations (31) and (32) to determine Aj and (33) and (34) to determine B1 will be discussed in Section 2.2. The above analysis is in terms of heave displacement. If heave acceleration is used, a factor of -to,, z appears in Equation (1) and follows through to appear on the righthand side of Equations (22)-(25). It appears as to4 in Equation (26). Equations (31)-(34) then have the signs of the R.H.S. changed. A "check ratio" R is also usually computed and for calculations starting with heave displacement is defined by I" R = k[cep
cH---H- / I/2 + cnRj
"
For calculations starting with heave acceleration in S.I. units
(37)
180
M.J. TUCKER
e :kl
CH--H-
/ 1/2
0~2 [c,~p + CRRJ
(38)
"
Note that in practice acceleration is usually specified as a proportion of g, in which case a factor of g appears on the right-hand side of Equation (38). For deep water, gk/¢o 2 = 1 and for shallow water gk/to 2 = 1/tanh(kh) where h is the water depth. From Equations (26)-(28) (39)
R = (rnn/rrr) in.
R is therefore a measure of the ratio of the heave and pitch amplitude responses of the buoy (or of errors in calibration!). However, the dispersion relationship for a moored buoy can be changed by Doppler effects due to currents changing the apparent phase velocity of the waves (Kuik and Holthuijsen, 1981) and this also modulates the check ratio. The departure of R from 1 at middle frequencies shown in Fig. 2b is almost certainly due to mooring restraints (it is too large to be due to currents), and a time series of R at, say, 0.1 Hz shows a strong semi-diurnal variation, as well as longer-term variations. When one of the three moorings broke, the value rose to about 2. Note that apart from the equations for R, the dispersion relationship does not enter into the above calculations, which are therefore valid for shallow as well as deep water. 2.2. The effect on the mean direction and spread parameters and methods of correction The components of the first and second angular Fourier harmonics are insufficient to allow a useful reconstruction of G(0) by simple superposition. Some workers favour fitting by maximum likelihood methods, which use the additional information that G(0) cannot be negative (Jefferys et al., 1981). Long and Hasselmann (1979) have developed a variational technique to minimise a "nastiness function". Kuik (1984) favours interpretation into 0M~ and or as described below, but with skewness and kurtosis parameters derived from the second angular harmonic instead of 0M2 and s2. However, the commonest practice is to follow Cartwright (1963) and assume that the directional spectrum is of the form G(0) = K cosz' 1/2(0 - 0M)
(40)
where s and 0M vary with frequency. Then separate estimates of 0M and s at each frequency can be obtained from the first and second angular harmonics: 0MI = arctan BI/AI
(41)
0Ma = 1/2 arctan B J A z
(42)
C1 where C12 = A1E + B12 s~ - l - C 1 1 + 3Cz + (1 + 14Ca + Ca2) 1/2 s~ =
2(l-G)
(43) where C22 = A22 + B22.
(44)
Measurement of directional wave spectrum
181
A further parameter tr is sometimes used, which for narrow directional distributions is the r.m.s, spread about 0M1 tr = (2 - 2C1) '/2.
(45)
On any particular occasion and in any particular frequency band, the similarity or otherwise between sl and s2 and between 0MI and 0M2 can be used as a measure of whether Equation (40) is a reasonable description of the directional spectrum. In low energy parts of the spectrum the parameters may be corrupted by noise (see Section
3). It is immediately clear from Equations (35) and (36) that A2 and B2 would be correctly computed using the classic equations, and so the computed values of 0M2 and s2 do not depend on buoy response. The same is true for 0M~, since the classic equation can be put in the form 0M~ = arctan(q,R/qHe) and from Equations (23) and (25) this still gives the correct result when the effect of buoy response is included. Note, however, that if et approaches 90 °, the result becomes noisy and so this is not in practice the best equation to use. However, sl and tr are affected because the value of C~ computed using the classic equations is actually C~ coset if the effect of buoy response is included. Equations (43) and (44) both include factors of 1 - C~, so that when s~ is high (that is, C1 is approaching 1) this effect can become critical. This problem, including its relationship to noise, will be discussed further below. Figure 2 shows that for DB2 et is significantly non-zero at high and low frequencies. For DB1 a can be significantly different from zero over the whole frequency range. What is more, analysing many occasions for the same buoy installation shows it varying also with time, presumably as the mooring restraints vary with wind, waves and currents. Thus, it is not practicable to fit and use a fixed relationship between ot and frequency. In principle, two estimates of a can be obtained at each frequency for each record. Using Equations (22) and (23) tan OIHR = CHR/qHR.
(46)
CHP/qHp.
(47)
Similarly tan OtHt, =
In practice, it is rare to get good values for both etHR and OtH,,. Figure 2 illustrates this: when the waves are coming from the west, there is little or no correlation between heave and N-S tilt, so that aHe is largely random. Returning to the practical computation of A~ and B~, it will be seen that the righthand sides of Equations (31) and (32) are the two components of a vector whose modulus is A~ and whose argument is ct, and similarly for B; and Equations (33) and (34). It is necessary to preserve the signs of A~ and B~ in order to resolve the 180° ambiguity in 0M~ (Equation 44)), so in most cases the simplest solutions are
I CH2+qHp2 /1/2 (Cpp+ CRR)J
A1 = sign(qHp) tcHH
(48)
182
M . J . TUCKER
(a)
l o g t o Slq) S ( f ) i n m~/Hz
"--"--~x,__ -2
I
I
i
~,._.
I
2
(b)
0
I
I
Check r a t i o
R
I
j',&
~/2
-~/2
I
I
~/2 (d)
------~./~,.~
-~12
I
0
0.1
~Ew
,~-_.,.~_.__----~/-..
I
I
0.2
0.3
0.4
F r e q u e n c y Hz.
FIG. 2 a - d .
Data
f r o m D B 2 11.00 hr, 21 March 1985. H~ = 9.50 m; T~ = 11.80 s e c .
/ c"R2+ qHR2 11'2 B~ = sign(quR) [Cull(C--------p-V-+C R R ) )
"
(49)
H o w e v e r , where et approaches 90 ° this could result in some erroneous signs, so if this seems likely to be a problem it might be better to rotate the vectors A ~ - c u t , sin et - q u v cos et
[cuu(c,.,. +
c,,,0l
(50)
and similarly for B1. In this case the value of ot used should be that corresponding to the c o m p o n e n t (pitch or roll), since this reduces the effect of any asymmetry of the buoy response. That is, for A l , tan a = CurJqHP and for Bt, tan ot = CHR/qHR.
Measurement of directional wave spectrum
183
40
(e)
s,
20
0
I
I
I
40
(f)
s2
20
0
I
I
I
S
(g)
E
eN=
N W S
I
I
I
S (h)
O~2
E
N W S
I
0
O.1
I
I
O.2
0.3
O.4
F r e q u e n c y Hz.
FtG.
2e-h.
In those cases where one or other of these has a large random fluctuation as described above, this is not of importance since the corresponding Aj or BI will be low. 3. THE D E T E C T I O N OF NOISE AND ITS EFFECTS
3.1. General Examination of records when the sea is comparatively calm indicates that for the three buoys mentioned in the introduction, there is little noise not generated in some way by buoy motion. However, other tests indicate that in higher sea-states there is noise present and that its effect is significant. In Section 3.2 the methods for detecting the presence of noise are described, in Section 3.3 the probable sources are identified, and in Section 3.4 the effects on the directional parameters are examined.
184
M . J . TUCKER
"Noise" in this context does not include the statistical instability of the estimates of the parameters: that is, the random errors in these estimates arising from the uncertainties inherent in using finite samples. These have been examined by Long (1980). One can often obtain a good feel for them in any particular case from the way the estimated parameters vary from frequency to frequency along the spectrum. One cannot expect Sl and s2 to agree closely in typical circumstances since the sea at exposed sites is a complex mixture of wave energy from various sources and it is unlikely to fit the simple cos 2s 1/2(0--0M) angular distribution. Similarly, 0M1 and 0M2 are unlikely to agree for much of the time, Thus, "discrepancies" in these parameters do not necessarily indicate anything wrong with the measuring system. However, as discussed below, there should be occasions when they agree quite closely, and so if the discrepancies are always present it does indicate something wrong. Ewing and Laing (1987) using numerical simulation show that with a bimodal sea, large discrepancies between sl and s2 occur only when the directions of the two systems differ by more than 90°. 3.2. Identifying the presence of noise In this context, noise can be defined as any signal present in the pitch, roll and heave outputs which is not generated by linear response to the waves. In general, it will therefore be uncorrelated with the first-order wave signals. The only really positive indication of the presence of noise arises because it has been established that there is usually very little wave energy in the sea at frequencies below 0.04 Hz. This was demonstrated by Munk et al. (1959) for Californian waters, and some unpublished work by the author's former colleague Ian Vassie in which he analysed long records from a pressure recorder in 50 m of water west of the Outer Hebrides, showed that in a storm with a significant waveheight of 6.7 m, the spectral density in the frequency range 0.2-0.4 Hz was three orders of magnitude below the peak spectral density, which was at approx. 0.6 Hz. On calmer occasions the ratio could approach four orders of magnitude. If acceleration or tilt spectra are used, then the ratios will be an order of magnitude greater (compare Figs 2a and 4 which show the same data: but note the different frequency scales). At frequencies below about 0.01 Hz the spectral density starts rising again, but this does not concern us here. In the North Atlantic Ocean, there will be perhaps one or two severe storm events during a year which generate significant spectral energy below 0.04 Hz. Thus, the presence of significant spectral density in the 0.01 and 0.03 Hz frequency bands is a positive indication of noise at the low frequency end of the spectrum. In modern buoys, the sampling frequency is usually sufficiently high to allow the analysis to extend to well above the buoy resonances, so that these and diffraction effects reduce the spectral density to low values at the upper end of the frequency range. The primary purpose is to avoid aliassing of high frequency energy into the main part of the spectrum. If the spectrum flattens out at the upper end more than would be expected by aliassing, this can indicate the presence of noise. The third method for detecting noise from internal evidence in the data is more subtle and requires some oceanographic judgement. Tucker (1987) has shown that if the wave energy is confined to a narrow range of directions (in practice, to an r.m.s.
Measurement of directional wave spectrum
185
beam width of less than about 20°), then independent of the shape of the angular distribution:
Cl --') 1 - 12/2 (?2""> 1 - 212
where l2 = f d+~(O-OM)2 O _ G(O) which is the mean square angular spread of 0 about 0M. From which Cl = (3 + C2)/4.
(51)
The exact relationship when G(0) = K cos2s 1/2(0-0M) is C 2 = Cl(2C
1 -
1)/(2 - Ct)
which reduces to Equation (51) as C~ tends to 1. Now at the low-frequency end of the wave spectrum it is common to find swell bands from distant storms which have narrow spreads in both frequency and direction. Thus, if one plots (?2 against C1 for, say, all frequencies below 0.1 Hz and for a considerable number of records, there should be some points falling near the above relationships for high values of Ct and (?2. However, if there is noise present, then, as will be shown later, this is likely to affect C~ more than C2 and so the points will fall at lower values of Cl. These effects are shown in Figs 5 and 6 (where the graph relating C~ and C2 using Equations (40) and (41) is included for general interest: as explained above, it tends to Equation (51) for high values of s). DB2 behaves as expected, but noise is obviously present in DB1 (the source of the noise will be identified later). Note that this type of diagram was first used by Cartwright (1963) but for a different purpose. 3.3. Sources of noise It is impossible to cover all possible sources of noise, but some which have been identified as important in practice will be mentioned.
a. Frequency spreading due to analysis of short record sub-sections. This is the main source of noise in DB1. This buoy was developed in 1973 and used a dedicated minicomputer to receive and analyse the telemetered time series of pitch, roll, heave displacement and compass bearing. The limited computing power available at that time made it necessary to divide the record into 100-sec lengths for Fourier analysis without tapering. Those components in the sea spectrum which do not have an exact number of waves in 100 sec have some of their energy spread to other frequencies by this process. The effect if well known, and an assessment of it for this case shows that it has the right order of magnitude to account for the observed low frequency noise levels in DB1 (Fig. 3). For reasons to do with the stochastic nature of the data, it is not practicable to correct for it. However, the original DB1 time-series have been kept and a small number of sample records have been re-analysed using Fourier transforms of the complete records without subdivision. These demonstrate a large reduction of noise
186
M . J . TUCKER
log S ( f )
-2
i
0
i
0.1
i
0.2
0.4
0.3
Frequency
Hz
FIG. 3. A heave displacement spectrum from D B I . 06.00 hr, 9 December 1979. SOt) is in m2/Hz; H~ = 5.25 m; Tz = 8.45 sec.
and an improved agreement between s~ and s 2. However, the most practically useful parameters of the data, Hs T~ and 0M1, are not significantly affected so it is probably not worth re-analysing the whole data set. The low-frequency noise levels in DB2 (Fig. 2a) are similar in relative magnitude to those in DB1, but in fact this seems to be a coincidence, and their significance i~ very different. In DB1 the heave displacement signals were Fourier transformed and the noise due to frequency spreading would rise into the main part of the spectrum. In the case Of DB2, 400-sec lengths the heave acceleration signal were tapered and then Fourier transformed. This resulted in negligible levels of spread energy in the acceleration spectrum, indicated by the low spectral levels at low frequency in Fig. 4. i
1
--2
log Cal
-4
-/a
a
0
i
i
0.2
i
0.4
Frequency
i
O.&
Hz
FIG. 4. A heave acceleration spectrum from DB2. 11.00 hr, 21 March 1985. C,, is in g2/Hz;H,. = 9.50 m; 7", = l l . 8 0 sec.
Measurement of directional wave spectrum
187
However, multiplying by 1/o04 to get the displacement spectrum greatly magnifies the levels at 0.01-0.03 Hz. This diagnosis is supported by the evidence from plotting C] against C2 discussed in Section 3.2 (Figs 5 and 6).
BoLow
Q.ln
+~ +*~4+ :+
HI
C2
+
+
%
7L+=
+ +
+ + + + + +4"+ .4.+ +++ ÷ ++
+
4-+
÷+
+ +
+ +
++
+++ +
+
+ +++
+ .
J
,
~
,
Cl FIG. 5. Plot of C, against C2 for all spectral estimates below 0.1 Hz which have R > 0.9. Five days of DB2 data: 16-21 March 1985.
+
++ +
l /
+
/ • el+ow •,l.
+H,.U.tF.,.tl~:~++ 4~, +
Nat
/
C2
++++++++; / +
+ +++
+ +,~ +
+,** ;÷
J
I CIL
FIG. 6. Plot of CI against C 2 for all spectral estimates below 0.1 Hz which have R > 0.8. Ten days of DB1 data: 1-9 December 1979.
188
M.J. TUCKER
b. Noise due to compass tilt. It is easiest to explain this effect initially in terms of a 3-component fluxgate compass fixed to the hull of the buoy. If the buoy is tilted by an angle +R in the E-W direction, it will introduce a component of the earth's vertical magnetic field Hv into the E-W sensor given by HE = Hv sin ~bR.
If the N-S component of the earth's field is HN, then this rotates the apparent bearing of the buoy by an angle 0E given by tan 0E = HE/HN = (Hv/HN) sin +n.
(52)
(We shall not worry about sign conventions here since the analysis is only qualitative.) In UK waters Hv/HN varies from about 2 in the south to 3 in the north, so this effect is considerable. By apparently rotating the buoy, it introduces a component ~bpE of the roll angle into the N-S (pitch) channel: d~eE = d~Rsin d~E.
For small angles, using Equation (52) gives dPpE = d~n2Hv/HN.
(53)
Being a square-law error, this introduces sum and difference frequency noise components across the whole spectrum. They are detectable at low frequencies using the first test described in Section 3.2. Tilting in the N-S direction does not rotate apparent north to first order. Carrying out a full analysis shows that some noise can be generated in the E-W tilt channel, depending on the directional properties of the wave system, but that in general it is less than that in the N-S channel. This noise can generate spuriously high values of s2 at the low-frequency end of the spectrum where the true spectral density is low. In the buoys known to the author, the compass is in fact gimballed, but the sideways accelerations of the buoy cause tilts of the same order of magnitude as if the compass were fixed rigidly to the hull. A further complication is that because of these effects, the designers smooth the response of the compass: by oil damping in the case of the card compass, and by electronic or computational means in the case of both card and fluxgate compasses. These circumstances make it impracticable to correct for the effect, but an approximate calculation shows that it produces low-frequency noise comparable to that observed in the tilt channel of DB2. Thus, the high low-frequency noise level in the tilt channels of DB2 (Fig. 7) is probably due to this effect, but it falls off through the spectrum, as indicated by the low spectral density at the high-frequency end. c. Errors in the heave sensor. These can develop for a number of reasons, one of the most common being a tilt of the stabilised platform supporting the accelerometer, usually resulting from a damaged suspension. They are detectable as low-frequency noise in the heave spectrum, or as zero wandering in the displacement time-history produced by integrating the heave acceleration signal twice. They are familiar from long experience with one-dimensional wave recording buoys and will not be discussed further here.
Measurement of directional wave spectrum J
I
I
i
i
189
I
I
i
i
-2
log T~(f:
-4
_~
i
0
0.2
,.
0.4
Frequency
O.&
Hz
FIG. 7. A total tilt spectrum from DB2. 11.00 hr, 21 March 1985; Tr(f) is c~ + c33 in rad2/Hz.
3.4. The effect of noise on the directional parameters First consider the effects of noise on the cross-spectral components. Define a noise voltage e(t) by
e(t) = ~ g, cos(to, t + ~J,,) . Then adding this to the heave channel vl(t) (Equation (9) with the sine and cosine components re-combined) gives
v~(O = ~ [rHa" cos(co, t + dp, + all) + g, COS(tO, t + 0,)]From which, by similar calculations to those in Section 2.1 the new value C'Hn is given by
C'HRAf = 1/2 ~ 1, rn rr ( a ' ) 2 sin (aT -- an) + ~'~ rrg, a" sin(~b, - ~b, - an). af
(54)
af
Now by definition of the noise, ~b, - ~b, is random in the interval 0-2~r, and so the expectation of the second term is zero, in which case Equation (54) reduces to the equation preceding Equation (11). The same is true if uncorrelated noise is also added to the roll channel, and similar results are obtained for qun, cup and que. Thus, noise does not affect the expectations of these four cross-spectral components. If large, it will increase the random variability of them, but even a signal/noise ratio of 1 is not likely to produce large errors. There are possible mechanisms by which the noise in one channel could be partially correlated with the noise in another channel, but as long as the noise is a small
190
M . J . TUCKER
proportion of the total signal this is a second-order effect and unlikely to be significant, Carrying through similar analysis for the auto-spectra gives cHu = SNn(f) + run f S(f,0) dO
(55)
CRR = SNR(f) + rrrk z I S(f,O) sin20 dO
(56)
Cpp = SNe(jz) + rrr k 2 f S(f,O) cos20 dO
(57)
where suH(f), SuR(f) and Suv(f) are respectively the spectra of the noise in the heave, roll and pitch channels. To sum up the above, the cross-spectral estimates are unlikely to be significantly affected by noise in the more energetic parts of the spectrum, but the auto-spectra are all biased upwards. Turning now to the effect on the directional parameters, it is perhaps easiest to see that 0M1 is unaffected by noise by using Equations (41), (32) and (34), which together give tan 0M1 = qnR/qnv. Since qnR and qnP are unaffected by noise, then so is 0M~. To assess the effect of noise on 0M2, it is best to consider A2 and B2 as the components of a vector whose argument is 20M2 (Equation (42)). Then Equation (35) shows that although A 2 can suffer large proportional changes due to noise when it is <<1, the effect on 0M2 will only be large if B2 is also small at the same time. In this case 0~2 will be largely indeterminate due to the statistical instability in the cross-power spectral components. Thus, though noise does affect 0M2, the. effect is likely to be of rather marginal practical importance in the more energetic parts of the spectrum. The situation is very different in the case of sl. In this case from Equation (43) ds, _ s,(1 + s 0
dCl
(58)
Cl
Putting Equations (51) and (52) together gives C l 2 = CHp 2 + q n p 2 + CHR 2 -.1--q n R 2
cnn (Cep + CRR)
(59)
Considering noise in the heave channel, which always increases cnn (see Equation (55)), differentiating the above gives dC1
C1
dcuH- 2cHn"
(60)
Taking this with Equation (58) gives ds--! = 1/2 (1 + S 0 dcn____y_H S1
CHH
(61)
Measurement of directional wave spectrum
191
A similar result is obtained if noise is added to (cee + CnR), which is the total tilt spectrum. While this equation is useful in assessing the confidence in measured values of s~, it is in some ways more instructive to consider the case where the true value of s is infinity, that is, CI = 1. Then adding 2.5% of noise to both CHILl and (cm, + Cnn) in Equation (59) reduces Ct to approx. 0.975 and hence from Equation (43), st to 40. Figure 2 shows a storm for which this is the estimated value at the peak of the spectrum, so clearly, even low levels of noise can be quite critical for s~. Considering s2, the situation is more complicated. For example, consider a very narrow beam sea coming from due north. Then without noise, CRn ~ 0 and cen "-->0 and combining Equations (35) and (36) gives (72 --~ 1 and s2 ---> infinity. Adding noise to cpp makes no difference to (72, but if a small amount of noise is also added to Cnn this does make a difference. If this noise has a spectral density of v'cm,, then (72 is reduced to approx. 1-2v. Putting v = 0.025 to allow a comparison with the calculation for s~ gives C2 = 0.95 and from Equation (44), s2 = 77.5. This demonstrates that s2 is generally less sensitive to noise than st, but the situation is rather more complicated if the noise is generated by the mechanisms described in Section 3.3(a) and (b), since the noise is not then equally divided between the two tilt channels. Ewing and Laing (1987) studied the effect of noise on sl and s2 by numerical simulation and came to the conclusion that s~ is very sensitive to noise and that s2 is the more reliable parameter when noise is present. 4. CONCLUSIONS It has been shown that the amplitude response of a pitch-roll-heave buoy does not affect the determination of directional spectral parameters, but that the phase difference between the heave and tilt responses must be taken into account when calculating the spread parameters derived from the first angular harmonic. In practice, the response may depend on mooring restraints which are varied by a number of factor s including current, so a method of correction is required which does not assume a constant response. Fortunately, the information required for correction is contained within the measured spectra. Ways of detecting the presence of noise in the records are described, and some probable sources identified. In particular, it has been shown that tilting of the compass can introduce significant noise into the tilt channels. The effects of noise on the directional parameters has been analysed, and it is shown that the mean direction 0mr derived from the first angular harmonic is very robust. 0m2 is less robust, but not likely to suffer large errors except when the spread parameter is low. s~ and the related parameter cr are very sensitive to noise, which always tends to decrease sl and increase e. It seems likely that the spread parameters at the peaks of storm spectra have been underestimated because of this, but it is difficult to quantify. s2 is less sensitive, by a factor in the region of 2 for high values. Some records show large differences between 0mr and Ore2, particularly at middle frequencies (Fig. 2 shows one such case). In the author's experience, s is low when this happens, indicating a complex sea, and the difference is probably genuine. In such cases it is sometimes difficult to resolve the inherent 180 ° ambiguity in 0m2, but when s is reasonably high, 0M1 and Ore2 agree well.
192
M . J . TUCKER
Acknowledgements--This paper is based on work supported by the U.K. Offshore Operators Association, to whom the author is grateful for permission to use the DB1 and DB2 data. A report to them dated 9 January 1987 was seen by a number of scientists. In particular, helpful comments were received from M. Bradley, R. Jeffereys and A.J. Kuik. The author would also like to thank J.A. Ewing of the I.O.S. Deacon Laboratory for his interest throughout the course of this work.
REFERENCES BARSTOW, S.F., UEta~qD, G. and FossoM, B.A. 1986. The wavescan second generation directional wave buoy. Proc. Oceanology Int. Conf., Brighton, U.K., March 1986. Society for Underwater Technology, London. CANaAn, H.J.S., CARTVCgIGm, D.E., GOODRICH, G.J. and HOGBEN, N. 1962. Seakeeping trials on O.W.S. "Weather Reporter". Trans. R. Instn nav. Archit. 1114, 447-492. CARrWRIGnT, D.E. 1963. The use of directional spectra in studying the output of a wave reorder on a moving ship. In Ocean Wave Spectra, p. 203. Prentice-Hall, Englewood Cliffs, N.J. CARrWRIGnr, D.E. and SmTn, N.D. 1964. Buoy techniques for obtaining directional wave spectra. Trans. 1964 Buoy Technology Syrup., Washington, 24--25 March 1964, pp. 112-121. Marine Technology Society, Washington, D.C. EWING, J.A. and LAlrqG, A.K. 1987. Directional spectra of seas near full development. J. phys. Oceanogr. 17, 1696--1706. FREATHY, P.E., HOOPER, A.G. and MACDONALD, H.W. 1981. Wind and wave directional data obtained from DB1 in the S.W. Approaches to the United Kingdom. Proc. Int. Conf. on Wind and Wave Directionality, Paris, 1981. Editions Technip., Paris. JEFFERYS, E.R., WAREHAM,G.T., RAMSDEN, N.A. and PLAI"rs, M.J. 1981. Measuring directional spectra with the M.L.M. Proc. Conf. on Directional Wave Spectra Applications, Berkeley, California, 14-16 September 1981. Am. Soc. Civil Engrs, New York. KUIK, A.J. 1984. Proposed methods for the routine analysis of pitch-roll buoy data. Proc. Symp. on Description and Modelling of Directional Seas, Paper No. A-5. 18-20 June 1984. Technical University of Denmark, Danish Hydraulic Institute. KUIK, A.J. and HOLTnUIJSEN, L.H. 1981. Buoy observations of directional wave parameters. Proc. Conf. on Directional Wave Spectra Applications, Berkeley, California, 14-16 September 1981. Am. Soc. Civil Engrs, New York. LONG, R.B. 1980. 3~he statistical evaluation of directional spectrum estimates derived from pitch/roll buoy data. J. phys. Oceanogr. 10, 944-952. LONG, R.B. and HASSELMANN, K. 1979. A variational technique for extracting directional spectra from multicomponent wave data. J. phys. Oceanogr. 9, 373-381. LONGUET-HIGGINS, M.S., CARTWRtGHT,D.E. and SMITH, N.D. 1963. Observations of the directional spectrum of sea waves using the motions of a floating buoy. In Ocean Wave Spectra, p. Ill. Prentice-Hall, Englewood Cliffs, N.J. MUNK, W.H., SNODGRASS,F.E. and TUCKER, M.J. 1959. Spectra of low-frequency ocean waves. Bull. Scripps Instn. Oceanogr. 7, 283-362. RusBY, J.S.M., KELLY, R.F., WALL, J., HUNTER, C.A. and BUTCHER, J. 1978. The construction and offshore testing of the U.K. Data Buoy (DB1 project). Proc. Oceanology Int. 1978 Tech. Session J., pp. 64-80. Society for Underwater Technology, London. TUCKER, M.J. 1987. Directional wave data: the interpretation of the spread factors. Deep-Sea Res. 3,633-636.