In t e r n a ti o n a l H ydrographie R e v ie w , M onaco, L V I (2), J u l y 1979.
SOME COMPARISONS OF RESPONSE
AND HARMONIC TIDE PREDICTIONS
b y B e rn a rd Z E T L E R
In s titu te of G eophysics a n d P la n e ta r y P hysics,
S crip p s In s titu tio n of O c ean o g rap h y , L a J o lla , C alifo rn ia, U.S.A.
D avid C A R T W R IG H T
I n s titu te of O c ean o g ra p h ic S ciences,
B id sto n O b serv ato ry , B irk e n h e a d , E n g la n d
a n d S au l BERKM AN
N a tio n a l O cean S urvey, R ockville, M d., U.S.A.
ABSTRACT
E m p iric a l te s ts co m p ared re sp o n se a n d h a rm o n ic tid e p re d ic tio n s fo r
A tlan tic C ity a n d P e n sa c o la (se m id iu rn a l a n d d iu r n a l tid a l re g im e s respec-:
tiv ely ). T h re e y e a rs of h o u rly h e ig h ts w ere an a ly z e d b y b o th m e th o d s in
th e fre q u e n c y ra n g e of one to six cycles p er day. T h e re s u lts w e re u sed
to p re d ic t a n o th e r th re e -y e a r p erio d , th e p re d ic tio n s w ere s u b tra c te d fro m
th e o b serv atio n s, a n d en erg y ca lc u la tio n s w e re m a d e fo r th e fre q u e n c y
b a n d s in e a c h of th e six tid a l species. O nce m o re, re sp o n se m eth o d s
w ere so m e w h a t b e tte r, b u t th e differences a re sm a ll co m p a re d to th e to ta l
(u n p re d ic ta b le ) c o n tin u u m . T he s tu d y d isclo sed : (1) th e n eed to in c lu d e
th ird -o rd e r n o n lin e a r in te ra c tio n s of d iu rn a l tid e s in re sp o n se p re d ic tio n s
fo r som e sta tio n s, (2) th e n eed fo r N a tio n al O cean S u rv ey to ex a m in e
c a re fu lly its re je c tio n lim it in an aly zed a m p litu d e s of 0.03 fo o t a n d a
p ra c tic e of in fe r rin g T 2 re g a rd le ss of its a m p litu d e , a n d (3) th e n ee d to
ex am in e a n a n n u a l m o d u la tio n of M 2 p re s u m a b ly d u e to som e local
seaso n al effect.
INTRODUCTION
T h e ‘re s p o n se m e th o d ’ of tid a l a n a ly sis w a s d esig n ed b y M u n k a n d
r i g h t (1966) p rim a rily a s a re s e a rc h tool a n d to a id p h y sic a l u n ­
d e rs ta n d in g of tid a l processes. It also p ro v id es v e ry good tid a l p re d ic tio n s,
Ca r t w
not only in the nearly linear regim es studied by M u n k and C a r t w r i g h t
(1966) but also for regim es w hich require fairly strong nonlinear terms,
as dem onstrated by C a r t w r i g h t (1968) and C a r t w r i g h t and R o s s i t e r
(1972). In fact, in all references ju st cited, and in others (e.g. Unesco,
1975), the residual variances from ‘response’ predictions have invariably
been shown to be less than those from modern ‘harm onic’ predictions.
The Unesco report included a caution, “It is like the difference between
a Kodak (harm onic m ethod) and a Hasselblad (response m ethod) : with
little input, the form er gives the better pictures, but w hen properly used,
the Hasselblad can improve the resu lt.” The response method has never
been used in the operational production of tide tables, partly because of
unfam iliarity w ith the technique, and partly because the harm onic method
gives good enough predictions for m ost purposes. The present experi­
m ents were designed to re-assess the relative accuracies of harm onic and
response predictions in the context of the standard operational proce­
dures of the U.S. National Ocean Survey. Because of C a r t w r i g h t ’s
greater experience in nonlinear response procedures, his offer to partici­
pate in the tests w as gratefully accepted by the other tw o authors.
The two stations chosen for the experim ent were A tlantic City, New
Jersey, on the United States east coast and Pensacola, Florida, on the
Gulf of Mexico. The A tlantic City tide is prim arily sem idiurnal,
(Kt + O ^/M a being about 0.3. Because of the broad continental shelf,
extrem e sea level fluctuations are alm ost as large as the tidal range (figure 1
from Z e t l e r and L e n n o n , 1967). This is m anifested by a high contin­
uum in the low -frequencies; the residual energy after a tidal analysis
exceeds 10 % of the observed total energy (in the frequency range of 0
DAILY VALUES OF MEAN SEA LEVEL FOR ATLANTIC CITY 1 939
"Tg,,
to 12 cycles p e r d ay ). In a sim ila r stu d y of S an F ra n c isc o tides, th e
re s id u a l e n e rg y w as ab o u t 3 % of th e o b served en erg y .
P e n sa c o la h a s a d o m in a n tly d iu r n a l tid e, (K t 4 - O ^ /M ^ being ab o u t
15. T h e m e a n d iu rn a l ra n g e is sm all, ab o u t 0.4 m , a n d a tid a l a n a ly sis
leaves a re s id u a l en erg y ab o u t 50 % of th e o bserved energy.
T h e p la n called fo r o p tim u m a n a ly se s of th re e y e a rs o f h o u rly h e ig h ts
by b o th m e th o d s, re sp o n se a n d h a rm o n ic p re d ic tio n s fo r a d iffe ren t th re e
y e a rs , s u b tra c tio n of p re d ic tio n s fro m o b serv atio n s, a n d d e te rm in a tio n s
of re s id u a l e n e rg y in six tid a l b a n d s, fro m one to six cycles p e r day.
In a sm u c h as h a rm o n ic m e th o d s o b ta in Sa a n d S sa fro m av erag ed m o n th ly
m ean s fo r as m a n y y ea rs as a re av ailab le, species 0 w as o m itte d fro m
th e c a lc u la tio n s. T h is decision also ru le d o u t th e u se of th e m o n th ly a n d
fo rtn ig h tly tid e s, M m , Mf a n d M sf, b u t exp erien ce h a s sh o w n t h a t th e s e
are n o t d efin ed s a tis fa c to rily by ro u tin e h a rm o n ic a n a ly sis b ecau se of th e ir
low s ig n a l/n o is e ra tio . T h e N atio n al O cean S u rv ey o rd in a rily solves fo r
M8, b u t th is is k n o w n to be sm all a t th e tw o s ta tio n s so it w as o m itte d
also, m a k in g 6 cp d th e h ig h e st fre q u e n c y resolved.
ANALYSIS PROCEDURES
An e d itin g p ro c e d u re fo r e lim in a tin g d a ta e r ro rs called fo r a q u in tic
p o ly n o m ia l as a te s t fo r sm o o th n ess. H ow ever, w h e n a ro u g h n e s s lim it
of 0.3 foot ( C a r t w r i g h t , 1968) w as ap p lied to A tla n tic C ity d a ta , ro u g h ly
2 % w ere flagged. It w as evident t h a t th is is d u e to th e larg e sea level
flu c tu a tio n s ; a la rg e r ro u g h n e ss lim it, 0.6 foot, w as n ec essary . U sing th e
la tte r, th e o n ly v alu es flagged w ere d u rin g a 1954 h u rr ic a n e ; no ch an g es
w ere m ad e to th ese d a ta p o in ts.
A lth o u g h sy n o d ic p e rio d s a re less n e c e ssa ry fo r d a ta a n a ly sis of lo n g
serie s by e ith e r th e re sp o n se o r th e le a st-sq u a re h a rm o n ic m e th o d (all
c o n s titu e n ts w ith in each species re so lv e d in one m a trix ) th a n in classical
h a rm o n ic p ro c e d u re s, it w as ag reed t h a t it w o u ld be p ru d e n t to allow fo r
th is aspect. T h e choice of a n o p tim u m p erio d w as b ase d on p ro x im ity to
in teg ers fo r th e n u m b e r of synodic p erio d s of S 2 a n d M2, M 2 a n d N 2 a n d
K j a n d O j. O n th is basis, 1107 d ay s (3 X 369 d) w as ch o se n in p re fe re n c e
to 1065 d ay s (3 X 355 d) or to 1093 d a y s (2 X 369 + 355 d ). T h e re sp o n se
an a ly sis w o u ld cover th e w hole p e rio d in one an a ly sis, th u s p e rm ittin g
th e se p a ra tio n of g ra v ita tio n a l a n d ra d ia tio n a l tid e s. Since N a tio n a l O cean
S u rv ey (NOS) p ro c e d u re s o rd in a rily u se 369 days, it w as ag re e d to av erage
h a rm o n ic c o n s ta n ts fro m th re e 369-day an aly ses.
NOS a n a ly s is o f 369-day series o rd in a rily in c lu d e s 37 c o n s titu e n ts ,
b u t w ith Sa, S sa, M m , Mf, M sf a n d M 8 o m itted , th is w o u ld leave 31.
O rd in a rily , a n y c o n s titu e n t w ith a n an aly zed a m p litu d e of less th a n 0.03
foot is n o t u s e d ; if a n in fe re n c e can be m ad e fro m one o r m o re n e a rb y
m a jo r c o n s titu e n ts , u sin g re la tio n s h ip s in th e e q u ilib riu m tid e o r sim ila r
re fere n ce tid e (n o t possible fo r sh a llo w -w a te r c o n s titu e n ts ) a n d th e in ­
fe rre d a m p litu d e is a t le a st 0.01 foot, th e n in fe rre d v alu es fo r th e c o n sti­
tu e n t a re u se d in fu tu re p re d ic tio n s. T 2 is alw a y s in fe r re d fro m S2.
A vailable R eference S eries for R esponse A nalysis
s p e c ie s
1 -6
SPECIES 1
G* ( 2 )
G3
0
)
c io
G3 ( 0 )
G* ( 0 )
G3 ( - O
Rî
R 2
(I) 2 " 1
(I) 1 +1 - 1
G ‘ (-1 )
(I)l +2 - 2
G* ( - 2 )
SPECIES
2
(2)
G3 (i)
G 2 (l)
G2 (0)
G 2 (0)
G2 (-1 )
G
2
(D l _ 0
(I)1 + °
R 2
?- 0
UJ
(1 ) 2 + 0
R4
(I)1+I
G 2 (-1 )
(I) 2 + , _ 1
(I)2 +l - 2
G2 ( - 2 )
SPECIES 3
G3 O )
R4
(I ) 2 +1
(I) 2 +2- ‘
G 3 (0 )
G3 ( - O
SPECIES 4
RI
( I ) 2 + 2 ( 0 ,0 ) , ( 0 , 'a ) , ( 'A, *4 ) ( 0 ,
-
V4), ( -
'/j, -
(I)2 + 2-°
(I)2+2 +°
(I)2+1+1
0 )2 +2+2 -2
SPECIES 5
d )2+2+1
(i)2+2+2_1
SPECIES 6
(I ) 2 + 2 +2 (0,0,0), (0,0, W), (0 ,¾ ¼ )^ . ■/*, Vz)
( 1) 2 + 2 + 2 - 0
(j )2 +2 +2+0
>/2)
O th e r g ro u n d ru le s fo r NOS a n a ly sis p ro c e d u re s w e re te s te d in th e co u rse
of o u r e x p e rim e n ts ; as a re su lt, som e p ro c e d u re s p re v io u sly ro u tin e ly
accep ted a re n o w b ein g q u estio n ed .
T a b le 1, e x tra c te d fro m C a r t w r i g h t a n d R o s s i t e r (1972), is a listin g
of v a rio u s p o ssible re fe re n c e series in a re sp o n se a n a ly s is fo r species 1
to 6 . T h e G;? series (m is o rd e r a n d n is degree) re p re s e n t com plex tim e
series of th e to ta l g ra v ita tio n a l p o te n tia l, th e R serie s a re c o m p a ra b le
ex p ressio n s of th e S u n ’s ra d ia tio n a l p o te n tia l, a n d th e (I)<:t; series a re
th e p ro d u c t of p r im a ry tid e p re d ic tio n s, species i a n d
c o n ju g a te fo r
— sign. Sym bols (I)J- ° re p re s e n t a n n u a l m o d u la tio n s to first o rd e r p re ­
d ictio n s of species / ; w e did n o t use th e m in th is w o rk b ecau se n o c o r­
re s p o n d in g c o n s titu e n ts are used in th e NOS h a rm o n ic p ro c e d u re . N u m ­
b ers in b ra c k e ts a re tim e lags in 2 -d a y u n its.
T ab le 2 lists re fe re n c e series u sed in th e A tla n tic C ity a n a ly sis. T im e
lags a re liste d h e re in h o u rs (ra th e r th a n 2 -day u n its ) a n d th e lag s a re
ce n te red o n tid a l ages fo r each species, r a th e r th a n zero, as su g g ested by
Z e t l e r a n d M u n k (1975). H ow ever, w h e n C a r t w r i g h t also e x p e rim e n te d
Table 2
R eference Series used in A tlan tic City A n a ly sis
(N u m b ers d en o te lag s in h o u rs)
SPECIES 1
G ‘ (72)
G3
(24)
Rj (24)
G* (24)
(I ) 2
”1
(I) 1+2~ 2
G*(—24)
SPECIES 2
G2 (120)
G2 (24)
R* (24)
(I) 2 +2 _ 2
G2
2 (72)
Gj (24)
G2 (-2 4 )
G22 (-7 2 )
SPECIES 3
G33 (24)
(I)2 +1
SPECIES 4 - 6
d )2+2
( 1)2 +2 + 1
( 1)2 +2 + 2
w ith lag s c e n te re d on zero, h e fo u n d no sig n ifican t differences in re s id u a l
v arian ces. T h e p rin c ip a l a d v a n ta g e in co n sid erin g th e tid a l age a p p e a rs
to a p p ly to an a ly se s u sin g few er th a n th e o p tim u m n u m b e r of w eights.
F o r f u r th e r refere n ce, n o te th a t (I ) 1+1-1 w as n o t used.
In c o m p u tin g re s id u a l v a ria n c e s in th e six species fre q u e n c y
su ccessiv ely w id e r lim its w e re u se d fo r fre q u e n c ie s h ig h e r th a n
2 becau se of th e g re a te r sp re a d im p lic it in n o n lin e a r in te ra c tio n s
p rin c ip a l c o n stitu e n ts. In c a lc u la tin g th e re sid u a ls for 355-day
th is involved th e follo w in g :
Cycles per day ± cycles per month
1
Species 1
±
4½
b an d s,
species
of th e
series,
Harmonics
285-401
Species 2
2
±
4½
628-744
Species 3
3
±
5½
958-1100
Species 4
4
±
6
½
1288-1456
Species 5
5
±
7½
1618-1812
Species 6
6
±
8
½
1948-2168
ATLANTIC CITY RESULTS
T ab le 3 show s re s id u a l en erg y in six fre q u e n c y b a n d s fro m IO S a n d
IG PP re s p o n se p re d ic tio n s a n d tw o sets of re sid u a ls fro m NOS h a rm o n ic
p re d ic tio n s. NOS #1 v alu es re p re s e n t ro u tin e p ro c ed u re s, in p a r tic u la r
s u b s titu tin g in fe rre d a m p litu d e s a n d p h a se s fo r a n y c o n s titu e n t w hose
an aly zed a m p litu d e is fo u n d to be less th a n 0.03 fo o t; in fe rre d c o n s titu e n ts
Table 3
A tla n tic C ity resu lts in c m * (3
X
355 days)
Residuals
Response
DATA
IOS
Harmonic
IGPP
NOS # 1
NOS # 2
79.47
3.90
3.86
3.93
3.85
SPECIES 2
1989.95
4.58
4.73
6.07
5.91
SPECIES 1
SPECIES 3
1.50
1.25
1.24
1.46
1.31
SPECIES 4
1.28
0.6 8
0.69
0.91
0.71
SPECIES 5
0.81
0.76
0.76
0.82
0.85
SPECIES 6
0.92
0 .68
0.70
0.82
0.84
Species total
. . . . 2073.92
Overall......... . . . . 2368.19
Non-tidal. .
294.27
a re u sed in p re d ic tio n only if th e in fe rre d a m p litu d e is a t le a st 0.01 foot.
T h e p ro c e d u re also in clu d es in fe rrin g T 2 re g a rd le ss of an a ly z ed a m p litu d e ;
sin ce T 2 a n d S 2 (used as th e base for th e in feren ce ) are b o th g ra v ita tio n a l
a n d ra d ia tio n a l, an in feren ce b ased on th e g ra v ita tio n a l p o te n tia l only
leaves so m eth in g to be desired. In NOS ¢2 tests, an a ly z ed v alu es o nly
a re u s e d ; of th e 31 c o n s titu e n ts resolved, o nly S 6 an d px a re re je c te d fo r
sm all a m p litu d e s a n d w id ely -v ary in g p h ases. T h e im p ro v e m e n t in re su lts
(sm aller re s id u a l energy) in d ic a te s a re c o n sid e ra tio n of NOS ro u tin e p ro ­
ce d u re s is advisable.
T h e re s id u a ls fro m IOS a n d IG P P re sp o n se p re d ic tio n s a re slig h tly
d iffe re n t; no a tte m p t w as m ad e to d e te rm in e th e cau se of th e sm all
d ifferen ces a n d e ith e r set m ay be ta k e n as re p re se n ta tiv e o f re sp o n se
a n a ly sis a n d p red ictio n . T h e p rin c ip a l ad v a n ta g e o f re sp o n se over h a r ­
m o n ic m e th o d s is fo u n d in th e species 2 b an d . A lth o u g h th e difference
a p p e a rs to be significant, it is less th a n 0.1 % of th e en e rg y in th e species
2 b an d . In th e lig h t of th e species 2 difference, it w as puzzlin g th a t th e
re s id u a ls in species 1 are a b o u t equal. O u r su b se q u e n t a n a ly se s of P e n ­
saco la tid es m ay have fu rn is h e d an ex p lan a tio n .
PENSACOLA RESULTS
T ab le 4 show s re s u lts o b tain ed w ith P e n sa c o la d a ta . R esponse an a ly sis
¢1 (at IG P P ) u sed a re d u ced set of re fe re n c e series:
Species 1: G^ (— 48,0,48), GJ ^O), R} (O)
Species 2: GS (— 48,0,48), G f (O), (1)1+ 1 (O)
Species 3: G® (O), ( 1 ) ^ + 1 (O).
Table 4
Pensacola R esu lts in c m 1 (3 x 355 d a y s )
Residuals
Data
SPECIES 1
152.24
Response # 1 Response #2 Harmonic # 1 Harmonic # 2
3.21*
1.43
3.64
3.02
SPECIES 2
2 .68
0.28
0.29
0.59
0.42
SPECIES 3
0.13
0 . 1 0
0.11
0.15
0.15
SPECIES 4
0 . 1 2
0.12
0.12
0.15
0 . 1 2
SPECIES S
0.08
0.08
0.09
0 .12
0 . 1 2
SPECIES 6
0.09
0.09
0.10
0.14
0.13
Species total, , . . .. 155.34
Overall........... ___ 307.35
Non-tidal.. . . . . .. 152.01
* This value was obtained using either 3 or 5 weights for G* reference series.
T h e decision to o m it species 4, 5 a n d 6 w as b ase d o n th e very sm a ll
a m p litu d e s in species 2 , h e n c e no sig n ific a n t o v ertides involving M 2 o r S 2
ca n be ex p ected. T h e re sid u a l e n e rg y in th e se species th e re fo re e q u a ls
th e d a ta en e rg y ; o th e r a n a ly se s did no b e tte r. T ests w e re also m ad e w ith
five w e ig h ts fo r GJ a n d G | resp ectiv ely . T h e re w as no difference in th e
m e a n re s id u a ls in species 1 ; species 2 a d m itta n c e s, u sin g 5 w eights, fluc­
tu a te d so w ild ly th a t th is p o rtio n of th e te s ts ended here.
R esp o n se a n a ly sis ¢ 2 (a t IOS) a d d e d th e re fe re n c e series (1) 1+ 1 - 1 (O)
a f te r n o tin g large re s id u a l lin es a t th e fre q u e n c ie s 2 0 K X (2 0 j — Kj) a n d
2K O x (2K X— O j). T h e re s u lts s h a rp ly d ec rea sed th e species 1 re sid u a l
en erg y . T ab le 5, sh o w in g a p p a re n t a d m itta n c e s fro m NOS h a rm o n ic
c o n s ta n ts , in d ic a te s cle a rly th e n e e d fo r a species 1 n o n lin e a r re fere n ce
series.
A lin e a r re sn
--------- ...
------------a----------1 on se- - a n a lv
^ s is ., re ax u ire d to be sm o o th over a n a rro w
fre q u e n c y b a n d , co u ld n o t reflect a d e q u a te ly th e s h a rp changes a t th e
e x tre m e s of th e b a n d . T h e p re sen ce of trip le in te ra c tio n s like (I ) 1+1_1
in d ic a te s fric tio n in th e tid a l system .
Table 5
Pensacola tidal a d m itta n c e s ( 1 cpd )
H a rm o n ic C o n sta n ts fro m NOS A n a ly sis — 3 y e a rs — 1952 to 1954
Ampl./
Coef.
Phase
Constituent
Speed
(°/h)
Ampl.
(ft)
2Q* (almost 2 0 ^ )
12.854
0.025
0.0097
2.58
268.7
0.090
1.23
1.20
311.4
311.2
Coef.
(°)
13.399
13.472
0.017
0.0730
0.0142
13.943
0.398
0.3771
1.06
323.2
14.497
0.013
0.0209
0.62
352.6
P,
St (radiational only)
14.959
0 . 1 2 0
0.1755
0 . 6 8
336.8
15.000
0.013
-
-
91.5
Ki
Ji
0 0 * * (same as 2KOt )
15.041
15.585
0.403
0.014
0.5305
0.0297
0.76
0.47
333.0
370.4
16.139
0.034
0.0163
2.09
325.3
Q,
Pi
o,
Mi
* Inferred 2Qj would be 0.010 ft, 313.4°
** Inferred OOj would be 0.017 ft, 342.8°
In re tro s p e c t, it is p o ssib le t h a t a fric tio n a l te rm could have slig h tly
im p ro v e d th e species 1 p re d ic tio n s fo r A tla n tic C ity as well. T able 6 is
a c o m p a ra b le ta b le fo r A tla n tic C ity. A lth o u g h n o t as p ro n o u n c ed , th e
a d m itta n c e s show n o n -sm o o th v a ria b ility a t th e e x tre m itie s of th e b a n d ­
w id th a n d th e re fo re su g g est a p o te n tia l im p ro v e m e n t if (I)i + 1 - i w as ad d e d
to th e species 1 re fe re n c e series. H ow ever, w ith a p re d o m in a n tly sem i­
d iu r n a l re g im e one w o u ld expect in te ra c tio n s involving th e species 2 tid e
to be th e m o re im p o rta n t, an d we do n o t co n sid er th is case w a rra n ts
fu r th e r com p lication.
Table 6
A tlantic City tidal adm ittan ces ( i cpd)
H a rm o n ic C o n stan ts from NOS A nalysis — 3 y ea rs — 1952 to 1954
Admittances
Constituent
2
Q* (almost
2 0
K j)
Q.
pi
°i
Speed
(°/h)
Ampl.
(ft)
Coef.
Ampl./
Coef.
Phase
(°)
12.854
0.007
0.0097
0.72
141.0
13.399
0.041
0.0730
0.56
95.0
13.472
0.009
0.0142
0.63
38.5
13.943
0.245
0.3771
0.65
90.9
14.497
0 . 0 1 0
0.0209
0.48
129.9
101.1
Pi
Sj (radiational only)
14.959
0.109
0.1755
0.62
15.000
0.035
-
-
K>
15.041
0.363
0.5305
0 .68
15.585
0.016
0.0297
0.54
92.3
16.139
0.015
0.0163
0.92
„ 7 .»
J,
OOf* (same as 2 KOj )
37.7
107.0
* Inferred 2Qj would be 0.006 ft, 74.6°
** Inferred 0 0 x would be 0.011 ft, 123.2°
H a rm o n ic ¢1 is th e tra d itio n a l NOS p ro c ed u re , u sin g a re je ctio n
lim it of 0.03 for analyzed values. H arm o n ic ¢2 averages all 31 c o n s titu e n ts
fo r 3 y e a rs an d uses th e m ean values fo r p re d ic tio n s. O nce ag a in , even
m o re obviously th is tim e, th e evidence suggests th a t th e tra d itio n a l re ­
je c tio n lim it decreases th e a c cu ra cy of fu tu re p re d ic tio n s. A lthough
h a rm o n ic an a ly sis does not im p licitly re q u ire a sm ooth a d m itta n c e w ith in
a species b a n d w id th , tab le 5 in d icates o th e r p ro b lem s in a h a rm o n ic p re ­
d ictio n . A t th e h igh freq u en c y end of th e 1 cpd ra n g e, O O j h a s exactly
th e sam e freq u en c y as 2KOx (the n o n lin e a r 2 K j— O j). T h e a n a ly z ed OOj
is re ally th e v ector su m of th e tw o a n d th e re fo re no d e c o rre c tio n s desig n ­
ed fo r O O i o n ly w ill be so m ew hat in a c c u ra te . At th e low fre q u e n c y end
of th e species sp ectru m , th e an aly zed 2QX m u s t be c o n ta m in a te d by 2 0 K X
(th e n o n lin e a r 2 0 x — Kt ) w h ich is 1 cycle p e r 4¾ y e a rs aw ay in freq u en cy .
It seem s re aso n ab le th a t h a rm o n ic p re d ic tio n s can be im p ro v ed slig h tly by
s u b s titu tin g in fe rre d h a rm o n ic c o n sta n ts fo r th e an aly zed 2 Qx values, th u s
rem o v in g th e effect of 2 0 K! sid eb an d s; w h e n this w as trie d , th e re sid u a l
v aria n ce fo r 1 cpd w as g re a te r th a n th a t sh o w n in ta b le 4 fo r H a rm o n ic
¢2. A n o th er test, e stim a tin g h a rm o n ic c o n s ta n ts fo r 2 0 K t fro m co m p u ted
sid eb an d s a t th e 2Qt freq u en c y for th re e consecutive y e a rs, also failed to
im prove th e tab le 4 re sid u a ls.
CONCLUSIONS
R esp o n se tid a l an a ly sis an d p re d ic tio n have once m ore been fo u n d
to p ro d u c e m ore a c c u ra te re s u lts th a n classical h a rm o n ic pro ced u res.
H ow ever, th e differences betw een the re s u lts are sm all co m p ared to th e
to ta l u n p re d ic ta b le v aria n ce w h ich is c o n c e n tra te d p rim a rily in freq u en cies
less th a n 1 cpd.
H a rm o n ic p re d ic tio n s w ere im p ro v ed by d e p a rtin g fro m u s u a l N a tio n al
O cean S u rv ey p ro c e d u re s; th e la tte r in clu d e re je c tin g th e h a rm o n ic co n ­
s ta n ts fo r th o se c o n s titu e n ts w hose an a ly z ed am p litu d e is less th a n 0.03
foot an d a p ra c tic e of in fe rrin g T 2 fro m S 2 re g ard less of th e analyzed T 2
a m p litu d e .
A n eed w as fo u n d fo r in c lu d in g th ird -o rd e r n o n lin e a r in te ra c tio n s of
d iu rn a l tid e s in re sp o n se p re d ic tio n s fo r som e statio n s.
MISCELLANEOUS RESULTS OF INTEREST
T h e re sid u a l b y b o th m eth o d s show ed an a n n u a l m o d u latio n of M 2
of a m p litu d e a b o u t 1 cm , as u su a l p re d o m in a n tly a t th e low er sid eb an d
fre q u e n c y (2 0-1). S u ch te rm s have p rev io u sly been identified, especially
a t p o rts in th e N o rth Sea, by C o r k a n (1934), C a r t w r i g h t (1968), a n d P u g h
a n d V a s s i e (1976). T h ey can be acco m m o d ated in a resp o n se p re d ic tio n
by (I)2“ ° as m e n tio n e d in re fere n ce to tab le 1 .
B o th an a ly sis p ro c ed u re s identified significant lines w ith am p litu d e s
of a b o u t 0.5 cm fo r S 3 an d S 5 at A tlan tic City, a r a th e r u n u s u a l situ a tio n ;
S4 a n d S 6 a re m u c h sm aller. Sj/K ^ at A tla n tic City is 0.096 w h e reas a t
P e n sa c o la th e ra tio is 0.032, th u s in d ic a tin g a re lativ ely la rg e r-th a n -n o rm a l
Sx. It h a s been su g g ested th a t th e S3 a n d S 5 am p litu d e s m ay be due to a
th e rm a l re sp o n se of th e tid e gauge to su n lig h t on its ho u sin g . E n v iro n ­
m e n ta l co n d itio n s a p p e a r to be a m ore p lau sib le re aso n . T he tide gauge
at A tla n tic C ity is lo cated a t th e fa r end of a p ier ex ten d in g in to th e ocean.
O n o ccasio n , th e th e rm o g ra p h a d ja c e n t to the tid e gauge h a s show n a
s u d d e n rise in te m p e ra tu re o f about 10 °F w ith in a n h o u r. An in v estig a­
tio n sh o w ed th e re is a v ery shallow in let ju s t n o rth of A tlan tic City a n d
t h a t th e su d d en change in te m p e ra tu re w as re la te d to th e tid a l c u r re n t
tr a n s p o r tin g th e h e a te d lagoon w a te r p a s t th e tide gauge a n d th e rm o g ra p h ;
th u s a d iu rn a l so lar fre q u e n c y in te ra c tin g w ith a species 2 tid a l c u r re n t
reg im e m a y ex p lain th e a n o m alo u s Ss a n d S 5 a m p litu d e s. It is open to
q u e stio n , how ever, w h e th e r an yone w ould w ish to in clu d e such sm all
a n d h ig h ly localized effects in a tid a l p red ictio n .
ACKNOWLEDGEMENTS
T h e a u th o rs w ish to ex p ress th e ir th a n k s to J. M . V a s s i e a t IOS,
B. M a a t IG P P , an d S. E. M c C o y and R. J. M i l l e r a t NOS fo r th e ir a ss ist­
an c e in th is stu d y . T he re se a rc h a t IG P P w as su p p o rte d by N atio n al
O cean S u rv e y an d th e Office of N aval R esearch .
REFERENCES
C a r t w r i g h t D.E. (1968) : A unified analysis of tides and surges round North
and East Britain, Phil. Trans, of the Royal Society of London, Series A,
No. 1134, Vol. 263, pp. 1-55.
C a r t w r i g h t D.E. & R o s s i t e r J.R. (1972) : A com parison of m odern tide predic­
tions for Southend Pier. National Institute of Oceanography Internal Report
No. A. 55.
C o r k a n R.H. (1934) : An annual perturbation in the range of tide. Proc. Royal
Society, A, 144, pp. 537-559.
M u n k W.H. & C a r t w r i g h t D.E. (1966) : Tidal spectroscopy and prediction. Phil.
Trans, of the Royal Society of London, Series A, No. 1105, Vol. 259,
pp. 533-581.
P u g h D.T. & V a s s ie J.M. (1976) : Tide and surge propagation offshore in the
Dowsing Region of the North Sea. Deutsche Hydrog. Zeitschr., 29, 5,
pp. 163-213.
UNESCO (1975) : An intercom parison of open sea tidal pressure sensors. Report
of SCOR W orking Group 27 : “Tides of the Open Sea”, Unesco Technical
Papers in Marine Science No. 21.
Z e t l e r B.D. & L e n n o n G.W. (1967) : Some com parative tests of tidal analytical
processes. Intern. Hydrog. Review, Vol. XLIV, No. 1, pp. 139-147.
Z e t l e r B.D. & M u n k W.H. (1975) : The optimum wiggliness of tidal adm ittances.
Journal of Marine Research, Supplement, 33, pp. 1-13.
*
**
Inform ation has been received that the National Ocean Survey has accepted
the values for T 2 from analyses of series of one year. They are now considering
adoption of the rem aining recom m endations. (Editor’s note).