Geophys. J. R. astr. Soc. (1980) 63, 467-478
On the use of the Proudman-Heaps tidal theorem
M. G. G. Foreman Institute ofocean Sciences, Patricia Bay, Sidney,
BC, Canada
L. M. Delves Department of Computational and Statistical Science,
University o f Liverpool, Liverpool
I. Barrodale Department of Computer Science, University of Victoria,
Victoria, BC, Canada
R. F. Henry Institute of Ocean Sciences, Patricia Bay, Sidney, BC, Canada
Received 1980 February 26; in original form 1979 December 31
Summary. The Proudman-Heaps tidal theorem provides, in principle, one
way of computing tides across the open boundary of a bay, given measurements round the shore; however, attempts to use the theorem numerically
have so far failed. We describe in this paper the results of a theoretical and
numerical investigation into the difficulties which are encountered. We show
first that:
(1) The mathematical model being solved is ill-posed; hence, M Zmethod,
~
and in particular a direct use of the tidal theorem, must expect numerical
difficulties.
( 2 ) The direct use of the tidal theorem yields the solution of a problem
close to that posed; thus, discrepancies in the solution reflect directly the
ill-posedness of the problem.
The ill-posedness can be countered by imposing suitable constraints,
which may be either mathematical (regularity conditions on the solution) or
physical (spot tidal measurements across the open mouth of the bay). We
investigate the extent to which using such constraints can improve the predictive power of the method; the results obtained are not encouraging.
1 Introduction
We consider in this paper the following problem: given a partially enclosed sea basin or bay
(see Fig. l), and tidal measurements around its shore; compute the tides across the open
mouth of the bay. This problem is discussed in Heaps (1969) and is formulated in terms of a
single harmonic constituent of speed u. A direct calculation would require the solution of
the tidal dynamical equations over the interior of the bay, and subject to boundary
468
M. G. G. Foreman et al.
Y
Figure 1. A typical bay with enclosing rectangle; notation as in Section 2.
conditions around the shore line. This problem is mathematically ill-posed: the boundary
conditions, along the shore line, are of initial-value type whde the defining equations are
elliptic. Any numerical method must either allow for the likely resulting instability, or
expect to encounter trouble.
An alternative approach was suggested by Proudman (1925) and extended to include
linear friction by Heaps (1969). In this approach a set of auxiliary equations is introduced,
together with a sequence of functions satisfying these equations with initial value boundary
conditions along the open boundary of the bay. Inserting these functions into an integral
relation yields eventually an expansion for the tidal heights and phases along the open
boundary. The details are repeated for convenience in Section 2.
We show below that this sequence of auxiliary problems is also ill-posed. Thus, to use the
Proudman-Heaps theorem as the basis for a numerical procedure replaces one ill-posed
problem by a sequence of ill-posed problems. However, there are practical advantages gained
in return:
(a) Each problem in the sequence involves the solution of a partial differential equation
only over a simple region (a rectangle) rather than over an irregular region (the bay).
(b) The auxiliary functions all satisfy the same differential equation, but with different
boundary conditions. As a result, N such functions can be produced at a cost much less
than N times the cost of the first, and the method remains fairly economical overall.
(c) Because the shape of the bay, and the measured tidal heights around the bay shore,
enter only at a late stage in the calculation, the effects of changes in these parameters can
be calculated relatively cheaply.
Despite these advantages practical experience with the Proudman theorem has not been
encouraging. The calculations of Fairbairn (1 954) yielded reasonable results only after he
had made ad hoc adjustments to the input data, while Sgibneva (1964) reported severe
numerical instability w h c h could not be controlled. In this paper we analyse the source of
these difficulties and discuss and test methods which might be used to alleviate or remove
t h e instabilities. The results are not encouraging.
The Proudman-Heaps tidal theorem
469
In the next section we establish the notation used, recall the Heaps form of the Proudman
theorem for a variable depth bay with linear friction; and show explicitly that the sequence
of subproblems introduced by the theorem is ill-posed. Section 3 then describes calculations
for a simple model basin, using a Galerkin method for solving the auxiliary equations. We
show that:
(i) The results obtained bear little relation to the (known) solution across the basin
mouth.
(ii) Nonetheless, there exists a neighbouring problem; that is, one with tides round the
shore very close to those assumed; which when solved with the Proudman theorem yields
the correct solution across the basin mouth. The gross errors obtained with a direct solution
therefore reflect only the ill-posed nature of the problem, and will inevitably be present with
any method of solution.
We can measure the degree of ill-conditioning of the problem by the sensitivity ratio (change
in tides across the mouth)/(change in assumed tides round the coast). The results given show
that (at least for the particular bay considered) the ratio is so large that changes in coastal
tides well within either experimental error or confidence in the model used, will produce
order of magnitude changes in the predicted bay mouth tides. Thus, the problem as stated is
numerically not solvable. We therefore consider two additional types of constraints w h c h
can be added to the problem formulation tc make it less ill-posed. The first is essentially
mathematical: we assume that the solution is smooth, in a sense defined below, and impose
additional equations which ensure this numerically. With such constraints, we no longer
obtain ridiculous answers; but the overall information content of the results is not
improved, The second constraint adds further physical information. We assume that one or
a few spot tidal measurements are available across the mouth of the bay, and include these
in the calculation. Numerical results show that, while adding sufficient such measurements
will indeed stabilize the results, the number required is too great to make this approach
significantly better than interpolation between the measurements.
The Proudman-Heaps theorem therefore appears in these calculations t o have only
neghgible predictive value. Left unanswered, however, is the question of the extent to which
our results are dependent on the shape of the bay.
2 The Proudman-Heaps tidal theorem
The dynamical equations for a tidal constituent of speed u are
where the notation is as in Heaps (1969); u and u are the components of mean-depth current
in the x and y directions; h(x, y ) is the sea depth; f is the tidal elevation; {' = 5 - 5, with
the equilibrium tide;f, u, g are constants defined in Heaps (1969); and F, G are frictional
terms which we here assume are linear in hu, hu:
F = Xhu,
G = Xhu,
X constant.
(2.2)
M. G. G. Foreman et al.
470
Defining a normal derivative v:
v = u cos (Y - u sin a
(2.3)
where a is the angle shown in Fig. 1, the physical boundary conditions associated with
Fig. 1, are
’ given on OBA
v = 0 on OBA.
Now let Z, U, V be auxiliary functions satisfying the equations
az
putfv=-g-,
ax
pv-fu=-g-,
az
(2.5)
aY
a
a
(hU) + - (hU) t ioZ = 0,
ax
aY
-
where
p = h t io.
Then, setting
N = U c o s a - Vsinol
and imposing the boundary conditions
Z ( x , 0) = 0,
h(x, O)N(x, 0 ) = k(x),
we find that the following identity holds:
ljt(x,
O)k(x)dx = io
J/oBAa
dx dY
-
(2.10)
Equation (2.10) is the Proudman-Heaps theorem. If Z, N are known, then the right side
may be computed to yield one piece of information on c’(x, 0), the tide along the open bay
boundary. In practice, we choose a sequence of boundary conditions
k(x)=k,(x),
n = 1,2, . . .
with corresponding solutions Z,. With k,(x) chosen so that the left sides of equation (2.10)
for n = 1 , 2 , . .. correspond to expansion coefficients of c’(x, 0) in some orthogonal basis,
f’(x, 0) can then be reconstructed from a knowledge of these coefficients. For a constant
depth sea, the choice
k,(x) = cos (nnxlb)
leads to the analytic solution for Z , (see Heaps 1969):
2, = A , sin 01, y ) cos (nnxlb)
(2.1 1)
where
pi = - n2.rr2/b2
- ia@ t f 2 ) / g h p
(2.12a)
The Proudman-Heaps tidal theorem
471
and
(2.12b)
An = (1-1’ +f2)/gI-11-1n.
Then equation (2.1 0) produces from the sequence (Z,) the half-range Fourier cosine
expansion of {’(x, 0).
Equation (2.11) demonstrates that the calculation of Z, is an ill-posed problem. To see
this, we note that equation (2.1 1) is a solution of the problem for any n , not necessarily
integer, and for fixed x and y is continuous in n. For large y , 2, increases exponentially:
Z,
- exp (nnylb),
y large
and hence, considering a ‘neighbouring’ solution n + n + E , a small change in the boundary
conditions leads to an arbitrarily large change in the solution of the problem.
3 Numerical calculations using the Proudman-Heaps theorem
We describe in this section and Section 4 some model calculations using the ProudmanHeaps theorem, which have the following aim:
(1) To assess how serious the ill-posedness is in practice.
(2) To test the effectiveness of methods for alleviating the ill-posedness.
Equation (2.5) with boundary conditions (2.8), (2.9) determine the functions U,, V,, Z ,
in the strip 0 Q x Q b, y > 0 . We assume that the bay lies wholly in the rectangle R: 0 G
x Q b, 0 Q y G y b , and introduce the scaled coordinates
t = 2nx/b,
71 = 2ny/yh.
(3.1)
Then eliminating U, V from equations (2.5), (2.8) and (2.9), we find that Z, satisfies the
equations
a2z,
LZ,=p+.--+--+
at2
-
a2z,
Y2 ar12
iab2cf2+ p2)
47%
[
ah f a h l a-+z ,
I-1-+--
at
Y 317
at
[
f ah
I-1 ah]
Y at
Y2 arl
---+--
az,
__
arl
z,=o
To relate the calculations as closely as possible to those made by previous authors, we
choose
=- b
2n
/2r{’(”,
0
2n
0) h ( % , 0) exp ( i n t ) d $
277
472
M. G. G. Foreman et al.
and we identify I, as a coefficient in the expansion
m
b<’(x,~ ) h ( x0,) =
1
I, exp (- int).
(3.7)
n=-w
Evaluation of I, via equation (2.10) requires that Z,(t, 1)) be known. Fairbairn (1954)
calculated 2, using a step-by-step method from the line 1) = 0; however, such an approach
makes it difficult to add side conditions of the type discussed below. Instead, we expand
Zn as a double Fourier series:
(3.8)
where Zo,(g, 1)) is a zeroth approximation which we take to be the exact solution for a basin
o f constant depth h:
where
pi = - y z [nzt iobZ(f2 t ~~))/4n~gl.lh].
(3.10)
We then replace equations (3.2)-(3.4) by the Galerkin equations
Jb’”j3
dr) exp [- i ( p t + q a ) l ~ z , ~7 ,) = 0 ,
p = - M, .. ., + M
(3.11)
4 = - N , . .. , + N
p = -M,.
. ., + M
p = - M , . . . , t M.
(3.12)
(3.13)
Equations (3.1 1)-(3.13) represent an overdetermined set of linear equations for the
coefficients Z?), of the form
AZ=b
(3.14)
where Z, b are P = ( 2 M t 1 ) ( 2 N t 1) and ( P t 4 M + 2 ) vectors respectively, and A is a (Pt
4 M t 2) x P matrix. The coefficient matrix can be related directly to the coefficients hzmin
a Fourier expansion of the sea depth:
(3.15)
and these in turn may be computed using Fast Fourier Transform techniques. However, we
restrict ourselves here to a sea of constant depth (2000 m) and consider a triangular bay
shown in Fig. 2. This consists of one corner of a rectangular ocean basin. The response of
the whole basin to a gravitational tidal force at the principal lunar semi-diurnal ( M z )
frequency was calculated using a linear finite difference model of the time- dependent
shallow water equations and the amplitude and phase of the periodic tidal elevation was
noted at t h e boundaries of the triangular portion of the basin discussed here. Using grid
values, linear extrapolation was used to obtain the 17 shoreline values shown in Table l(a)
The PI-oudman-Heaps tidal theorem
473
I
I
9
?
% Accuracy 3
3
2
I
I
I
I
I
Figure 2. The triangular bay used in numerical tests. Points marked
represent shoreline tidal
elevations from a numerical model used as input for the Proudman-Heaps theorem; tides evaluated using
the theorem were compared with the model results across the bay mouth indicated by circles e.g. 0.
Results of the comparison are shown as percentage error.
and linear interpolation for results along the basin mouth for comparison with those calculated using the Proudman-Heaps theorem. T h s procedure reflects reasonably well the
practical situation in which discrete tidal measurements of h i t e d accuracy are available
at points around the coast. We have estimated the accuracy of the model results by computing the ratio
where { = { ' + f is the calculated tide and
where k = udf2 - SL2)/ghSL and SL = u - 21.The ratio e gives a measure of the (percentage)
accuracy with which the tidal equation is satisfied by the finite difference model; calculated
Table 1. (a) 'Exact' coastal tidal data for the model of Fig. 2.
Point
Amp.A (m)
Phase I$(deg)
Point
Amp. A (m)
Phase @ (deg)
0.0 1059
0.02922
0.04484
0.05 87 3
0.07103
0.08154
0.08990
0.09557
0.09974
64.09
62.48
51.07
41.73
33.26
25.58
18.91
13.84
9.76
10
11
12
13
14
15
16
17
0.09663
0.09281
0.08832
0.08475
0.08385
0.08702
0.09483
0.10692
4.13
- 3.52
- 14.95
-29.63
-46.75
- 64.97
- 80.82
-99.22
474
M. G. G. Foreman et al.
Table 1. @) ‘Exact’ bay-mouth tides for the model of Fig. 2. The ‘Fourier’ values are those obtained
b y summing the truncated Fourier series (3.7) with the exact In values of Table 2.
Point
Fraction of
bay width
from coast
(m)
0.0625
0.1875
0.3125
0.4375
0.5625
0.6875
0.8125
0.9375
0.0193
0.0278
0.0323
0.0339
0.0357
0.0434
0.0615
0.0902
Amp. and phase
Fourier sum
Fourier sum
In1 4 1
lnIG2
(deg)
53.07
32.56
18.64
1.99
- 21.64
- 50.04
- 74.68
-92.80
0.0215
0.0217
0.0386
0.0390
0.0393
0.0542
0.0625
0.0496
-66.36
27.98
36.25
16.12
-28.57
-63.81
-78.95
-83.48
0.0106
0.0372
0.0332
0.0327
0.0345
0.0427
0.0752
0.0670
-40.40
48.48
17.58
-12.03
-9.99
-46.52
-80.42
-89.86
Table 2. Exact and computed values of Z,, for model calculation 1.
Exact
Calculated
n
Re pt
Im pt
Re pt
-3
-2
-1
0
1
2
3
-0.503 X l o 8
-0.729
- 1.54
1.22
0.653
0.489
0.425
-0.05175 X l o 8
-0.00195
0.357
1.09
0.5775
0.196
0.112
-3.44 X 10’
0.0773
- 1.59
1.17
1.17
- 6.45
93.7
Im pt
48.0 X 10’
-2.77
0.563
1.10
0.466
4.46
- 107.0
-
values of e are shown in Fig. 2, and suggest that e 1-3 per cent. For the basin considered,
{ and { ‘ are of the same order of magnitude, so that the errors in {‘ are also expected to be
1-3 per cent. Numerical values round the coast and across the bay mouth are shown in
Table l(a) and (b); the ‘exact’ Fourier coefficients In implied by the bay mouth tides are
shown in Table 2 .
Both in the finite difference model and subsequent numerical calculations, the friction
coefficient h was set equal to 0.5 x
which is a typical value for coastal models.
Given as input the simulated tides around the coast, we carry out the following
calculations :
Calculation 1
We set M = 4, N = 8 and h = h = 2000 m in equation (3.8); then ZOn yields formally the
exact solution to this problem, and equations (3.1 1)-(3.13) should have the solution Z(”) =
0. Setting u p and solving these equations in the least squares sense, using the technique
described in Delves & Barrodale (1979), thls was found to be true numerically. The
computed coefficients, Z,,, therefore come wholly from Zen; they are shown in Table 2 ,
together with the ‘exact’ values derived from the simulation model. For n = 0, the agreement
is very good; the results for n = k 1 are recognizably similar; while for n = k 2, 3 there is no
correlation between the two sets of numbers at all, the calculated 1Z1, increasing rapidly with
n while the exact values decrease as expected.
For larger values of n(ln( 2 ) we cannot expect good agreement, since the accuracy of
the simulation results is not high enough to define ,
Z very well; we estimate that the real and
The Proudman-Heaps tidal theorem
475
imaginary parts of the ‘exact’ I, have an absolute accuracy of about f 1 x lo’, so that the
smaller values are not well determined at all. Nonetheless, the results for I , , , I,, are
obviously ridiculous.
These poor results cannot reflect the behaviour of the Calerkin scheme used to calculate
Z since Z = 0 in this calculation. Nor does it stem from errors in the numerical integrations
used to compute I, from Z,,,; we estimate the accuracy of the I,, given the input data, as
5 per cent at worst. Rather, it reflects the physical ill-conditioning of the problem being
solved: the ‘exact’ values round the shore are not compatible with the ‘exact’ I,, but there
exists a set of shore tides which differ from these prescribed by amounts smaller than
the errors in the simulation model, and which are compatible with the ‘exact’&.
*
Gdculation 2
To show this explicitly, we perform a sensitivity analysis as follows. The measured coastal
tides do not affect the computed Z,, and enter linearly into the calculation of 1, through the
line integral in equation (2.10).
Noting that in fact only a finite number Q of measured tidal values tl,.. . , (Qare used,
equation (2.10) may be written in the form
(3.15)
where the coefficients A,i can be readily calculated. We now allow the t j to vary, and seek
the smallest variation such that equation (3.15) is an identity when the ‘exact’ I, are
inserted.
Specifically, we seek a set of coastal tides 5‘ and pose the problem:
Choose Z, and 5’ to minimize
II g - g’ I1
(3.16)
subject to
Table 3 shows the percentage changes required to the coastal tide values given in Table 1(a),
calculated from solving equation (3.16) in the L2 and L 1 norms. The adjustments needed are
in general much less than 3 per cent, with a maximum change of about 4.5 per cent. These
changes are within the expected accuracy of the simulation model used, and indeed within
the accuracy expected for practical tidal measurements. We conclude that the Proudman
technique is in fact solving the given problem as well as can be expected, but that the
sensitivity of the bay mouth tides to small changes in the coastal tides is so great that experimental errors, and model uncertainties, will in practice make it impossible to derive
information on the bay mouth tides merely from a direct solution of the tidal equations.
4 Regularization techniques
It follows from these results that any method designed to predict the tides across an open
bay mouth must take special action to counter the ill-conditioning of the physical problem.
476
M. G. G. Foreman et al.
Table 3. Percentage changes in the coastal tides needed
to reproduce the 'exact' values of I,, n = - 2 , 1 , . . . , 2 .
Notation: 5' = A exp (i@), 5' = A ' exp (i@');
A ' ) X I O O / A ; W = ( @ - @ ' ) X loo/@.
L , fit
Point
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
6 A = (A -
L , fit
6A
60
6A
4.3
3.2
2.2
1.6
1.2
1.o
I .6
2.5
- 1.5
- 2.5
0.00
0.55
0.03
-0.59
-0.79
-0.64
-0.22
-0.18
-0.15
-0.00
0.09
0.09
- 0.09
-0.45
-0.87
-0.11
0.43
0.20
0.21
0.23
0.17
0.07
-0.01
-0.03
0.00
4.47
0.00
0.00
0.7 1
0.00
3.05
1.54
- 1.54
-2.93
-0.05
0.00
-3.15
0.00
0.00
0.00
0.00
60
0.00
- I .22
0.00
0.00
0.17
0.00
-0.16
-1.11
0.04
0.35
0.14
0.00
0.96
0.00
0.00
0.00
0.00
We consider two techniques for doing this:
Calculation 3: Imposing smoothness conditions
Provided the function {'(x, O)k(x, 0) is piecewise continuous, the expansion (3.7) converges
a t least in the mean, and hence certainly
lim I,, = 0.
n-00
The expected behaviour of I,, as n increases can be characterized in terms of the analytic
behaviour of {'k in the interval 0 s x G b (see, e.g. Mead & Delves 1973; Bain & Delves
1977): there exist constants CA, C R ,CI, r, such that for all n:
IReZ,,l-t lImZ,,I
G
C,(lnl+
(4.1)
In practice we expect r = 2 or 3; CA, or CR, C,, can be estimated roughly from the expected
size of the tides. Inequalities (4.1) and (4.2) are certainly violated by the results of
calculation 1. We therefore impose them as additional constraints, yielding the procedure :
Choose Z t o minimize
II A Z - b I( (see equation 3.14)
(4.3)
subject to inequalities (4.1) or (4.2).
Problem (4.3) can be solved efficiently in the L 1 norm, being reducible in that case to a
standard linear programming problem. Table 4 shows some representative results obtained
The Proudman-Heaps tidal theorem
477
Table 4. Calculated values of the coefficients I, computed from the constrained
problem (4.3). Entries marked with an asterisk (*) satisfy (4.1) or (4.2) exactly
as an equality.
Constraints
Results
I-, x
I_,X10-'
I,X
I, x to-'
I, x l o - @
(4.2)
r 2; CR = 0.6 x 10
c 1 ~ 0 . 2 X4 l o 9
0.7* -i0.3*
-1. 5*+i0. 5
1.1 + i l . l
1.2 +i0.5
-0.7* + i0.3*
(4.1)
=2
C ~ ~ 0X .l o69
(4.1)
r=1
cA=o.4 x l o 9
Y
(0.0 - i0.7)*
(-1.1 +iO.4)*
1.2 + i l . l
(1.1 + i0.4)*
(0.0 + i0.7)*
(0.0 - i 1.3)*
+ i0.6
1.2 + i l . l
1.2 + i0.5
(0.0 + i 1.3)*
- 1.5
with this technique. The results all lie, as they must, within the bounds set by equations
(4.1) or (4.2); and so they look more reasonable. But those values which were badly in error
before, now (mostly) lie hard up against the constraints.
Since the constraints by their nature contain no information about the exact values of
the I,, the total information content of the calculation has not been improved. Imposing
constraints of this type cannot in general improve the accuracy of an ill-conditioned calculation; it does, however, stabilize it in the sense of avoiding the disastrous blowing up of the
error which otherwise occurs (see Table 1) when In I is increased in an attempt to increase
the accuracy of the predicted tides.
Chlculation 4: Adding known tidal values
The constraints imposed in calculation 3 are mathematical in origin. We can also hope,
perhaps more reasonably, to stabihze the calculation by adding further physical information.
In particular, tidal measurements might in practice well be available at one or more points
across the bay mouth; we can then impose the condition that the calculated I, reproduce
these 'spot' values. Such constraints can be added very easily in the formalism used here.
Equations (3.7), evaluated at a point g j with known left side, can be rewritten as a linear
equation involving the unknowns Z - M , . . . , Z, and either treated as an equality constraint
during the solution of equation (3.14), or, more simply, given a suitable large weight and
added to the overdetermined system. In either case, the effective size of the problem is
increased since now the defining equations for different Z, are coupled; however, in the L z
norm this coupling can be treated explicitly and the problem solved as a sequence of uncoupled problems. For brevity, we omit computational details.
Table 5. Results for calculation 4: The coefficients (I, X
to the constraint that one or more 'exact' tides be reproduced.
n
-2
Calculation
R.P
No tides
1 tide
2 tides
3 tides
4 tides
5 tides
Exact
0.008-i0.277
-0.108-i0.361
-0.171 -i0.422
-0.072-i0.389
0.219 +iO.O84
-0.061 +i0.006
-0.073-iO.0002
0
-1
1.P
R.P
computed from the tidal equations subject
1.P
-0.159 +i0.056
-0.037 -iO.O18
-0.014+i0.090
0.111 + i0.069
-0.143 -i0.082
-0.154 + i0.033
-0.154 +i0.036
R.P.
1
I.P.
0.117 +iO.110
0.139 +i0.251
0.139 +iO.120
0.285 +iO.130
0.086 -i0.004
0.122 +iO.106
0.122 +i0.109
R.P.
2
I.P.
0.117 +i0.047
-0.022 + i0.013
-0.043 +iO.123
0.079 +iO.161
-0.010 -i0.033
0.065 +iO.O55
0.065 +i0.058
R.P.
-0.645
-0.561
-0.500
-0.398
-0.143
0.039
0.049
1.P
+i0.446
+ i0.331
+iO.268
+i0.249
+iO.258
+iO.O31
+iO.O20
478
M. G. G. Foreman et al.
Table 5 gives results obtained for a number of runs including one to five ‘measured’ tides
across the bay mouth; these measured tides are those obtained by summing the truncated
series (3.7) with the ‘exact’ values of I,, and hence are compatible with the simulation model
results we are using. We make the following comments on these results:
(1) The results with no measured tides are those obtained in calculation 1, which gave
very bad predictions for In I > 1.
(2) As a minimum, we would hope that adding a small number of spot tidal measurements would at least improve the prediction for n = 1 , f 2. The results for 1 , 2 , 3 , 4added
tides are therefore very disappointing; no particular improvement is obtained overall with
these calculations.
(3) The results with five added tides are dramatically better, and are in rather good agreement with the exact values. Unfortunately, this agreement reflects only a mathematical
identity inherent in the calculation when P tides are used to help predict P Fourier
coefficients I,. Here, P = 5, and the residual differences between ‘calculated’ and ‘exact’
values stem from the discretization errors of the calculation.
(4) Given these results, it is clear that we could do just as well by interpolating bay
mouth tides from the available spot measured values. We conclude from these calculations
that, at least for the model bay considered, the problem of computing bay mouth tides from
coastal values is too ill-posed to yield physically interesting predictions.
*
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