RELATIVE ACCURACY OF SOME METHODS
OF HARMONIC ANALYSIS OF TIDES
by Vice A dm iral A. Dos S a n t o s
B razilian Navy.
F
ran co
,
1. — Introduction
In this study we seek to establish the relative accuracy of some methods
of harm onic tidal analysis, involving observations of a m onth’s duration.
It is obvious that the developments w hich we shall make w ill be of an
entirely general character; however we shall here lim it ourselves to shortperiod analyses.
If hourly heights of a tide curve are analysed by the least squares
method, the problem of determ ining the accuracy of the results is a
standard one whose solution is more than 100 years old. In fact it w ill
be necessary to compute first the residuals v w hich w ill be the differences
between the actual tide curve and the curve predicted w ith the help of
harm onic constants derived from the analysis, so as to be able then to find
the m ean error of u n it weight, given by the form ula :
m v = V [»t>] / (N — 2Q)
In this fo rm u la N is the num ber of observations and Q
constituents under consideration.
(la )
that of the
The errors in the unknow ns R cos r and R sin r w ill be given by the
general relations :
mi = m a V Laa],
where [aa],
m2 = m„ VTPPI
........
(lb)
etc. are the weight numbers.
If for the various methods under consideration we can determ ine m
in terms of m v, the relative accuracy of these methods can be inferred.
2. — Mean errors in phase and amplitude
A lthough it
the m ean errors
the m ean errors
now know that
is
in
in
R
possible to compare the methods by merely com paring
R cos r and R sin r, it appears interesting to express
R and r in terms of those in R cos r and R sin r. W e
and r are computed from expressions :
R = y / (R cos r ) 2 + (R sin r ) 2
(2a)
r = ta n -1 (R sin r) / (R cos r)
(2b)
R cos r = a
(2c)
R sin r = b
(2d)
R = V a'- + b2
r — ta n - 1 b / a
(2e)
(2f)
and
•where, if we put :
and
we obtain :
In order to apply the general form ula for propagation of errors to these
expressions, the partial derivatives <9R/da, <3R/db, d r/d a and d r/d b m ust
firstly be found. From (2e) and (2f) we then have :
d H /d a
dR /db
d r/d a
d r /d b
=
a /R
—
fc/R
= — a/R2
=
b /R 2
The mean errors in R cos r and R sin r being designated by m a and m b,
respectively, through the application of the general form ula for the
propagation of errors, we obtain :
m R — — -v/ a2m 2 -4- b2m 2
R
R
v
6
and
m r = — a / a2m 2 + b2m 2
r
R 2
«
6
Now, in the classical methods, we generally have m a « m b for unknow ns a
and b corresponding to the same constituent. If we then put m a — m b = m,
these last expressions w ill take the form :
mR — m
m r = m /R
w hich shows that while the am plitude is determined w ith the same
accuracy as the unknow s R cos r and R sin r, the phase accuracy is
inversely proportional to the constituent’s am plitude. This is the reason
it is always difficult to determine the sm all constituents w ith accuracy.
3. — Determination of m for the Doodson methods
To express m in terms of m y, it w ill be necessary to study the
propagation of the errors of the hourly heights through the daily and
m onthly processes.
W e know that the daily process gives us functions X„ and Y„ . If we
denote these functions in a general w ay by F„, the daily process can be
w ritten in the form :
F . = E D (gt
(3a)
The propagation form ula applied to this expression gives us :
= I D7m'l
(3b)
t
D f being the hourly m ultipliers in the daily process, I D7 can be computed
t
by using the well-known tables of these m ultipliers f1'.
Institute’s m ethod of analysis, we find :
Table
In
the
Tidal
3-1
X0
X,. Y ,
X ,, Y2
X3
X 4, Y 4
X6
I D? 42
52
84
54
28
92
t
For the so-called British A dm iralty method, duly developed by Doodson
to give the same constituents as the T idal Institute method, we see,
according to table IA (2>, that using form ula (3b), for all the functions
except X 4 we shall have :
m |n = 24 m l
(3c)
and for X 4
m x4 = 16 m l
(3d)
In the two methods under discussion, Doodson did not fin d it
necessary to employ functions Y„ for n — 0, 3 and 6. Thus, for these
species, the daily process im m ediately supplies the know n terms of the
set of equations in R cos r and R sin r. These know n terms m ay then be
represented by
:
X np = 1
Xn
(3e)
d
and
X np. = Z D ^'X „
(3 f)
<1
the first expression representing the know n terms of the set of equations
in R cos r a nd the second in the R sin r equations. In these form ulas D 'd
represents the combinations designated by a num erical symbol (here
expressed by p ), and D £ the com binations designated by a literal symbol
called p' having the same num erical order in the alphabet as p.
As for d iurnal, semi-diurnal and fourth-diurnal tides, the know n terms
in the equations are all of the general form :
(Lnp or L np.) = ± X np ± Y„p. ± X np. ± Y„p
for the T idal Institute method, and :
= X np ± Ynp.
(3g)
(3h)
and
— X np*
^ np
(3i)
for the A d m iralty method.
As w ill be seen, the signs have no im portance as far as the propaga­
tion of the X„p, X np. , Y np and Y np. errors on L is concerned.
Therefore
relation (3g) m ay be expressed in the form :
L = E (± D i ± D i') X„ + E (± DJ ± DJT) Y„
d
d
(1) I.H.R., Vol. XXXI, No. 1, M a y 1954, pag e 68.
(2) Ib id ., p a g e 84.
(3j)
D£ and D d always corresponding to combinations p and p' of the same
order. By next applying the general form ula for the propagation of
errors to (3e), (3f) and (3j), we respectively obtain :
mh P = 2 D?
(3k)
d
m £*-np' = L D M
d
(31)
and
= m lnp, = 2 ( D ? ± 2 D a'D ^ ' + D ''* ,
+ 2 CD? ± 2D£ D„"4- D D
However, in tables II <3> and IIA <3>, we see that colum n vectors B'd and
are all orthogonal, giving us 2 D d D d = 0 and the last expression then
becomes *
^
m i. =
= 2 (D22 + ~Dd2) (m|n + m ^n)
(3m)
d
This expression w ill also be obtained from (3h) and (3i).
W e can now draw up a table of the values of 2 B'dz, 2 D ^'2 and
2 (D ^2
D P ) for the two methods of analysis.
d
d
d
T
able
3-II(a)
Tidal Institute Method
P
2 DJi
p'
2 D 'd'2
I (T>d2 + W )
d
0
29
1
58
a
62
2
70
b
64
3
66
c
62
4
70
d
60
5
56
e
66
6
62
f
60
7
58
§
62
29
120
134
128
130
122
122
120
4
3
always 29
5
6
7
T
able
3-II(b)
British A dm iralty Method
P
2 D5P
d
P'
2 Dd2
d
2 (Df» + D i'2)
d
0
29
1
2
a
28
b
24
c
28
d
24
e
28
f
28
g
28
57
53
57
53
57
57
57
In the Tidal Institute method we always have the same values of
2 D? (table 3-1) for X„ and for Y„ (n being the same for both functions).
In these conditions, from (3b), (3k), (31) and (3m), we may write :
mi
= 2 Dd
' 2 ■2 D ?/n2
(3n)
n,,
t
t
m l , = 2 Di'2- 2 D ?m 2
(3o)
d
t
m l np = m l np/ = 2 2 (D ? + D ’P ) 2 D? m j
d
*
(3) Ib id ., pages 69 a n d 85.
(3p)
W ith the exception of L 4p and L 4p. , these same expressions can be used
in the case of the B ritish A dm iralty method. As expressions (3c) and (3d)
show, for L 4p and L 4p. , expression (3 j) in this case gives :
mL P=
= 40 '
<W2 +
(3q)
d
It is now quite easy to find the num erical coefficient of m 2 for all the
know n terms. For the T idal Institute method it is only necessary to
m u ltip ly suitably the values contained in tables 3-1 and 3-II(a). In order
to find the coefficient corresponding to a function X np, the value of £ D?
t
corresponding to X„ m ust be looked u p in table 3-1 and the value of E D ^2
d
corresponding to p in table 3-II(a). The product of these two quantities
w ill be the num erical coefficient of mg in expression (3n). To find the
num erical coefficient of
in expression (3o) it w ill suffice to look up the
value of £ D a2 corresponding to p ’ in table 3-II(a) and to m u ltip ly it by
i
the same value of £ D? already found for X n . For functions L np and L np. ,
t
the value of £ (D<P + D^'2) w ill be fo und in either the p or p' colum n
d
of table 3-II(a) and should be m ultiplie d by twice the value of E D?
t
corresponding to X„ and Y n in table 3-1. This w ill give the result of the
com putation of expression (3p). A ll the vector colum ns of table 3-III(a)
have been computed according to the instructions ju s t set forth.
In the case of the B ritish A dm iralty method, it is m uch easier to find
the num erical coefficients in expressions (3n) to (3p). In fact, w ith the
exception of the fourth-diurnal constituents for w hich expression (3q) m ust
be applied, we shall always have £ Df = 24, w hich allows us to confine
f
our search to table 3-II(b) where the values of E D^2, E D^/2and E (D^--f- D£'2)
i
d
d
according to the second subscripts p and p' w ill be found. The vector
colum ns of table 3-III(b) have been computed from these instructions,
noting that for the fourth-diurnal constituents these values m ust be
m ultip lie d by 40. Here there is only one vector colum n for the R cos r
equations and another for R sin r since all the unknow ns are derived from
equations containing all six species of the tide under consideration.
For the application of the T idal Institute method, on exam ining
table IV <4) we shall see that it consists of the inverse matrices required
for the com putation of R cos r and R sin r. From the w ay in w hich the
values are set out, the unknow ns in each set of equations, designated by I,
are expressed as a function of the know n terms, designated by L, in the
general form ula :
I = EaL
(3r)
According to the previously employed propagation form ula, we shall have :
m 2 = E a 2 mg
(3s)
The values of E a 2 have already been computed for the two methods in
order to apply form ula (3s), taking the values of mg from tables 3-III(a)
(4) Ibid., p a g e 71.
and 3-III(b). Table 5-1 already contains the values of m for all unknowns,
for both the semi-graphic and the least squares method.
T
able
3-III
Coefficients of mg
(b) British A dm iralty method
(a) Tidal Institute m ethod
f
I
1218 -j
X 01 2436 y
X02 2940 J
r
Xoo
3016
D io
12480
12 1 3 9 3 6
r> n
D i2
13312
D l3
C 10
C lC
f X 0a 2604 \
\ X 0b 2688 J
C l3
C 14 1 3 5 2 0
{
14
12480
Cla
D lb
13936
Q lb
c 20
4872
D>>.o
C 21 2 0 1 6 0
c 22 2 2 5 1 2
D 2i
C 23 2 1 5 0 4
D 22
D 2s
C 24 2 1 8 4 0
D 24
D 2b 2 2 5 1 2
C 2b
C 12 1 3 9 3 6
D 12
X32 3780
X 33 35 64
X 34 3780
C42
C43
C44
C45
X 62
X64
X65
X 88
X67
d
D la
6440
6440
5152
5704
53 3 6
{
75 04
7168
72 80
68 32
X 3b 3466
X 3c 3348
X 3d 3240
D 42
D 43
D 44
D 45
X 6b
X 6d
X 6e
X 6(
X 6g
5888
5520
6072
5520
5704
x 00 696 'i
x 01 696
X<,2 696
Cio 1392
C n 2736
^12 2544
Cjg 2736
C14 2544
Dla 2736
Dlb 2544
C20 1392
c 2i 2736
C22 2544
^23
C24
D 2b
X 32
X 33
X 34
C42
C43
C44
G45
X 62
2736
2544
2544
696
696
696
2120
2280
2120
2280
696
x 04 696
696
696
* x 67 696 ^
X(ja
X 0b
Dio
Du
D 12
D 13
d 14
Cla
Clb
D 2o
D2i
D 22
D 23
d 24
*■
C2b
X 3b
X 3c
x ,d
D 4b
D 4e
D 4a
D4e
x 6b
Xea
x 6e
X«5
X 08
L
X 61
x «,
672
576
1392
2736
2544
2736
2544
2736
2544
1392
2736
2544
2736
2544
2544
576
672
576
2120
2280
2120
2280
576
576
672
672
672
4. — Determination of m in the semi-graphic method
W e know that in the semi-graphic method large disturbances arising
from meteorological phenom ena m ay be reduced by means of smoothing
the contours w hich represent the tidal surface. In these conditions we
m ay assume that the analysed ordinates are only affected by random
smoothing errors.
To be able to study the propagation of errors in ordinates for the tidal
surface in known terms of the equations to be solved, we shall follow the
steps set forth in m y article published in the International Hydrographic
Review (Vol. XL, No. 2, Ju ly 1963). Once the ordinates of the tidal surface
have been obtained they w ill be combined according to the expressions :
X. = Z D
t
and
(4a)
X n. = I D f y t
t
where D " represents the m ultipliers in table 4-II 151 w hich consist of the
colum n vectors indicated by the figures n = 1,2, 3, 4, 6, and D f represents
the m ultipliers in the same table whose vector columns are indicated by
the letters n ' = a, b, c, e, f. To isolate one species, the subscripts of
functions X n and X n. are of the same order, i.e. the num erical order in
the alphabet of letter n ' is equal lo n. The two equations (4a) are sufficient
for the isolation of the third-diurnal and sixth-diurnal species, but for the
other species form 4-A (6> shows that it is necessary to find the functions :
U x = X , + X 3/3
U 3 = X„ — X c/ 3
U2 = X 2 + X e/3
U b = X fr — X//3
Ud = 1.15 X d
From the general relations (4a) we m ay write the above expressions as :
(4b)
Ui = Z ( D > + D ? / 3 ) yt
t
(4c)
u tt = I (D t°— D£ / 3) g t
I
u 2 = Z ( D ? + D ? / 3 ) yt
(4d)
t
U b = S (D2 — D / / 3 ) yt
t
Ud = 1.15 I D f y t
*
Then expressions (4a) to (4f) are all of the form :
(4e)
(4f)
F„ = 1 8 t yt
(4g)
t
to w hich we can apply the general form ula for propagation of errors.
W e can therefore w rite :
(4h)
*
The values of 8t for expressions (4a), w hich should be applied in the case
of the third- and sixth-diurnal constituents, are respectively m ultipliers
(n = 3, n' = c) and (n = 6, n ' = f ) from table 4-II <7>. The sum of the
(5) I.H .R ., V o l. X L , N o. 2, J u l y 1963, pag e 80.
(6) Ibid., page 83.
(7) Ibid.. page 80.
squares of these m ultipliers gives the values of 2 St for X 3, X c, X 6 and X ,
as seen in table 4-1. As for the value
T able
2 8*2
t
X0
24
u t, u a
20
u 2, u b
18.67
4-1
x 8, x e
18
x4
24
ua
21.12
x 6, x f
12
of 2 8? for U4, expression (4f) shows that it w ill be sufficient to m ultiply
t
the sum of the squares of the m ultipliers in colum n (d) table 4-1, by the
square of 1.15. The result is entered in table 4-1 under Ud.
To find m ultipliers 2 8? corresponding to equations (4b) to (4e) the
t
values of Dp and Df m ust be taken from the same line in table 4-II (7)
and then combined according to the expressions of the coefficients of yt
in (4b) to (4e). W e thus obtain a series of m ultipliers 8t w hich w ill
directly give U u U„, U 2 and U 6. The sums of the squares of the m ultipliers
so obtained are those to be seen in table 4-1 for the same functions.
E xam ining expressions (5f) (8) and (5g) (9), we find that all the know n
terms of the sets of equations are obtained by the sum or the difference
of pairs of values of F„ : F„ being any one of functions Ui, Uffi, U2, U 6> X 3,
X c, etc. The m ean errors of the know n terms are then equal to those
of F* m ultiplied by
Let
5-III <10)
form ula
gives an
w hich, according to (4h), can be w ritten :
m l = 2 2 8? m|
(4i)
t
us now investigate the propagation of m Xl on the unknow ns. Table
shows that all the unknow ns are com puted from the general
(3r). Therefore the propagation form ula applied to this expression
expression sim ilar to (3s), i.e. according to (4i) :
= 2 E g? S a
(4j)
t
Table 4-1 shows that all the values of 2 5? are equal w hen the num ber n
t
is equal to the num erical order of the letter n ' in the alphabet except for
n = 4 and n ' — d. Though even in this case the mean of the values 24
and 21.12, equal to 22.56, can be taken as an adequate approxim ation.
Thus for all the constituents, w ith the exception of the fourth-diurnal, we
can find the num erical coefficients of
in expression (4j ) by m ultiplying
the double of the values of 2 §? in table 4-1 by the values of 2 a 2 computed
t
from the values of a for each constituent, given in table 5-III '10). In the
case of the fourth-diurnal constituents, we write 2 2 8? = 45.12. The square
t
roots of the values thus found are entered in the S.G. columns of table 5-1.
T
able
5-1
Mean errors in the unknow ns in units of 10~s m v
L.S.
So
...........
...........
M S ,...........
Q i ...........
Ox ..........
Mi ...........
Ka ..........
Ji
...........
O O i ..........
|*2 ...........
N2 ..........
Mo ..........
1-2 ...........
S2 ...........
2SM2 ____
m o3 .. . .
m3
MK,
....
MN4 ____
m4
sn4
____
m s4 ____
2MNe ___
m6
m s n 6 ___
2MS8 ____
2SMe ____
27
1ft
R cos r
T.I.
B.A.
S.G.
40
57
57
68
72
73
64
79
92
73
73
72
73
71
75
62
63
66
67
69
67
66
81
78
76
75
73
66
75
79
104
104
104
104
104
104
108
106
104
106
107
101
104
104
104
106
107
107
107
103
104
104
104
104
38
59
62
94
91
95
85
102
105
95
93
92
91
84
99
95
94
104
90
90
87
86
97
96
91
91
95
L.S.
R sin r
T.I.
B.A.
S.G.
60
59
70
70
72
83
80
88
74
72
72
72
71
74
63
63
63
68
67
66
65
80
81
79
76
72
72
68
104
104
104
104
104
104
108
106
104
106
107
101
104
104
104
106
107
107
107
103
104
104
104
104
62
55
94
90
95
85
102
106
95
93
93
91
79
98
93
91
87
90
89
87
87
93
90
88
86
85
5. — Conclusion
To draw the final conclusions, the sets of norm al equations for the
determination of R cos r and R sin r were established according to the
Horn method <11», taking into account 28 constituents. The matrices of
these systems both have very strong diagonal elements in comparison
w ith others. Under these conditions we are of the opinion that for
an accuracy study it w ill be sufficient to consider the inverse matrices as
diagonal matrices whose values are the inverse of those of the equations’
diagonals. Since the values of the data of the equations’ diagonals are
all close to 347.5, the inverse matrices w ill have the terms of their diagonals
close to 1/347.5. W e know, however, that the weight numbers [aa], [J3J3],
(11) I.H .R ., V o l. X X X V II , N o. 2, J u l y 1960, pp. 65-84.
[yy],
etc. are exactly the elements of the diagonals of the inverse
matrices in the sets of normal equations, which allows us to write :
[aa] ~
[pp] -
[yy] -
..... «
0 .00 287 8
This being so expression (lb) w ill give :
m
=
m,
m
=
0.054 niy
V 0.002878
or
for all the unknow ns.
By comparing the values entered in table 5-1, we note that the Tidal
Institute method is the most accurate of the three and that there is little
difference between the British A dm iralty and semi-graphic methods.
The random errors are already reduced before the analysis since,
generally speaking, all the analyses carried out by electronic computers
are preceded by a sm oothing of the curve and the observations themselves
are very accurately carried out. This being so, we feel that unless the
analysis has a strictly scientific purpose there w ill be no need to apply
the least squares method. In hydrographic offices not possessing all the
resources of more well-equipped organizations, perfectly valid results w ill
be obtained by applying the Tidal Institute method. It is interesting to
note that Mr. Lennon of the Tidal Institute has draw n up a programme
for using the IBM 1620 computer in the analysis of a one year’s series of
observations, w hich includes sm oothing the curve in less than fifteen
m inutes. This m ay be considered as a remarkable achievement.
A u th or's n ote. — T he m a j o r p a r t o f th is a r tic le ’s s u b je c t w as d isc ussed b y
D o o d s o n a n d L e n n o n a t th e 3rd In te r n a tio n a ] S y m p o s iu m on E a r t h T id e s, T rie ste, 195!).
T he a u t h o r w as u n a w a r e o f th is fa c t a t th e tim e th is a r tic le w as w r itte n .