HARMONIC TIDAL ANALYSES OF

LONG TIME SERIES

by M .G .G . FOREMAN (*) and E.T. NEUFELD (**)

Abstract

A harmonic tided analysis program is developed for observational records of18.61 years or longer. The amplitudes and phases of over five hundred astrono­

mical and shallow water constituents are calculated using a least squares approach. The program is tested with 38 years of hourly observations at Victoria

and the amplitudes and phases of satellite constituents whose amplitudes lie above

the background noise level are generally found to be consistent with potential theory. Predictions based on the results of a 19-year analysis are found to be

only slightly better than those based on averages from 19 one-year analyses,

thereby confirming the accuracy of G odin’s [1972] satellite correction algorithm and satellite inference based on potential theory relationships. However it is

demonstrated with constituents NOv J ]t N2, and L , that results from the 38-year analysis can be used to improve the satellite inference calculations in shorter

analyses. Based on the stability of the 38 one-year analyses, recommendations

are also made for the inclusion of additional constituents in the standard prediction of tides at Victoria.

INTRODUCTION

The harmonic analysis of tides requires calculating the amplitudes and

phases of a finite number of sinusoidal functions with known frequencies from a time series of observations. Although tidal potential theory (e.g., D o o d s o n , 1921;

C a r t w r ig h t and T a y l e r , 1971) predicts hundreds of these frequencies, many of

them are so close that they cannot be adequately separated in analyses of a few

(*) Institute of Ocean Sciences, P.O. Box 6000, Sidney, B.C., V8 L 4B2, Canada.(**) Department of Computer Science, University of Victoria, Victoria, B.C., V8 W 2Y2, Canada.

years or less. Attempts to do so result in an ill-conditioned matrix equation and

large confidence limits around the computed constituent amplitudes and phases.

Consequently, conventional harmonic tide analyses and predictions (e.g., Godin,

1972) are actually quasi-harmonic (Zetler, Long, and Ku, 1985) in the sense

that they modify constituent amplitudes and phases to account for the satellite

constituents that are not directly included in the analyses.

However many permanent tide stations now have records longer than

18.61 years, the period of rotation of the moon’s node, and it is feasible to

consider a harmonic analysis that does not require satellite adjustments. Such an

analysis would directly resolve constituents differing by multiples of the 8.85-year and 18.61-year basic tidal periods and thereby provide, both a check of the

accuracy of the satellite adjustment calculations in shorter analyses, and, in cases

where a shallow water constituent and astronomical satellite have the same

frequency, more accurate inference parameters for these adjustments.

Recently Franco and Harari (1988) presented a tidal analysis technique

for long time series that is based on Fourier transforms. They computed tidal harmonics for Cananéia (Brazil) from the Fourier coefficients of a series of M J>

5 successive transforms of 214 hourly observations. (214 hours is approximately

1.869 years.) Zetler et al. (1985) and Amin (1976) have also calculated tidal harmonics from long time series. Zetler et al. found that harmonics calculated

from a Fourier analysis of 18.61 years of Seattle data gave slightly better predictions than those produced by three conventional quasi-harmonic techniques.

Amin (1976), following an approach originated by Cartwright and Tayler

(1971), first band-pass filtered a long time series from Southend (England) into

separate tidal groups, and then showed that the / and u nodal parameters differed

from those computed using the equilibrium tide.

Though the technique presented here is much simpler than those of Franco

and Harari, Zetler et al., and Amin, it has the advantages of easily handling

missing data and being applicable to any record lengths greater than 18.61

years. In fact, even though the version described here is for regularly sampled

data, the same approach could also be used for irregularly sampled data. In this

sense, this technique is more versatile than Fourier-based approaches.

However, it must be emphasized that long harmonic analyses are only

possible because of the increased speed and memory of present-day computers. A decade ago, the large matrix equations that arise from these analyses could only

be solved on a select few super-computers. With the proliferation of faster and

cheaper computers, it is now feasible to develop a program that can be used on a

wide variety of machines. Indeed, it is planned that the program described here will become part of the widely distributed Institute of Ocean Sciences package of

tidal programs and test data.

THE NUMERICAL TECHNIQUE

Doodson (1921) showed that all tidal constituents have frequencies that are

linear combinations, termed harmonics, of the rates of change of r, mean lunar

time, and the following five astronomical variables that uniquely specify the

position of the sun and moon: s, the mean longitude of the moon; h, the mean

longitude of the sun; p, the mean longitude of the lunar perigee; n', the negative

of the longitude of the moon’s ascending node; and p ' the mean longitude of the

solar perigee. The approximate periods for these six variables are 24.84 hours,

27.3 days, 365.24 days, 8.85 years, 18.61 years, and 20932 years respec­tively. For each constituent, the integer coefficients of these six harmonics are

called the Doodson numbers.

The tidal frequencies used in this harmonic analysis are taken from the

Cartwright and Tayler (1971), and Cartwright and Edden (1973), update of

Doodson’s calculations. Care was taken that pairs of constituents whose

frequencies differed by integer multiples of p ' would not both be included in an

analysis. In such cases, only the constituent with the larger tidal potential

amplitude is included. Each analysis included 474 astronomical constituents and

55 shallow water constituents, but the latter number could be easily increased if

nonlinear interactions were thought to be more important. A criterion (see Godin,

1972) to select constituents for the analysis has also been used to ensure that

neighbouring constituents are separated by at least one cycle over the analysis

period. For example, the frequencies 2n' and p require approximately 179 years

for separation. So constituents whose last fourth and fifth Doodson numbers differ

by ±(-1,2) are handled in the same manner as those differing only in the sixth

Doodson number.

The least squares technique employed in the analysis is identical to that

described by G o d in (1972) and F o r e m a n (1977). It is briefly reviewed as follows.

Assume that a selection procedure has chosen M constituents for inclusion in the

analysis. We then wish to solve the system of equations

My,- = A0 + X AjCOs{a)jtj~ ¢,) ( 1 )

) i

for the unknowns A j and , j = 1 ,M. A p ojj, <pj are the amplitude, frequency,

and phase of constituent j; y, , i = 1 ,/V, is the observation at time f, ; and in

accordance with Doodson, each frequency to, can be written as

ûJj = /, r + l2s + l3h + /4p + l5n ' + /6p', (2)for integers lk , k = 1,6. Each equation can be made linear in the new unknowns Cj and Sj by rewriting

A j COs{a>jtj - <t>j) = Cj COS{(Ojtj) + Sj sin(<Wyfy), (3)

where

A j = (C j + S f )'/2 and <f>j = arctan(5y /C;).

As the number of equations, N, is greater than the number of unknowns,

2M + 1, the system of equations is overdetermined and all the equations cannot be solved exactly. The least squares technique calculates the solution that mini­

mizes the sum of the squares of the residuals. The original set of equations given by (1) is reformulated as the normal equations and solved efficiently and stably

with the Cholesky algorithm (e.g. O r t e g a , 1972).

All calculations with the new program were done in single precision on a

VAX-785 computer. Each analysis solved a 1057 x 1057 matrix and the 38-

year analysis required approximately four hours of computer time. Were there no

gaps in the hourly time series, the normal matrix could have been partitioned and the computation time would have been reduced by a factor of four.

Results

The harmonic analysis program was initially applied to hourly Victoria data from 0100 PST January 1, 1939, to 2400 December 31, 1976. Only 624 and

720 values are missing in October 1950 and March 1973 respectively. In

addition to a 38-year analysis, the two 19-year sections of data were also

analysed. A sequence of one-year analyses using G o d in ’s (1972) satellite

adjustment technique as also done and averages and standard deviations were computed for the amplitudes and phases of all constituents.

Tables I, II, IV, and V are analogous to Table I in Z etler et aJ.. They

compare the results of the two successive 19-year analyses, the 38-year analysis, and the average values from the 38 one-year analyses. Values shown in brackets

in the averages/inferred columns in Tables I, II, IV, and V are the inferred satellite amplitudes and phases that were calculated separately using potential theory and the 38-year analysis results. Values not enclosed in brackets in the

same column are averages from the 38 one-year analyses. The columns entitled

o provide a measure of the stability of these yearly analysis results. For each

constituent /, the standard deviation, , is defined (C raw ford , 1982) as

(4)where

_ 38 _ 38Cy= X C A j / 38 , Sj= X 5/, j /3 8 ,

/=1 IA

and Q j, S/ j are as defined by equation (3) for year 1.

As Z etler et al. suggest a rejection limit (due to contamination from back­

ground noise) of 0.25 to 0.5 cm for a 19-year analysis, only constituents with

amplitudes greater than 0.1 cm in the 38-year analysis are listed in Tables I and

II. Due to lower and higher levels of background noise in other frequency bands, rejection limits of 0.05 cm and 0.5 cm were chosen in Tables IV and V

respectively.

Figure 1 gives a visual definition of the background noise level in the low frequency band. It shows the Fourier amplitudes for a 184320 hour (21.03 year)

segment of the Victoria data with no gaps. The fact that the tidal amplitudes

.000 .005 .010

FREQUENCY (eycle»/hr)

FlC. 1.— Fourier amplitudes for V ictoria in the frequency range 0 .0 to 0.01 cycles per hour.

barely rise above the general background level indicates that the harmonic

method can be expected to have difficulty in calculating stable tidal energies.

(This is confirmed by the results in Table V and will be discussed later). Better

estimates of the tidal contribution to this low frequency energy could be obtained

by using the C r a w f o r d (1982) technique for reducing the atmospheric effects.

Figures 2, 3, 4, and 5 show the analogous plots for the diurnal, semi­diurnal, and high frequency bands. Amplitudes larger than 5 cm in Figures 1, 2

and 3, and 2 cm in Figures 4 and 5, have been cut off in order to provide a

better view of the low energy spectra. Notice that many tidal amplitudes are

significantly larger than the background noise in these bands. The presence of

tidal cusps around the large constituents has been studied by M u n k , Z e t l e r , and G r o v e s (1965).

As pointed out by Z e t l e r et al., Godin (1972), and Amin (1976), the

presence of non-tidal energy (background noise) in the observations and variations

in the tides themselves due to nonlinear interactions and changing physical regimes (e.g., silting of harbours, density variations, seasonal ice) may have the

FREQUENCY (cycles/hr)

FlG. 2 .— Fourier amplitudes for V ictoria in the frequency range 0 .03 to 0.05 cycles per hour.

effect of falsely distributing energy to neighbouring frequencies. It is therefore

revealing to compare amplitude ratios obtained by the long harmonic analysis with those computed from the tidal potential. Ratios that are significantly different

may denote such an energy distribution or the presence of a shallow water

constituent. It should be pointed out that leakage of tidal energy to lesser

constituents or satellites is not a phenomenon that is unique to harmonic analyses. Fourier techniques will exhibit the same behaviour since they are, in fact,

harmonic methods with the number of unknowns equal to the number of

observations and the frequencies dictated by the record length rather than tidal

theory.

The two 19-year analyses are included in Tables 1, II, IV, and V to

evaluate stability of the satellite amplitudes and phases. Notice that the harmonic constants for many of the smaller satellite constituents change dramatically from

one analysis to the next. This suggests that the tidal signal has been significantly

masked by the background noise and the calculated amplitudes and phases are not primarily of tidal origin. However most of the harmonic constants for the

major constituents and satellites are relatively constant.

FREQUENCY (eyeI**/hr)

F ig . 3 .— Fourier amplitudes for Victoria in the frequency range 0 .07 to 0 .09 cycles per hour.

A comparison of amplitudes and phases in the 1939-76 and averages/

inferred columns in Tables I and 11 shows how well the satellite harmonic

constants correspond to potential theory predictions. The large satellites of Ku O,

have amplitudes and phases that are very close to what is predicted using

potential theory and the main Kx and values. However, the correlation

generally deteriorates as the amplitude of the smaller satellites approaches the

background noise level.

The satellite constituents marked with an asterisk in Tables I and II denote third-order terms in the tidal potential expansion. They deserve special mention as

they often do not seem to occur as predicted by theory, and are consequently

omitted from nodal correction calculations (e.g. FOREMAN, 1977). The Qu Ou Ku

Nz, M2, and Z , third order satellites exhibit high variability and should probably not be included in any predictions. As seen in Figures 2 and 3 their amplitudes

are not sufficiently large to emerge from the background spectra. Consequently,

the energy attributed to them has a large non-tidal component. The and

satellites are relatively consistent for the two 19-year analyses, however their

1.80-

1. 60 -

).40

1.20

§ 1.00

0 . 80 -

0-60

0.40

0.20

0.00

MO,

NO,

^SOj

SK,MNi

•NK,

N4

M

ms4

2MKs

mk4

sn4

p SK4fuL—

m n k 52M 05MNO,

Ml

\

2MP./ 5m s k 5

2SK5

JjL0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21

FREQUENCY (cycles/hr)

FlC. 4 .— Fourier amplitudes for V ictoria in the frequency range 0.11 to 0.21 cycles per hour.

amplitudes and especially phases, are quite different than predicted by potential

theory.

The results in Tables I and II can be used to assess the expected success

of nodal corrections in the light of G o d in ’ s (1986) recommendations. G o d in

remarks that nodal corrections are effective for Ku Ou and K2. We concur. Our

analyses show that all these constituents have large satellites whose amplitudes

and phases are both stable and in accordance with tidal potential theory. He also states that nodal corrections should be successful for Qlr /,, OOu and NOu

where the tide is primarily linear and third order effects are minimal. Discounting

the third order satellites for Qu and OOu we agree here also. Although G o d in

does not recommend nodal corrections for either N2, or L?, we don’t feel that

they should be disgarded completely. Although both third order satellites seem to

be unreliable and should not be included in nodal corrections, consistency of the other satellite harmonics suggest that their inclusion should help predictions.

However it does not appear that one should use potential theory amplitude ratios

and phase differences in the nodal correction calculations, as they are significantly

different from the analysis results.

FREQUENCY (cycles/hr)

Fig, 5.— Fourier amplitudes for Victoria in the frequency range 0.23 to 0.33 cycles per hour.

Notice that some of the satellites around large constituents such as M2, and

K\ have amplitudes that are much larger than predicted by potential theory. This

probably indicates a leakage of the main constituent energy due to radiational

effects, tide gauge problems, or a changing physical environment (e.g. harbour dredging).

Many shallow water constituents have the same frequency as an astrono­

mical constituent. (Table III provides a partial list.) Consequently in cases where

the harmonic analysis has computed a much larger amplitude for a major

constituent (e.g., Qi) relative to neighbour (e.g., O ,) than is predicted by potential

theory, it is likely that a substantial portion of the major constituent energy is

actually due to a shallow water constituent (e.g., NKi). The precise shallow water

contribution can be calculated by admittance functions along the same lines as is

done by G o d in and G u t ie r r e z (1986). A crude indication of the nonlinear activity around the diurnal frequencies is seen with the following calculations. Using /C, as

a reference and assuming a constant admittance function across the entire

diurnal band, potential theory predicts amplitudes of 8.70, 45.44, 0.59, 3.57,

21.15, 3.57, and 0.59 cm respectively for ¢),, 0,, r,, NO,, P,, -A, and SO,.

The first six of these values are 37%, 22¾. 44%, 26%, 6%, and 6% larger than is

calculated by the 38-year analysis whereas the last is 40% smaller. So it is likely

that there are substantial nonlinear contributions to both K , and these seven

constituents. Similar calculations could be done for some of the semi-diurnal

constituents but they would be of dubious value due to the presence of several amphidromes near Victoria. In the vicinity of these amphidromes, amplitudes vary

significantly and the assumption of a smooth admittance function is not valid.

In Tables I, II, IV, and V, values of a that are relatively large with respect to the amplitude denote cases where either background noise or the satellite

adjustment calculation has caused the analysed signal to be unstable. For many

constituents where there are substantial nonlinear contributions, this stability can

be improved significantly if the inference parameters used in G o d in ’s adjustment calculation are based on the results of the 38-year analysis, rather than potential

theory. For example, the o values for NOu J u N2, and L2, are reduced from those in Tables I and II to 0.35, 0.27, 0.26, and 0.26 respectively, when inference parameters are based on the 38-year analysis results. Consequently, an

important role of long harmonic analyses will be to obtain better inference parameters for satellite adjustment calculations in shorter analyses at the same (or

a nearby) location.

Table IV shows analysis results for the terdiurnal and higher frequency

constituents whose amplitudes were found to be greater than 0.05 cm. The two 19-year analyses are remarkably consistent, demonstrating little contamination by

background noise or tidal signals that are not included in the analysis. The inferred amplitudes were calculated using tidal potential theory and assuming just

one interaction. For example, M 03 is assumed to arise solely from M2 » O ,. The

extent to which these inferred values differ from the 1939-76 analysis results is an indication of the importance of other interactions. M 03 may also arise from

M4 - Ku /u2 * P\, and S2 + or,. As seen in Table IV, it is likely that at least one

of these additional interactions has caused a larger than expected value (based on the size of MKZ) for M 03 (3-10000). Notice that since the constituent selection

criterion does not permit the inclusion of both M3 and NK3 in one-year analyses,

the larger constituent, NK3, has been chosen.

Table V shows the analysis results for the slow constituents. In this case,

there is significant background noise (see Fig. 1), predominantly in the form of

meteorological effects, that causes large variability in the results of the two 19- year analyses. Only the results of constituents whose amplitudes are larger than

0.5 cm are shown. Notice that even among the relatively large signals, there is

significant variability in almost all the constituents. In fact, probably only 5a

(00100-1), Ssa (002000), and Msf (02-2000) are sufficiently stable that they

could be considered for inclusion in a prediction. The o values for Mf, Msf, Mm, and Msm are quite close to those published by C r a w f o r d , 1982 (Table 2) for

nineteen years of Victoria data, and are all much larger than the average

amplitudes.

Table VI shows the slightly improved predictive capability with harmonic

constants from the new program. Amplitudes and phases from an analysis of the

first 19 years of Victoria data were used to predict elevations for the second 19

years. Root mean square residuals were then computed from the actual

observations for this time period. Similar predictions were also made using

average amplitudes and phases from one-year analyses over the first 19 years.

(For these predictions, the / and u factors were assumed constant for each month

and set equal to their values on the 15'* day of that month.) Based on the

stability of the harmonic constants in the two 19-year analyses, only a constant

term and constituents with amplitudes greater than .1 cm in the diurnal and

higher frequency bands were included in all the predictions.

Notice that on average, predictions based on results of the 19 year analysis

are only slightly better than those based on the average of the 19 yearly

analyses. In fact, the former predictions are worse for five of the nineteen years.

There are two reasons why the predictions based on the 19 year analysis are not

as good as those shown by Z e t i j . r et al. for Seattle. The first reason is that

Z e t le r et al. predicted for the same 19-year period that was analysed. As seen

by the fact that there are differences in the two nineteen year analyses in Tables

1, II, IV, and V, the values in both columns A and B of Table VI would

probably decrease if the same strategy were followed here. The second reason is that Zetler compared his 19 year predictions with predictions using harmonics

from Form 444 (presumably a standard set of amplitudes and phases) rather than

averages of yearly analyses. It is likely that yearly-average predictions would be

more accurate.

It should be pointed out that the yearly-average predictions actually infer

values for all the satellites of each major constituent, whereas the predictions

based on the 19 year analysis have only included those satellites whose

amplitudes are larger than the threshold of 0.1 cm. However this discrepancy

makes very little difference to the results. When the yearly-average predictions are

repeated using the same satellites that are included in the column B predictions,

the mean rms becomes 14.610.

The set of constituents used for the yearly-average predictions in Table VI

has 31 more entries than the standard set used to produce the Canadian Tide

and Current Table (1989) values for Victoria. (Constituents included in this

standard set are designated with the superscript • m Tables I, II, IV, and V.)

The o values shown in Tables I, II, IV, and V demonstrate that some of these

constituents — notably e2, H u H2, NOit NK:i, 2MOs, and 2MPS, and 2MKS —

are both large enough and sufficiently stable to warrant inclusion in future predictions. Figures 3 and 4 show the relative importance of these constituents.

Column C in Table VI shows the residuals obtained from predictions using

averages of the 19 yearly analyses for only those constituents included in the

Tide Table set. The residuals are consistently, though not appreciably, worse than

those in column A. This indicates that slightly better Tide Table predictions could

be obtained with an expanded set of constituents.

The closeness of columns A and B in Table VI confirms the accuracy of

G o d in ’s [ 1 9 7 2 ] satellite adjustment calculation and inferences based on tidal

potential theory. Although approximations within these calculations are most valid

when the analysis period is one year, it is not likely that predictions based on,

say, an average of 6 month analyses, would be much different. Even though we

have seen that the inference parameters for NOu Ju N2, and L2 can be improved

by using values from the 38-year analysis, these satellites are so small in

amplitude that they scarcely make any difference to subsequent tidal predictions.

CONCLUSIONS

The preceding discussion has summarized the development and testing of a

new harmonic tidal analysis program for time series of 18.61 years or longer. The method is more versatile than Fourier-based approaches in the sense that it

can easily handle missing data and is applicable to all record lengths greater than

18.61 years.

Results obtained from 38 and 19-year analyses of Victoria observations demonstrate that, provided a tidal constituent amplitude lies above the

background noise level, its relative amplitude and phase generally conform to potential theory predictions. However this background noise may be sufficiently

large in the low frequency band that even long period analyses may not provide

accurate estimates of constituents such as Sa, Ssa, Msm, Mm, Msf, and Mf. In such cases, additional measures (e.g. C r a w f o r d , 1982) may be required to

reduce atmospheric effects prior to the tidal analysis.

Predictions based on results from a 19-year analysis were seen to be only

slightly better than conventional predictions using Godin’s technique for satellite

adjustment. This is an indication that: a) the harmonic constants corresponding to

most of the satellite constituents that are now included directly in the long analysis are not significantly different from potential theory predictions; and b) the

approximations made with Godin’s satellite adjustment calculation are quite

accurate.

It was demonstrated with constituents NOx, J u Nz, and L,, that an important role of long harmonic analyses will be to obtain better inference

parameters for satellite adjustment calculations in shorter analyses at the same (or

a nearby) location. For Victoria, the stability of these constituents is improved

significantly by using satellite inference parameters from a 38-year analysis.

Differences between columns A and C in Table VI suggest that the

standard set of constituents for producing the Canadian Tide and Current Tables (1989) at Victoria could be expanded. Standard deviations calculated from the

38 yearly analyses demonstrate that many of the presently omitted constituents

(such as t2, H x, H>, N 03, NK:i, 2MO,, 2MP5, and 2MK5) are both large enough and sufficiently stable to be included in future predictions. It is likely that similar

improvements are possible at other sites where long records permit analyses of

the type performed here.

Acknowledgements

We thank Bill C r a w f o r d and Gabriel G o d in for helpful comments, and Patricia KlMBER for drafting the figures.

References

[1] AMIN, M. (1976): The fine resolution of tidal harmonics. Geophys. J.R . astr. Soc. 44,

pp. 293-310.

[2] Ca n a d ia n T ide and C urrent Ta b l e s , Vol. 5, Juan de Fuca and Georgia Straits

(1989). Department of Fisheries and Oceans, Ottawa.

[3] CARTWRIGHT, D.E and R.J. TAYLER (1971): New computations of the tide-generating potential. Geophys. J.R. astr. Soc., 23, pp. 45-74.

[4] CARTWRIGHT, D .E and A .C . E dden (1973): Corrected tables of tidal harmonics.

Geophys J.R. astr. Soc., 33, pp. 253-264.

[5] CRAWFORD, W .R. (1982): Analysis of fortnightly and monthly tides. International

Hydrographic Review, LIX(l), pp. 131-141.

[6 ] DOODSON, A.T. (1921): The harmonic development of the tide-generating potential.

Proc. Roy. Soc. Series A ., 100, pp. 306-323. Re-issued in the International Hydrographic Review, May 1954.

[7] FOREMAN, M.G.G. (1977): Manual for tidal heights analysis and prediction. Pacific

Marine Science Report 77-10, Institute of Ocean Sciences, Patricia Bay, Sidney, B.C., 97 pp. Unpublished manuscript.

[8 ] FRANCO, A.S., and J. H A RA R I (1988): Tidal analysis of long series. International

Hydrographic Review, LXV(l), pp. 141-158.

[9] GODIN, G . (1972): The Analysis of Tides. University of Toronto Press, 264 pp.

[10] GODIN, G . (1986): The use of nodal corrections in the calculation of harmonic constants.

International Hydrographic Review, LXIII(2), pp. 143-162

[11] GODIN, G . and G . G iJTIÉRREZ (1986): Non-linear effects in the tide of the Bay of Fundy.

Continental Shelf Research, 5(3), pp. 379-402.

[12] MUNK, W.H., B. ZETLER, and G .W . G roves (1965): Tided cusps. Geophys. J., 10, p.

211-219.

[13] ORTEGA, J.M. (1972): Numerical Analysis — A Second Course. Academic Press, New York, 193 pp.

[14] ZETLER, B.D., E.E. L o n g , and L.F. Ku, (1985): Tide predictions using satellite

constituents. International Hydrographic Review, pp. 135-142.

Tab

le

la¢0-oc<0

-o"D *2

00C

-oc¢0

!! !•* i fc-f ‘ T: o <u roo: çu

r 2i O) _! O i u a;> « jef _o —: c .jsI "O S1 fl) -t-; « oJ2 ai w -a: c ca> S

, <u -21 3 Vi

' J *,¾ Si f s•â 05

L S.®• V) COi « as

» 2 4JL £ -C I O +-1''■3 E i â p ! 0) ' h.. a: _Q, «0

«j : cGO*5■u-O

CoU

a

(cm

)

0.2

4

0.2

9

0.2

7

0.3

7

0.3

5

aver

ages

/inf

erre

d |

S 'FT ce V)"S 9- 1

74.5

(136.8

)

136.7

(156.4

)

154.4

(130.9

)

(130.9

)

131.1

(129.7

130.4

I I 0.1

9

(0.1

4)

0.7

7 eor-H

d 0.6

7

(0.1

6)

(1.1

9)

6.3

3

(0.2

4)

1.2

7

1939-1976 *>

«a"S 9* 1

75.0

192.4

128.7

136.8

163.1

156.4

6S

6I 5

6.0

296.5

131.8

130.9

145.9

12

0.2

129.7

136.4

242.4

1 1OCM

O

O

o 0.1

0

0.7

5

91

0

0.7

1

31

0

0.1

4

0.1

3

1.2

4

6.3

4

0.1

4

0.2

6

1.2

5

ZY

0

0.2

3

1958-1976

phase

(°PST

)

179.6

198.3

108.3

136.4

166.9

157.4

194.1

42.3

285.7

133.3

130.8

86

.8

114.1

129.7

132.1

245.9

1 1CM

d 0.0

9

0.0

8

0.7

8

0.1

5

0.6

7

0.0

7

0.1

4

0.1

6

1.2

0

6.2

5

iro

0.3

2

1.2

5

0.1

3 0 0

d

1939-1957 8 »-¾ 9* û. ^

66

91

187.3

141.2

137.3

160.5

155.4

197.0

68

.6

305.6

130.1

131.0 O

0 0

128.9

129.7 O

oLOO

CM

I I

61

0

iro

0.1

3

0.7

2

0.1

7

0.7

6

0.1

7

sro

0.1

1

1.2

7

6.4

3

0.1

2

0.1

9

1.2

6 iro

0.2

7

Const

ituent

Doodso

n

num

bers

1-

42

10

0 OO

i

1-3

0 2

1 0

1-3

0 2

0 0 O

toCM

eni

1-3

2 0

0 0

1-3

3 0

0-1 O

O

o

CO1

1-2

0 0

0 0

1-2

01

-1

0

1-

20

10

0

1-

21

1

0-1

1-2

2-

1-1

0

1-2

2 1

1 0

1-

1-2

2 0

0

O

opH1

r>H1

nam

e

S < 5 $CM CM

b*

• ^to O ’ O ’ O* a o.

Tab

le

la (c

ontinued)

CM<£>o 0

.46

0.3

9

1.0

4

0.3

2

0.3

3

to S ' rX> S ' S ' q CO 5 T in •—« t-N 0 •—< q qd d d d d CM* d CO d d d d 0 d CO 00 CO COCO CO CO CO CO CO CO CM CM Tf Tf in CO CO m inCM CM •w-

/*v ? o /*» y“V /~vCM in ■4i CO 00 O f-H O CM un 0 r- CM f- 4CM O o CM Tf O «—* O m 00 in 0 CMo o o O O O O O *•* 0 CM* O O O 0co n"

00 CO q CO r-~ q CM 0 5 r- 00 CM CO CM CO CO 0 5

d co CO d d 06 CO 00 r- d r- d d in d 00 Tj" 00 coCM CO CO CMLO CO f-H l-H r- CM 00 CO CO 0 5 Tf CO CM inCO CM CM CM

o LO , 00 in CM , T}* co T* Oi 00 O Tf 00 CM ID in O ,CM o CM CO CM *—> t* *—< CM CM «*■ 00 in •—* m •—> —< CMd o d d O d © d O 0 d •—« d CM 0 O 0 d dCO

°î LO q Tf o 0 0 't q Ol Ol O 00 q 00 CM q m r>-CM d d o6 CO r-< 0 CM 0 d in CO d in 00 CM* 00 in«-H CO CO 00 CO CO CM CO 0 5 CO 00 CM CO 0 CO Tt CO inCM CM CM CM CM

IT) ,0 5

, t 0¾ CM r— 0 3 CO r- rj- 00 <£> ID 00 CM CO co , 0 5CM 00 CMCO CM •*r «—• O *—1 CM CM CO r— in «—1 in 1-M *— 1

o d o o © 0 0 O d 0 O O O CM d O 0 O 0CO

q o o r- tj- CO oq r- t— q q O 1— 1 in 00 0 5 0 5 CMCO d CM iri 0 5 CO CO d d CO 1^ O d CO d d CM

CO CO in CO CO 0 O CM 00 CO 0 5 •*fr CO 1-* inCO CM CM CM

r- r- 0 5 O) O in in r» <X> oo CO CM 0 5 CM 0 X) 0 5 CO1—N CM q CM CO CM «■H CO CM q in cr> O CMo d o d O d 0 d d O 0 0 CM d d 0 d O

CO

o o o o o O 0 0 0 -< 0 O 0 O 0 0 0 w* -CM o o 0 O 0 | 0 O O 0 O O 0 0

1 1

o © o CM 0 O 0 0 —• 0 r- —■ r- O 0o o o o O CM CM CO CM CMI 0 O 0 O O CM CM CO1 co

1

1 1 1 | 1 1 1 O O O 0 O 0 O O O O Oi-< -H

O O o O O c ÛÏ oa Ï I X >< fc b

b (cm

)

0.4

7

1.0

8 0000

o 0.5

4

zr

o

0.36

|

-ou>—

V.a:

*0

q . ^ (147.5

)

147.5

(147.5

)

128.4

(116.6

)

(149.3

)

(149.3

)

(149.3

)

149.3

(149.3

)

(149.3

)

(149.3

)

162.2

(162.4

)

157.1

149.3

<si-4 00 CM4) S- 'c' CNJ S Ç0 CO 00 H T f N Z t O O J — |s«> 5 S cm u ! o • LO O O CM • <£> — O . ©(Q

d 2 o ° * o o o ^ $ g o d o° é

© d

£ PCO LO CM <£> 05 © <£> CO CO CM CO 05 T f CM in oo

<£>3 ^ cm co d d 00 00 CM d CO 00 T t CM d d

0¾ -C CX, co r- *-H »—1 N CM T f T f T f ^ ;Q in T f- - CM CM — — — — CM — —

a i

O iQ , ^ QO ^ O 00 <n CM CM 05 © O CM © oo in 00 ©

5 ÊCM GO —; <£> — — — 05 ID CM — 05 — r- <x>

S ^ d a i d — d d d — ’ co oo d d © © d ©<£>

S P1/5 — <£> in LO ^ r- T f ©

cts ST> in d 00 Tf* d in o d d ^ i>^ in co dr--

-5 ^ CO ^ ^ — CM oo T f T t r f co in <T> 00 in in05 Ci- ^ - - CM CM — — — 05 CM — —

OCLO05

^ 7 > r- r"~ ct> LO O CO 05 in CM CM oo o <X> © r-

s §co — O CO CM CM CO CM in CM CM 0*3 CM r - LO

<3 s i o d d — o d o < co oo d © d d © d<£)

^ O 'Vi ^ <£> ^ a> r^. r t CM — CM — © tD CM CO CM <£> <£>

r- !§ Sfi in n co '« t 00 CM © CO d d CM d — 00 inLD

~S ‘S 'CM r * r r — o — 05 Tj* Tj* ^ LO in in in

05 -S- — — CO — CO CO CM — — — — —

05cOO i O ) — oo — in CM © CO CM 00 CM 05 © 05 CO

P - E— o o (£5 o O — O Tt- r^. CM — 05 — r>- tD

1 1 o d dCM

^ d © © — ^ cd © d d © © d

o o o l »—H © © © © © © © *—« © ©

§ e— o ©

1© — — © — © — CM —

1 |© — © ©

J ï s a "S

O O CM O O — — o o o o — o © o —

c4> o | CM CM CM © © © © © © © — — CM CM3

C l c — — — — J . ^ ^ ^ ^ ^ ^ ^ - , — CM

co

Uu

5!« C c C û T c/7c

Tab

le

Ib (c

ontinued)

0.4

9

81

0

0.2

0

0.3

0 00 ^ H

o

T f *** o T f 05 T f CD Tf *—<

r— h- r- r- CM CO CM CM CM CM CO TfCD CO CD CD © <D cD 00 cO 00 00 © ©I'H »-* CO CM CM 1-* CM CM

o CO h» T f © 00 05 CO f-M CM o r^. CO»—1 CO CD CM CO 05 l-H CO CM T f CO CO CM

o CO O © © © © © CM © © ©' ^ ' ' w w

CD 00 T t CO CM 05 o CD CD ■*t 05 oo

h- Is- CO T f CM in CM © CM CM CM © T f T fLO CD CD CM r- © i—* CD in <-4 r- 00 00 05 CO © ©

CM CO co CM CM CM CO CM CM

CO 00 o CM © 05 CD 0> 00 in in CM r— CM © CD 00CO r- CM «—i CM 05 *—< CM CM T f CO CO

o CO o O © © © © o o © CM © © © ©

CM o r— CM © CM © f- © © in © in © T f © CO

•■n r*. T f ID O T f CO © lO CD CM CM CM ©LO CD CD CM in © CO to in *—< 00 on 05 CO © ©T-H CM CO CO CM CM CM CO CM CM

in 05 05 T f 00 CO © 00 T f oO CO 00 in T f 00'—I CO CD CM *—• CM »—I © ■ »—» CM CO *—i CO CM

o CO O © © © © © © © CM © © © ©

CM CO CD CM 00 05 © © CM CD 05 © h- 00

r- h- co CO © 05 T f in in 05 T f CM CD © 00<D CD CD CM 00 05 © CD T f © 00 00 00 05 CO © ©

CM CM CO CM CM CM rH co CM CM

CO 05 o l-N r*. 00 CD h» © CD CO CD*—• CO h» CM CO CM 05 1-* CM CO T f CO © CO f-H

o CO © © © © © © © © © CM © © © ©

o O © O © —© o © © © © o © ©—O -, © © © © © © © © CM © ©

——© © © © © © CM © © ©o O © © T f CO CM CM —© © © © CM © ©CM CM CM CM CM CO CO CO CO CO CO CO CO CO T f 'tf T f

—-H — —<-H— -H —•—•

*<«"S' fCM

O OCO CO

oOCfo o o o o

Tab

le

II

Sem

i-D

iurn

al

Har

monie

Const

ants

fo

r V

icto

ria

b

(cm

)

0.2

2

0.1

3

0.6

2 00d 0

.66

CMO 0

.41

I 0.

39

|

-o0) P r- to Go in ST CM O CO tD COu

'c

Q"S. £

CO dinCO

<£>CO

oO 05 CMCM CM<£>

0r-

0ss CM

05 05

"-s.V5<U

>-V> P1?

0 CMc£>

0 x>in

S 'in

COCO

0000 in

00*O

OtT

CO0 0CO

0 O f- 0 CM 0 0 00 O O 0

0) p C£) Tt in 00 0 CM r- in in in in ,IDr-(X "S. 1

CO 05CO

r»-co

00 00 <DCM

<£>CO

<£>CMCO

r—m

CM<£>

0 0in

CMr-

OCM

CO00

0¾inCM

05

Oi<.yJ05

g- rO

<£itO

*“H in05CM CM

CMmCM

m00 inin

0 *■—»

CMTf l-H CO

CQ O 0 0 O CM O O O 0 00 r- 0 O O 0 O

<t> p O CO 0 05 O ID m 05 <x> 05 <d m 0 CO 00 m CO

f-Oi

CQ“S. 1

CMCM

0inCO

CM <£>CO

O 00CM

OCO CO

CO05*<*

CM<r>

0Is-

rfin CM

inCM

COf-

00r-CM

CM

OCi/;Ü5

1 ? 05O

00 c© CM 05 OCO

r-CM

r- 00CM

<x>r-

Ttin

COCM

cr>CO

«S 0 O O O O CM 0 O O O 00 0 O O O O

incn p

hase P

? 6.0

348.3 V

L

37.2

CO 18.3

|

29.0

42.7

301.9

64.9

61.8

70.8

48.5

101.5

15.8

92.8

239.4

16.4

oi05 & I

. CM<D

m 050 05in

00CM CM

r--O

CMCM

in05

toin

000 o>Tf

CO CMCM

5ec O O O r-i 0 CM O O O O 00 0 0 O O O

O O O O 0 O -. — O O 0 0 0 0 O O —

e ?» O O O O O O O O F-. 0 0 CM *-« O O O

■fl4)-O CO — CM 0 O O — O — — -7 O 0 O CM O O

cD 0 0 CM O CM CM CO —< O O 0 CM CM CM CM CMÇ CO CO CO CM CM CM CM — — — O O O O O O

</!C CM CM CM CM CM CM CM CM CMCM CM CM CM CM CM CM CM CM

U

COO

CNJ'V g

wa.

• CMa.

* CNJ _C\] ?CNI •5

C\1_CM _CN9? & O O

CN£■ 5;

Tab

le

II (c

ontinued)

0.7

0

0.3

7

0.3

5 CMCM

© 0.5

01

©CM

o 0.2

3

sro

0.2

2

0.1

2

0.1

2

60

0

(85.7

)

(85.7

)

(85.7

)

85.7

(85.7

)

255.1

301.2

198.1 © in

LO CO 00 00

(183.5

)

114.9

(93.7

)

93.6

103.6

99.3

(99.6

)

67.1

107.7

69.2

(00

0) (0

.02)

(1.3

9)

37.2

9

©©

on

ZVO

S9

0

1.0

3

(0.4

0)

(0.2

6)

0.5

6 CM©

d 10.4

1

0.1

9

1.9

6

(0.5

9)

0.3

9

0.1

4

0.3

2

o CO © T f 0¾ 00 © CO TT © m r- CM © r- © © in CO r- CO f"-

r- on in rsi T f I1 H © co 00 T f m © CO 00 © CO r- © ©

OCM

CO r - 00 inCM

©CO

COCO

©CM 2

COCM

00 CM CM

© oo © © © © © © © T fCM

CM © co m r- o CM CM © © T f CM ^ © © © © T f 00 © T f CMCM CO »—i »—1 Tf «—i © © ^ *—i m >—1 Tf CM © CD CO CO

o © — r>-CO

© — ' © © © © © O © © © © © © © © © ©

tT r- © r- T f © © r- 00 m © T f m m © 00 T f CM © 00 CO © CO

nn rf) eg in T f CM © in CM © © CM OO 00 in © CO 00 © © © r- ©oo © 1 i r- 00

COinCM

©CM

COCO

©CM

© CMCM 2 ©

CM© © © © © © © T f

CO

cn © \n c> <n CM in in 1ft CM CO © © © r>- 00 T f © © in CMCM CM T f «— H CO <—< © — \n CO CM © in CO CO CM

O ©co

© © © © © © — o © © © © © © © © ©©

in © 00 in in 00 CM CO T f oo © © CM © r- OO © © © © © CO © f-H

<r-> r— in f^- in © 00 f—« 00 in © Tt* T f CO 00 © 00 r- © CO

co© 00 00 ■*t ©

CM©CO

CMCO

©CM

©CM

COCM

o o m^ CM i ■<

CO © © © © © © r- T fCO

in T f in © Of) o m © CO tO T f © © oo CO T f © 00 00 CO CM ©o CM in ■ i © CO © © 1-H © — « «— m © T f CM © © CO *—• CO 1"‘No © h*

CO© © © © © © ^ © © © © © © CM © © © © ©

© © © o © r-, © — © — © © © — © © © © © © © ©

CM f H© © © © © © © © © © © © © © © —« © © © ©

— © © © — © © © — © — © © © © © © © ©

© © © © © CM CO CO CM © © © CO CM CM © © CM © CM

© © © © © © © © — CM CM CM CM CM CM CO CO T f T f

CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM

* C*45 a? I *s<

• .CO Û?

k

1 ' 1CM

Table IIIAstronomical and shallow water constituents with the same frequency

Astronomical Constituent Shallow Water Constituent

Name Doodson numbers Name Composition

Q, 1 -2 0 1 0 0 NK, N2 - K,

o , 1 -1 0 0 0 0 MKt M2-K ,

T\ 1 -1 2 0 0 0 MP, Af2- P ,

NO , 1 0 0 1 0 0 NO, N2- 0 ,

Pi 1 1 2 0 0 0 SK, S t- K ,

1 1 0 0 0 0 M O , M2 - 0,

A 1 2 0 -1 0 0 AfQ, M2- Q ,

s o , 1 3 -2 0 0 0 SO, s2- 0,

2 -3 2 1 0 0 m n s 2 M2 + N2 - S2

M2 2 -2 2 0 0 0 2MS2 M2 + M2 - S2

n 2 2 -1 0 1 0 0 k q 2 * . + <?.

m 2 2 0 0 0 0 0 k o 2 * , + 0 ,

\2 2 1 -2 1 0 0 s n m 2

2 1 0 -1 0 0 2MN2 m 2 + m 2 - yv2

s2 2 2 -2 0 0 0 k p 2 K, + P,

k l 2 2 0 0 0 0 Kz Kt + Kt

V2 2 3 0 -1 0 0 k j2 K ,+ J ,

Tab

le

IVHig

h Fre

que

ncy

H

arm

onic

Const

ants

fo

r V

icto

ria

a

(cm

) 60

0

00

o

80

0

0.1

2

0.1

7

or

o

90

0

00oo 0

.15

60

0

0.1

3

0.0

7

60

0

©

d

I S

00

-oV£

c

"«r4)

phase

(°P

ST

)

T? LO

d d CM CM

(4

1.1

)

41

.1

9.6

2.9

(2.6

)

44

.2

25

.7

(2

5.7

)

37

.6

61

.3

(6

1.0

)

30

6.5

33

0.8

35

6.9

34

1.7

11

.9

25

.4

(25

.6)

35

9.2

2<v>CT3

I ' s

(0

.17

)

0.9

2

(0

.40

)

2.0

9

0.2

7

0.6

1

(0

.08

)

1.2

1

2.0

3

(0

.27

)

0.2

8

0.8

3

(0

.11

)

0.2

0

0.8

4)

1.8

3 oCM

o 1.0

2

0.5

3

(0

.15

)

0.2

0

19

39

-1

97

6 * £03 V)

17

.5

20

.4

28

.2

41

.1

9.9

11

.3 q cm csi ^

44

.5

25

.7

24

.3

37

.6

L'Z

L

ri9

30

7.1

33

1.0

35

7.0

34

2.1

12

.0

25

.6

28

.1

35

9.2

§ SÜ 0.1

1

0.9

3

0.2

2

2.1

1

0.2

7

so

o

60

0

19

0

1.2

2

2.0

2

0.3

0

0.2

8

0.8

3

0.1

3 OCM

d00

d 1.8

3 oCM

o 1.0

2

er

o

IS

O

0.2

0

19

58

-1

97

6

phase

(°P

ST

)

15

.3

20

.4

27

.9

40

.9

9.0

Tf

q q

cm’ oo

44

.8

26

.0

29

.5

38

.1

61

.0

76

.8

30

8.8

33

2.4

35

8.1

34

2.2

12

.7

24

.4

32

.3CO

I I 0.1

0

0.9

2

0.1

9

2.0

9

0.2

6 m©d 0

.61

0.1

0

1.2

3

2.0

0

0.3

2

0.2

8

0.8

3

0.1

4

0.2

0

00d 1.8

2

OZ

O

1.0

3

0.4

8

0.1

2

0.2

0

19

39

-1

95

7

phase

(°P

ST

)

19

.3

20

.4

28.2

41

.4

10

.7

17

.6

2.2

35

9.2

44

.3

25

.4

18.1

37

.1

61

.2

68

.2

30

5.5

32

9.6

35

5.8

34

1.9

11

.3

r— r- d tj* CM CM 3

57

.0

I I 0.1

2

0.9

3

0.2

5

2.1

2

0.2

8

SO

O

0.6

1

0.0

9

1.2

1

2.0

4

0.2

8

0.2

8

0.8

4

0.1

3

CMd 0

.85

1.8

4

0.2

1

1.0

2

0.5

3

0.1

5

61

0

I C

onstitu

ent

Doodso

n

num

bers

3 2

0 1

1 0

3-

20

10

0

3-1

0 0-1

0

3-1

0 0

0 0 ©

o

CMi

o

3 0

0 0

0 0

3 0

0 1

0 0

3 0

0 1

1 0

3 1-

2 0

0 0

3 1

0 0

0 0

3 10

0 1

0

3 3-4

0 0

0

3 3-2

0 0

0

:3 3-2

0 1

0 o

o

CM

oCM

1

O

O

o

I

o

o

o

o

o

Tf 4 1

-2

10

0

4 2-2

0 0

0

4 2

0 0

0 0

4 2

0 0

1 0

4 4-

4 0

0 0

6<0c

-JO

§ §? 5 "

u s V < ?

• n P n

cC? « . «

5 ? Ï•

S I f ^5 5

Tab

le

IV (c

ontinued)

i nO

© CO dto <£>

cm r-CM O

d o

T f CO

d d r- r-

Tt co 00 00 0> 05

O CM«—« m

d o

00 UO CO O

o o

Tf t"-o oo oCM CM

in mCM CM

to m co O

r— Tfoi oCM COCM CM

& ^CO

o o

« r o

O CMin mCM CM

m inO

o o

CDco

00 ID 05 r—

00 CO 05 O — CM

O 05o oCM CM

m ^ r f inCM CM

CM O CM CM CM

cm inCM O

o o

O CMin

o o

r— ooCO O

o o

CD CD CO ©

d ©

T t CD CD 00

oo in o>

00 oo

05 © CM

cm inCM ©

<J5 CM© m

go r- <£> m

co m cm ©

d ©

© ©© ■o ©CM cm

CO V )

00 CD CO ©

d ©

© CM*—« in

d ©

© ©©

© © © ©

i

§ §cm <N

0 5 CO © © ©

© ©

© ^

© ©

© ©in in

cd in

© oC© ^CM CM

f p

© ^© ©CM CM

r- oo ©

© ©o ^© ©© ©

§ 5cm <N

CO * -J

T t 05rj* inCM CM

m r— co © © ©

r » 05

in cd

CM CM

r- m c0 © © ©

© ©O —<© ©CM CM

i i

CO CO

in in

_ L

§ §

CD

© cO CO CO CM CM

CD

CM CMm cd

CDC0

in cd

^ P © ©

00

© coCO Tj*CM CM

in cd

in r-CM CM

CD © CO

© ©

0 5 ^cri dCM CM CM CM

CD co CO ^

d d

© ©

© —'o ©© ©CM CM

CD CD

cd in © ©

o d d

CM CMin mCM CM

in cdO

© ©

© —© ©CM CM

I ITt Tf<D CD

Tab

le

VLo

w Fre

que

ncy

H

arm

onic

Const

ants

fo

r V

icto

ria

<7

(cm

)

1 3.0

6

4.4

6

3.6

2

2.3

7

2.3

8 O00

1.8

5

avera

ges/infe

rred

phase

(°P

ST

)

00

(35

6.0

)

35

5.8

(22

2.9

)

(22

2.9

)

223.4

(22

2.9

)

281.3

168.4

(171

.7)

188.2

(18

7.7

)

150.1

(15

0.4

)

II oo (0.0

7)

8.5

7 M O irt ^q q i ^ o

£ 0.3

5

1.1

0

(0.0

2)

0.8

5

(0.0

0)

1.2

8

(0.5

1)

1939-1976 «5 SO 0.0

299.5

342.3

14.8

304.0

356.0 tO 03 Ol tD

^ <\j © in CM ©

ro ro m ^ 111.3

277.2

171.7

29.0

327.1

187.7

229.2

318.3

150.4

175.5

II

187.4

0

1.0

7

1.0

3

1.1

1

1.0

8

8.5

6 CM 03 CO Oin in r*- r» d d cm o 0

.67

! 0.4

9

1.1

2

0.7

5

0.5

8

0.8

1

0.5

0

ISO

1.2

4

0.7

4

1958-1976

phase

(°P

ST

)

0.0

199.9

293.2

356.9

341.2

354.5

356.1

338.2

238.0

63.3

98.3

343.2

181.3

11.2 r"-

CMCO 1

90.1

213.2

307.6

164.3

180.9

II

188.2

9

1.4

3

1.1

2

1.1

1

1.8

3

9.1

3

0.7

7

1.0

4

2.6

4

0.3

4

0.6

4

S9

0

0.8

9

0.3

6

0.3

2

0.7

9

0.6

8

0.7

3

0.8

5

0.6

4

1939-1957

phase

(°P

ST

)

0.0

269.2

14.5

37.9

24

3.0

357.5

344.3

62.4

209.0

111.0

122.5

239.9

165.8

34.4

325.5

186.7

261.1

343.3

144.9

168.5

II

186.5

3

1.1

8

1.5

6

1.1

0

1.2

5

7.9

9

0.3

2

0.2

7

3.0

0

1.1

1

0.7

7

96

0

1.3

6

1.2

1 CO00O 0

.86

0.4

2

0.3

5

1.6

4

0.8

3

Constitu

ent

Doodso

n

num

bers

0 0

0 0

0 0

0 0

0 0

1 0

0 0

0 0

2 0

0 0

0 2

1 0

0 0

1 0-1-1

0 0

1 0

0-1

0 0

2-2

0 0

0 0

2.1

0

0

0 0

2 0

0 0

0 0

2 0

2 0 CM

i

o

o

Tf

o

o 0 1-2-1-1

0

0 1

0-

10

0

0 10

11

0

0 1

3-1

0-1

0 2-

2 0

0 0

0 2

-2

10

0

o

o

CM

o

O O

o

o o

o o

CM CM

o o

nam

e

? 0 . 0 . 0 . 0 N N M N

« *<B «a Vi V)co cn co 1 *E £ S S

§ 5

Table VIRoot Mean Square Residuals (cm) for Victoria

A: averages of yearly analyses for 1939^1957;

B: 1939—1957 nineteen year analysis;

C: as in A but with the same diurnal and higher frequency constituents

used in the Canadian Tide and Current Tables (1989).

Predictions based on

Year A B C

1958 16.447 16.408 16.578

1959 13.784 13.752 13.930

1960 13.634 13.591 13.783

1961 14.667 14.630 14.810

1962 12.983 12.972 13.125

1963 13.847 13.881 13.981

1964 15.813 15.850 15.943

1965 15.248 15.256 15.359

1966 16.010 16.009 16.146

1967 13.766 13.820 13.919

1968 15.831 15.864 15.982

1969 16.021 16.003 16.159

1970 16.259 16.230 16.421

1971 12.714 12.679 12.879

1972 14.250 14.218 14.440

1973 16.986 16.962 17.086

1974 14.788 14.762 14.950

1975 14.020 13.992 14.123

1976 10.542 10.474 10.672

Mean 14.611 14.598 14.752