The harmonic analysis of tidal model time series

M. G. G. Foreman and R. F. Henry

Institute of Ocean Sciences, P.O. Box 6000, Sidney, B.C., V8L 4B2, Canada

The harmonic analysis of tides is reviewed and guidelines for the analysis of tidal model time series are presented. The analysis technique introduced by Godin and circulated by Foreman is summarized. The related issues of constituent selection, time series length, confidence regions, and inference are discussed. It is shown that since numerical model time series have a much lower non-tidal content than observational time series, constituents can be separated with shorter record lengths.

I N T R O D U C T I O N

Over the centuries many theories have been proposed to explain and predict oceanic tides. The major break- through in our present understanding of tides came about with the development of Newton's 1 theory of gravitation. He not only showed that tides originate from the gravitational forces of the sun and moon acting on a rotating earth, but he also expressed the variations of these forces by means of the equilibrium tide which would result under certain idealized conditions. Laplace 2'3 made further major contributions. He derived the mathematical expression for the astronomical forces that cause tides, the tidal potential. He also formulated the equations of motion for tides on a rotating earth and solved them for the special case of an ocean completely covering the earth.

The harmonic method for tidal analysis and prediction was independently and almost simultaneously developed by Thomson (later known as Lord Kelvin) and Ferrel. (See Doodson and Warburg 4 for further details and a more complete historical review.) This method was expanded by Darwin s and further refinements were made by Harris 6 and Doodson 7. The harmonic analysis technique introduced by Godin s and presented here is based on Doodson's work.

Doodson's 7 tidal potential expression reveals that the astronomical forcing for tides can be written as a linear combination of sinusoidal terms, each having a distinct amplitude, phase, and temporal frequency. The oceanic response to this forcing may be expressed in the same manner, where each sinusoid is referred to as a tidal constituent. Although the constituent frequencies are the same as those in the tidal potential, the amplitudes and phases of the constituents can vary greatly due to the irregular coastal boundaries and bathymetry of the earth's oceans. For example, the natural frequency of the Gulf of Maine and Bay of Fundy region is very close to the frequency of constituent M 2. This causes resonance and very large tides. On the other hand, in the southern North Sea an interference of M z signals causes an amphidrome where the constituent amplitude is zero.

There are two types of ocean tides. Components of the tide that can be traced to the gravitational forcing of the sun and moon are termed astronomical. Nonlinear

Paper accepted March 1989. Discussion closes February 1990.

interactions of astronomical tides can produce secondary tidal components. These components are often referred to as compound tides or overtides. However since the nonlinear interactions often arise in regions with shallow depths, the term shallow water tides is also commonly used. Details on the generation of these additional components can be found in Godin s (pp. 154-164).

All tidal frequencies 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 - p', the mean longitude of the solar perigee.

The approximate periods for these six variables are 24.84 hours, 27 days, 1 year, 8.8 years, 18.6 years, and 21 000 years respectively. For each constituent, the integer coefficients of these 6 harmonics are called the Doodson numbers. For example, the Doodson numbers o f M 2 and K 1, the largest semi-diurnal and diurnal constituents (approximate periods of 12 and 24 hours respectively), are (2, 0, 0, 0, 0, 0) and (1, 1, 0, 0, 0, 0) respectively.

In essence, a harmonic analysis requires calculating the amplitudes and phases for a finite number of sinusoidal functions with known frequencies. The precise manner in which this is done will now be outlined.

GODIN 'S APPROACH

As there are hundreds of tidal frequencies, many so close that several years are required to separate neighbours by one cycle, and others which always have very small amplitudes, it is neither practical nor mathematically feasible to include all constituents in every analysis. Godin 8 resolved this dilemma by defining constituent clusters. Specifically he lumped together constituents with the same first three Doodson numbers and assumed that for the first stage of the analysis, each cluster can be replaced by a single sinusoid having the same frequency as the major constituent (in terms of tidal potential amplitude) in the cluster. Consequently, constituents in

"~) 1989 Computational Mechanics Publications Adv. Water Resources, 1989, Volume 12, September 109

The harmonic analysis of tidal model time series: M. G. G. Foreman and R. F. Henry

the same cluster have frequencies that differ by linear multiples of the rates of change of p, n', and p'. The major constituent gives its name to the cluster while the lesser constituents are referred to as satellites. For example, Godin's M 2 cluster contains the major constituent M2, with Doodson numbers (2, 0, 0, 0, 0, 0); and seven satellites with Doodson numbers (2, 0, 0, - 1 , 0, 0), (2, 0, 0, 0 , - 2 , 0), (2, 0, 0, 0, - 1 , 0), (2, 0, 0, 1, 0, 0), (2,0,0, 1, 1,0), (2,0,0, 2,0, 0),and (2,0,0,2, 1,0). Given a time series, Godin's harmonic analysis first calculates the amplitude and phase for the sinusoid representing each cluster. Then it accounts for the presence of the satellites and calculates the amplitude and phase of each major constituent.

With short records, it is not wise from a mathematical perspective to include all possible constituent clusters in a harmonic analysis. Although linear algebra only requires that the number of unknowns (two for each constituent cluster, and amplitude and phase) be no larger than the number of equations (one for each observation), short records and close frequencies result in an ill-conditioned matrix equation. Although such equations can be solved, the confidence intervals for the solutions are large. In particular, a small change in the observations can produce a large change in the solutions.

In order to narrow the confidence intervals, a criterion is needed to restrict the inclusion of constituent clusters in the analysis. Godin uses the Rayleigh separation equation

l a 2 - a l [ r > R (1)

where (71 and a2 are major constituent frequencies (cycles/unit time), T is the record length, and R is the Rayleigh constant. This criterion includes both constitu- ents in the analysis if the number of cycles separating al and a2 over the period T is greater than R. For observational time series, R is commonly set to 1. However it can be smaller if the non-tidal content of the signal is small. Such is generally the case with tidal model

time series. This point will be discussed in more detail later.

In addition to a criterion that restricts constituents according to the length of record, a complementary criterion is also required for deciding the order of inclusion. For example, if the Rayleigh criterion has determined that the constituents associated with frequencies al and a2 cannot both be included in the analysis, a second criterion is then needed to decide which one should be included. Godin makes this second decision on the basis of tidal potential amplitude. That is, the constituent with the larger expected amplitude is chosen first. The decision trees used by Foreman 9 for constituent inclusion in the diurnal and semi-diurnal species are shown in Tables 1 and 2. The tidal potential amplitudes in these tables do not include a latitudinal dependence factor which is constant throughout each species. At latitude ~b, the factor is sin 2q~ for diurnal constituents and cos 2 q~ for semi-diurnal constituents. See Doodson ¢ for further details. The values in the record length columns are calculated from equation (1) with R = l.

THE LEAST SQUARES ANALYSIS

Assume that a selection procedure has chosen M constituents for inclusion in the analysis. The next stage is to calculate the amplitude and phase for the sinusoids representing each constituent cluster. This is done by solving a system of linear equations. For a one dimensional time series (such as would arise from surface elevations), the analysis objective is to solve the system of equations

M

yi=Ao + ~ Ajcos(ajti-(I)j) (2) j=1

for the unknowns Aj and (:I)j, j = 1, M. A j, aj, (I)j are the amplitude, frequency and phase of constituent j; and Yi,

Table 1. Order of Diurnal Constituent Selection in the Foreman 9 harmonic analysis programs

', L ink Royleigh Comparison P a i r s ( ) T i d a l Potent ia l Amplitudes for Main Constituents

Length of Record (hr)

Required for

Constituent Inclusion

2 4

3 2 8

6 5 1

6 6 2

7 6 4

4 3 8 3

4 9 4 2

8 7 6 7

Frequency Differences ( c y c l e s / h r ) X l O 3 Between

cq _ 03 0 0 0 ~ 04 CO o,I 0 ',~ ~" 0 r,~ N re) (M ~ N o N N ~b --

-' 6 - 6 -:

ALP~ (278)

(955) (7217) 2QI - OI "\ \

(1152) (13711

;1[

(376941 01-=

/ / . ' ~

/

Neighbouring

o o o

- o 6 o 6 o 0 o 0 T A U I ~ B E T / N O I ~ C H I I ~ P I I '~ PI U S i ~ K' ~ P S I I P H I '

(53011)

_._.._--- /

\ TAU I {493)

/ / /

\ P'/ (17543

CHI I / 1567: /

I P[JI (~o28) I L

Constituents oJ P-- co 03 to ~ h- cO 0 0 C~ o o~ ~ ~

(1624) oo i

\ / t ,, ~ ( 2 9 6 4 ) / /

' , j I *

/ / sd,

(2964 N O I '

/ \ /

BETI (278)

J

(755)i /

ITHEI

SI PS I I r 5671

(416) (422}

{3,r) UPSli

110 Adv. Water Resources, 1989, Volume 12, September

Table 2.

The harmonic analysis of tidal model time series: M. G. G. Foreman and R. F. Henry

Order of Semi-Diurnal Constituent Selection in the Foreman 9 harmonic analysis programs

L i n k R a y l e i g h C o m p a r i s o n P a i r s

( ) T i d a l P o t e n t i a l A m p l i t u d e s f o r M a i n C o n s t i t u e n t s

L e n g t h of F r e q u e n c y D i f f e r e n c e s ( c y c l e s / h r ) × i O 3 B e t w e e n N e i g h b o u r i n g C o n s t i t u e n t s

R e c o r d ( h r ) P.- o~ ~- o~ ~- o ¢o = oo ¢~ ~ ~ _ r~ ~ ~ r~ t~ ,~. ~ O 0 0 tD ~ 0 0 0 ~ - g R e q u i r e d ~ o~ N o~ oJ F- Qo ,~ ,¢ ~ ao ~ ' * ,¢ '~ co

f o r o o o o o o o = : - - N 0~ - - - e~

C o n s t i t u e n t I n c l u s i o n 002 IEPS2 2N2 M R2 K2 MSN?-- I

( 13

355

662

764

' ) / / / / 4 38 3 S2 SN~

4 9 4 2 (259) (2sol) (3s02) ,

HI H 2 T 2 R 2 8767 ( s = 3 ) { 2 7 7 ) ( 2 4 7 6 ) ( 3 5 5 )

GAM'~ I I :3 2 6 ( 2 7 3 }

i = 1, N, is the observation at time t~. Each equation can be made linear in the new unknowns C~ and S t by rewriting

Ajcos(cTjti-(Yl)/)=Cjcos(ajti)+ Sjsin(~yjtl) (3)

where

A~ = (C 2 + 82) 1/2 and (I)j = arctan(SflC).

As the number of equations, N, is greater than the number of unknowns, 2 M + l , the system of equations is overdetermined and all the equations cannot be solved exactly. The solution must therefore be calculated by minimizing the equation residuals in some sense. The least squares approach, wherein the sum of the squares of the residuals is minimized, has been adopted by Godin. This approach produces the so-called normal equations which are efficiently solved with the Cholesky algorithm (see Ortega1°).

It should be mentioned that a harmonic analysis which uses the least squares approach permits the time series to be irregularly sampled, or to be regularly sampled and contain gaps. Such would not be the case if a Fourier transform were used to calculate the amplitudes and phases. However, if the time series is regularly sampled and has no gaps (as would be the case with tidal model time series), the terms in the normal equations matrix can be calculated efficiently and accurately and the matrix can be partitioned. This partitioning reduces the solution time by a factor of four. See Foreman 9 for further details.

The recommended sampling interval for the time series is one hour. In addition to convenience, such an interval permits at least two observations per period for the high frequency shallow water tides such as M4, M6, and M s. It also avoids aliasin9 wherein constituents that are sampled fewer than twice per cycle become irretrievably confused with lower frequency constituents. With one hour

sampling, the M 2 harmonic whose frequency is closest but still less than the Nyquist threshold, is m]2.

STATISTICS OF TH E LEAST SQUARES S O L U T I O N

The statistical properties of the least squares solution can be found in any analysis of variance or regression model text (e.g., Draper and Smith11). They are summarized as follows.

The system of equations given by (2) and (3) can be re-expressed in matrix form as

y = Ax + e (4)

where y, x, e are vectors of the observations, the C i and Sj coefficients, and the residuals. The normal equations that result from minimizin9 e'e are

AtAx = Aty (5)

and their solution is

x s = ( A t A ) tA'y. (6)

The total sum of squares is y'y and the sum of squares due to regression is x'sAty. Their difference is the residual error sum of squares, and this difference divided by the degrees of freedom is the residual mean square error, a2mse. With an analysis of N observations that includes M constituents and the constant term Zo, a,,~e2 is then

2 - - yty _ xts Aty a,,~e - N - 2M - 1 (7)

If it is assumed that e is distributed normally with 0 mean and 0-2I variance (where I is the unit diagonal matrix), then the variance ofxs is (AtA) - ~a z. Since ~rz~e is

Adv. Water Resources, 1989, Volume 12, September 111

The harmonic analysis of tidal model time series: M. G. G. Foreman and R. F. Henry

E 0

0 o r-

I I I I I I I

. . . . . . . . . . . . . . . . . . . . . .

Cosine Coefficient

Fig. 1. An example of confidence intervals for Cj and Sj and the associated confidence ellipse for the complex amplitude Aj e- ~,

an unbiased estimator of a 2, an estimate of the standard deviation of xi, the ith element of xs is then

ai = (/~(A'A)- l ~ a 2 m s e ) l / 2 (8)

where/~ is the vector with 1 in the ith position and O's elsewhere. Such estimates are easily calculated (c.g., in the program associated with Foreman 9) as (AtA) - 1/~ is the solution of the modified normal equations A~Ax =/~.

Confidence intervals can be computed from these estimates. For example, the 1 - ~ confidence limits for xi, the ith element in x, are

x i ++ t(v, 1 - ½oOai (9)

where al is given by equation (8) and t(v, 1 - : ) is the Student's t distribution coefficient at the 1-½~ percentile for v degrees of freedom.

When Cj and S i are combined as in equation (3), it is generally not correct to simply take the cross product of the one dimensional confidence intervals to form a confidence rectangle for the complex amplitude A~ e -i*j. Such a combination does not take into account either the covariance of C i and Sj or the covariance of these two parameters with their counterparts arising from the other constituents that were included in the least squares analysis. The correct joint 1 - ~ confidence region for all parameters in the least squares analysis is given by

(x-xs)~AtA(x-xs)<-..pa2seF(p, v, 1 - ~ ) (10)

where F ( p , v , l - s ) is the upper c¢ point of the F distribution with p parameters and v degrees of freedom. This inequality describes a p-dimensional ellipse whose shape will depend on the relative magnitudes of the entries in the covariance matrix (AtA) -1. With a harmonic analysis of regularly sampled data and a time origin at the center of the analysis period, matrix A~A can be partitioned so that the Cj and S~ parameters can be solved separately. Thus

cov(C~, S j) = 0. (11 )

Furthermore, if the analysis period is sufficiently long to resolve constituents j and i, then both cov(Cj, C~) and coy(S j, Si) will be small and the (C j, S j) confidence region will be essentially unaffected by Ci and S~. This means that the joint (C i, S j) confidence region will be an ellipse whose

major and minor axes are approximately equal to the Cj and S~ confidence intervals. Figure 1 illustrates these points.

CORRECTIONS TO THE LEAST SQUARES SOLUTION

In order to calculate the amplitude and phase of the major constituent from the representative sinusoid for each constituent cluster, at least two corrections to the least squares solutions are required.

The first is usually termed a nodal correction and it accounts for the fact that the amplitudes and phases of the reresentative sinusoid that arise from the least squares solution actually represent the cumulative effect of all constituents in the cluster. Since these constituents differ slightly in frequency, the amplitude and phase of the replacement sinusoid vary slowly in time and thus do not provide a basis for prediction at a subsequent time. In order to avoid this difficulty, the time-invariant amplitude and phase of the major constituent in each cluster are calculated from those of the replacement sinusoid.

So to summarize, the outcome of the least squares analysis is an amplitude and phase pair for each constituent cluster. The nodal correction calculation transforms this pair into an amplitude and phase for the major constituent. When a prediction is required, the reverse process occurs. That is, the major constituent amplitudes and phases are first transformed into cluster values, and these in turn are used to produce the time series.

Algebraically, the nodal correction process re-expresses the least squares result for each representative sinusoid A cos(at-@) as

f(t)a cos(at + u(t)- 0). (12)

f(t) and u(t) are the amplitude and phase correction factors that account for the presence of satellites in the cluster; and a, 0 are the amplitude and phase of the major constituent in the cluster. The nodal modulation terms f(t) and u(t) vary slowly with time and, for records up to one year in length, very little error is introduced by assuming them to be constant and equal to their value at the midpoint of the record.

With appropriate assumptions one can solve for f and u. Godin assumes that the amplitude ratios and phase differences between the satellites and the major constituent are the same as predicted by the tidal potential, and that the satellite constituents do not interfere with the major constituent. This latter assumption loses validity with analyses over periods longer than one year. Further details of these assumptions and the calculation of f and u can be found on pages 167-171 of Godin s.

It should be mentioned that with Godin's harmonic analysis, the term nodal correction is a misnomer. The phrase and the symbols f and u were first used before the advent of modern computers to designate corrections for the moon's nodal progression that were not incorporated into the astronomical argument calculations for the main constituents. However, now the term satellite modulation is more appropriate because the correction is not only for the effects of the moon's ascending node, but also for the effects of the solar and lunar perigee.

The second correction to the least squares solution is

112 Adv. Water Resources, 1989, Volume 12, September

The harmonic analysis of tidal model time series: M. G. G. Foreman and R. F. Henry

referred to as the astronomical argument correction. It simply re-expresses phase lags with respect to an absolute time and space origin. Instead of regarding each tidal constituent as the result of a particular component in the tidal potential, an artificial causal agent can be attributed to each constituent in the form of a fictitious star which travels around the equator with an angular speed equal to that of its corresponding constituent. Making use of this conceptual aid and setting t o to be a reference time and er to be the frequency, the astronomical argument of a given tidal constituent,

V(t)=•(t-to)+ V(to) (13)

can then be viewed as the angular position (longitude) of its fictitious star. For historical reasons, all such arguments or longitudes are expressed relative to the Greenwich meridian and can consequently be expressed as functions of time only. The phase lag for a particular constituent can then be viewed as the time between the overhead passage of the fictitious star and the time of maximum tide. So the contribution of each major constituent can be expressed as

a cos(a(t - to) + V(to)-9) (14)

where 9 is a phase lag that is measured with respect to an absolute time and space origin.

It should be pointed out that many types of phase lag can be used in harmonic analyses depending on whether or not one wishes to take into account the longitude and time zone of the observation site. Schureman 12 lpp. 75 78) discusses several common phase lags. The analysis programs distributed by Foreman 9 calculate G, the Greenwich epoch, if the time series is in Greenwich lime and g, the modified epoch, if the time series is in local lime. They do not employ any longitude correction. Consequently, the analysis results from neighbouring sites that were recorded in the same time zone can be compared directly. And the conversion of a phase lag from one time zone to another is straightforward. For example, if g is calculated from observations that were recorded in PST, then

G=g+8a (15)

for a constituent with frequency a measured in degrees~hour.

I N F E R E N C E

A final correction that is sometimes made to the least squares solution is referred to as inference. It involves the extraction of important constituents, excluded from the analysis because the record was too short, but deduced afterwards from known relationships with constituents that were included. If it is done properly, inference serves the dual purpose of further reducing the residual error in the least squares solution, and eliminating a periodic behaviour in the estimated amplitudes and phases of the reference constituents. For example, a 182-day time series is required to separate P~ from K 1 (the third largest and largest diurnal constituents), and K z from S z (the fourth and second largest semi-diurnal constituents) by one cycle. Without inference, a series of monthly analyses would show a periodic behaviour in the amplitudes and phases o fK 1 and S z and have substantial contributions to the residual errors due to the P1 and K 2 constituents.

With good inference parameters, that is accurate amplitude ratios and phase differences for P~/K 1 and Kz/S z, both these problems would be greatly reduced.

The question of when constituents should be included directly in the least squares analysis and when they should be inferred is not easily answered. The presumption that a record of length T is required to distinguish constituents with a frequency separation of T - ~ (i.e., equation (1) with R = 1) is incomplete. In fact it conflicts with the algebraic viewpoint whereby, for any n independent observations, one can obtain n equations and solve for n unknowns (e.g., n/2 amplitudes and n/2 phases) regardless of the frequency separation.

The missing consideration in both of these viewpoints is that the observations contain other non-tidal contribu- tions, henceforth referred to as noise, in addition to the discrete tidal signals. With actual observations, this noise may be comprised of meteorological effects, non-tidal waves, and random errors in recording the observations. With tidal model time series, the noise may be comprised of transient solutions and rounding errors. Taking these effects into account, Munk and Hasselmann 13 showed that meaningful statements can be made about a~ and a2 provided

standard deviation of noise

''laZ-allT>standard deviation of signal (16)

(Notice that this equation gives a guideline to the choice of R in equation (1).) Munk and Hasselmann refer to any resolution improvement over and above the frequency separation T-1 as super-resolution. They also show that for two neighbouring spectral lines at frequencies fo and f0 + N', the variance in the estimate of either amplitude is

3 S( f o)/(zr 2 (Af)2 T 3 ) (17)

where S(f) is the underlying power spectrum of the noise. Munk and Bullard 14 estimate S ( f ) = l cm2/cycle/day based on measurements of the noise spectrum well to one side or the other of the tidal line clusters. However when cusps in the noise spectrum are taken into account, estimates of S(f) increases dramatically to as much as 300cm2/cycle/day (Munk and Hasselmann 13) within a band of _+0.01 cycles/day around the M 2 frequency.

With S( f )= 300 cmZ/cycle/day and frequencies separ- ated by Af= 1/180 cycles/day (the approximate separ- ation between K 1 and P1, and S 2 and K 2) , the amplitude variances for each constituent arising from analyses of 30, 60, 90, and 180 days, are 109, 13.7, 4.05, and 0.51 cm 2 respectively. However with tidal model observations, both S(f) and these variance estimates will be much smaller.

Uncertainties in the analysis results can also be viewed from the perspective of matrix theory (see Ortega ~° for example). Assume B x = b and Bx '=b ' are the normal equations (see equation (5)) arising from a least squares fit for the amplitudes and phases of several tidal constituents. In particular, assume the right-hand sides b and b' are respectively calculated from observations without and with background noise. That is, b assumes a pure tidal time series whereas b' assumes tide and noise. If for some norm II'l], K(B)= [IBII lIB-11] is the condition number of matrix B, then

[Ix--x'll ~<K(B)Ilb-h'[I (18) rlxll Ilbll

Adv. Water Resources, 1989, Volume 12, September 113

The harmonic analysis of tidal model time series: M. G. G. Foreman and R. F. Henry

Table 3. Harmonic analysis results with and without inference. Amplitudes are in centimeters and phases are in degrees. Analysis type A denotes the direct inclusion of all constituents; type B denotes the direct inclusion of only Zo, K1, M2,' type C denotes the direct inclusion Zo, K1, M2, and the inference of 0 l and S 2. Analyses I ~ 4 were with the time series comprised of only tidal signals whereas analyses 5 --* 8 were with the time series comprised of tides and random noise

test hours

nO.

1 72O

2 6O

3 60

4 60

5 720

6 60

t

7 ] 60

i

8 60

, i

type statistic

A

i l l e an

A [ max

I min [ I s . d .

m e a n

B m a x

rain

s .d .

mean

C m a x

min ! ! s . d .

A

mean

A l max

[ rain

i s.d. p r m e a n

g max

min

s .d .

mean

C m a x

lllii1

s .d .

Zo O1 $2 amp amp ph amp ph

25.000 3.700 297.5 4.225 69.9

25.000 3.700 297.5 4.225 69.9

25.000 3.702 297.6 4.229 69.9

25.000 3.698 297.5 4.222 69.8

.000 .002 .03 .002 .03

25.044 I

25.118 F

24.990

.055

25.044 3.754 298.3

25.118 4.197 303.9

24•990 3.594 296.3

• 055 .169 2.13

25.074 3.832 295.7

25.070 3.530 297.3

25.370 5.143 310.1

24.665 2.291 272.2

• 244 .840 12.3

25.118

25.429

24.743

.252

4.224 69.7

4.310 70.4

4.151 69.1

.058 .42

4.280 69.6

4.487 69.3

5.388 82.7

3.157 58.3

.615 8.] F

25.118 3.681 298.0

25.429 4.052 310.3

24.743 3.350 292.4

.252 .244 4.8

K1

amp ph

5.250 179.1

5.250 179.1

5.251 179.1

5.247 179.1

.001 .02

5.901 176.8

8.515 223.6

2.308 136.6

2.285 31.3

5.327 179.9

5.955 185.4

5.099 177.9

• 240 2,13

5.138 179•0

5.005 174.9

6.121 188.3

4 ; 1 : : 161.2

• 9.9

I 5.856 176.5

!8 .722 226.8

~2.131 i 132.2 i 2.398 ! 34.2

i 5 . 2 2 3 t 1 7 9 . 6 r

i 5 . 7 5 0 i 191.9

j 4.753 , 174.0 I

i .346 4.8

amp ph

9.065 220.3

9.066 220.3

9.069 220.3

9.062 220.3

.002 .02

9.494 220.5

12.845 244.0

5.404 194.2

2.873 19.5

9.062 220.2

9.247 220.8

8.901 219.5

.125 .42

9.078 220.3

9.188 220.0

9.917 227.0

8.148 214.2

.506 4.5

9.483 220.5

13.000 245.1

5.158 193.5

2.974 19.5

9.014 220.2

9.323 221.3

8.561 218.3

.206 1.0

4.201 69.8

4.345 70.9

3.990 67.9

o96J lO

The effect of seeking amplitudes and phases corre- sponding to frequencies al and a2 that are relatively close, i.e. [a l-a2[ < T-1, is to make the appropriate rows in B more linearly dependent and increase K(B). So in the presence of substantial background noise, one can expect significant differences between the calculated set of parameters x' and their true values x. If accurate inference parameters are available, inference in such a case should yield better results, because solving for the parameters of only one frequency, either al or a2, would remove the rows that are almost linearly dependent and reduce K(B).

On the other hand, if the noise level were very small the effect of a large K(B) would be counteracted and a reasonably accurate set of parameters x' could be expected without inference. This should be the case with time series from numerical tidal models that have been run sufficiently long that the transient solutions have essentially died away. (Such models are usually said to have reached a steady state.)

Table 3 gives the results of tests designed to demonstrate these points. Two 30-day records of hourly tidal heights were simulated, one using only the

constituents Z o, 01, K 1, M2, and $2, and the other with these same constituents plus some random background noise. Time periods of 328 hours and 355 hours are required to respectively separate O 1 from K 1 and S 2 from M 2 by one cycle. (See Tables 1 and 2.) The tidal amplitudes (cm) and phases (degrees) were chosen to be (25.000,0.0), (3.70,297.5), (5.25, 179.1), (9.065,220.3), and (4.225, 69.9), for constituents Z o, O1, K1, M2, and S 2 respectively; while the noise was uniformly distributed in the range [ -1 .0 , 1.0]. Nodal corrections were not included in any of the time series syntheses or analyses. Three sets of twelve consecutive 60-hour harmonic analyses were executed: the first searching directly for all constituents; the second searching for only Z o, K l, and M2; and the third extending the second by inferring O 1 and S 2 from K 1 and m 2 respectively. (Inference parameters were calculated from the true amplitudes and phases.) In order to compare performances, means, maximum and minimum values, and the standard deviations were calculated for each amplitude and phase from the twelve analysis results in each series.

The standard deviation ratio of noise to signal (without

114 Adv. Water Resources, 1989, Volume 12, September

The harmonic analysis of tidal model time series: M. G. G. Foreman and R. F. Henry

Zo) in the second time series is approximately 0.07. So with [0" 2 --0"1[ ~ 1/355 hours in equation (16), meaningful results should be possible with analyses of only T = 60 hours.

Results from the analyses of the tidal record with no background noise (Tests 1 to 4) demonstrate a clear advantage to including all constituents directly in the least squares fit. The small non-zero standard deviations in Test 2 are wholly attributable to the fact that the data were rounded to four digits, thereby making Hb-b'll slightly larger than zero. The standard deviations for the runs with inference (Test 4) are not zero because of simplifying assumptions in the inference calculations. (See Foreman 9 for further details.) The semi-diurnal standard deviations are generally smaller than the diurnal values because they have more cycles sampled over the analysis period.

However, when random noise is included in the time series (Tests 5 to 8), the standard deviations for the inference runs (Test 8) are consistently less than those obtained by the direct inclusion of all constituents in the least squares fit (Test 6). This is a consequence of reducing K(B) (using the l~ norm) from 120.1 for Test 6 to 2.884 for Test 7.

The largest standard deviations in Tests 3 and 7 should also be noted. The presence of O1 and $2 in the time series but not in the harmonic analysis, has caused significant fluctuations in the amplitudes and phases of K 1 and M 2. However, as demonstrated by the standard deviations in Tests 4 and 8, these fluctuations can be significantly reduced through inference.

The standard deviations in test 6 can be compared to those estimated by equation (17). With the uniformly distributed noise applied to hourly values, S ( f ) = 1/72cm2/cycles/day and the standard deviation asso- ciated with the amplitude estimates are approximately 0.22 cm for O 1 and K 1, and 0.24 cm for M 2 and S 2. The larger values in Table 3 are probably due to interference between the two constituent pairs.

With reference to equation 10, it is interesting to note the covariance estimates arising with the different tests. As all the tests involved a uniformly sampled time series whose origin t o could be defined (for the purpose of the least squares analysis) at the center of the analysis period, cov(Cj, S i )=0 for all j, i. For Tests 1 and 5, B was diagonally dominant with the diagonal elements at least a factor of ten larger than the off-diagonal elements. In particular, cov(Cj, Ci)/var(Cj)~0.1 and cov(Sj, Si)/ var(Si)~0.1 within species and these ratios were much smaller between species. However for Tests 2 and 6, the covariances and variances were approximately equal in magnitude within species and slightly smaller between species. So the effect of trying to separate constituent O 1 from K 1 and S 2 from M 2 with too short a record is to increase the covariance estimates and joint confidence regions of the C and S parameters.

NORTH

Fig. 2. Current ellipse notation

current contribution for each constituent can also be represented as a vector whose tip traces out an ellipse over the period of the constituent, it is customary to re-express the amplitudes and phases for the two components in terms of four ellipse parameters. As illustrated in Fig. 2, these are: the amplitudes of the major and minor semi-axes (Ma, Mi); the angle of inclination of the northern major semi-axis, 0; and the phase lag of the maximum current behind the maximum tidal potential of the individual constituent, 9. Since an ellipse is invariant through rotations of 180 °, there is an ambiguity of 180 ° in the ellipse orientation. Godin avoids this ambiguity by requiring 0 to be in the range 0 to 180 °. This means that as 0 passes through the value 180 °, both 0 and 9 will be altered by 180 ° .

One can also consider the current contribution for each constituent as the sum of two rotating vectors, one counterclockwise with amplitude and phase a + and 9+, and the other clockwise with amplitude and phase a - and g- . For example, Crawford and Thomson a5 use g- and the ratio a+/a - to determine the shelf wave contribution of the diurnal constituents off Vancouver Island. See Foreman 16 for further details on these various current representations.

TIDAL CURRENTS

If the time series is two dimensional, such as occurs with horizontal tidal currents, an extension of the one dimensional harmonic analysis procedure can be carried out. Specifically each component (usually in the north/south and east/west directions) is analysed separately as a one dimensional time series. Since the tidal

C O M M E N T S ON TIDAL M O D E L ANALYSES

The preceding general discussion is applicable to the harmonic analysis of time series arising either from observations or numerical models. The following specific comments are intended for analyses of tidal model records. They apply directly to analyses with Foreman's 9.16 set of computer programs. In all cases, it is assumed

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The harmonic analysis of tidal model time series: M. G. G. Foreman and R. F. Henry

that the tidal model has either been forced directly with astronomical body forcing or indirectly by specifying tides, either surface elevations or currents, on some of the model boundaries. It is also assumed that the tidal model has been run sufficiently long for it to have reached an approximately steady state. This is probably best determined by producing several time series plots at representative points in the model domain.

The first comment is that it is important to maintain consistency between model forcing and the harmonic analysis. This not only means that the same constituent frequency values (i.e., to the same precision) should be used in both the model and the analysis, but also that the nature of the forcing should be consistent. Specifically, if single major constituents (rather than clusters) are used in the forcing, then nodal corrections should not be included in the analysis. On the other hand, if the model results are to be compared with observations or to predict tides, the satellite effects should be included, though their contribution is minor for many constituents. This can be done in either of two ways. The first way is to force the model with the representative sinusoids for the constituent clusters, rather than just major constituents. The second way is to use the major constituents for the forcing and analysis and adjust the harmonic analysis results so that conventional nodal modulation correc- tions are applied to account for the presence of the other constituents in the cluster.

A second important point is that the constituents to be included in the harmonic analysis should be chosen by the user and not by the analysis program. Specifically, the constituents in the analysis should be those included in the forcing and those generated in the model interior by nonlinear interactions. Constituents chosen by the Rayleigh selection criterion in the analysis program may not necessarily include all of the constituents in the model time series. For example, the simulation period may not be sufficiently long that the selection criterion includes P~. Consequently, some of the energy from the constituents that are not included in the analysis will spill over and contaminate the results of the constituents that have been included.

As discussed in Lynch and Werner 17, it is also preferable to include in the boundary forcing the significant shallow water constituents generated by the model. If this is not done, the model assumes zero boundary forcing for these constituents thereby making the interior amplitudes and phases incorrect.

It should also be mentioned that some shallow water constituents have the same (or almost the same) frequency as those of astronomical origin (see Table 4). Foreman's harmonic analysis program generally looks for the astronomical constituent, although it could be easily altered to look for the shallow water twin. If the tidal model has not been forced with the astronomical constituent but there is energy at its frequency, obviously the signal must then be due to shallow water activity. As the nodal corrections for the astronomical and shallow water constituents differ slightly, it is important to correctly identify the energy source.

Time series arising from tidal models obviously have a low non-tidal content. Consequently, as discussed in the preceding Inference section, a record long enough to separate all frequencies by one cycle may not be required. For example, although approximately 182 days are

required to separate P1 from K 1 and g 2 from S 2 by a full cycle, these constituents can be adequately separated with much shorter time series. However it is important to note that the non-tidal content in the time series is partially determined by the constituents chosen for the analysis, and these may only be a subset of the constituents in the model. For example, if the advective terms are included in a model simulation but the compound tides generated by these terms are not included in the analysis, the compound tide energy is effectively noise.

These points are now illustrated with two runs using Flather's18 Vancouver Island model. The advective terms and the forcing constituents Q1, O1, P~, K1, N2, m2, $2, and K 2 are included in the first run, but only the forcing constituents are included in the analysis. Figure 3 shows the P~ amplitude contours based on a 30-day time series analysis. (The model was run for six days prior to this analysis period in order to permit the transient solutions to decay.) The wiggliness of the contours indicates an unsatisfactory variability in the analysis results. As predicted by equation (18), a combination of too short a record and too much noise has perturbed the results from their correct values. This wiggliness decreases slightly when thirteen shallow water constituents are also included in the analysis, but the results are still unsatisfactory. However, as seen in Fig. 3, with a 60-day harmonic analysis the contours become much smoother. When the model is rerun without the advective terms, the time series are less noisy and P~ and K~ (and K: and Sz) are adequately separated with a 30-day analysis. Nevertheless, from a statistical viewpoint a longer record will reduce the confidence intervals around the solutions. So, computing resources permitting, a longer simulation is always preferable,

In summary, ifa model can be run sufficiently long that all transients have decayed below machine precision at

Table 4. Shallow water constituent and astronomical constituent twins

Water Constituenl~ Name Frequency (cycles/hr) Composition |

Q1 N~ - K1 |

Astronomical Constituent [ Shallow 1

NaYglc

.03721850 Nh-1 O1 .03873065 r~ .03895881

NO~ .04026859 PI .04155259 K1 .04178075 J1 .04329290

SO .04460268 OQ .07597495

e: .07617732 2N .07748710 #2 .07768947 N2 .07899925 72 .08030903 M~ .08051140 L: .08202355 $2 .08333333 K2 .08356149 q2 .08507364

MOa .11924206 M3 .12076710

* M1 is a satellite of NO1.

MK~ M P1 NOz SK1 M0I MQI

OQ2 MNS2

02 2MS~ KQ. O ~ K02

2MA_~ KP2 E2

K J: MOa NKa

M2 t(1 M2 P1 iV,_ Oi $2- K1 M2 - 0 1

M2 O~ & 01

M2 + N2 $2 01 + 0i

M, + M~ - S~ k-~ + Ol 0 1 + P 1 K~ + 01

M2 ~ 2142 - N~ K~ =P1 K1 + K1 K1 + ./1 M2 + 01 N2 + K1

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The harmonic analysis of tidal model time series: M. G. G. Foreman and R. F. Henry

15.0

15.0

~ b,x':

% 14.5

PACIFIC OCEAN

Fig. 3. P1 amplitude (cm) contours based on a 30-day analysis of Flather's TM model with advection

PACIFIC \ ( OCEAN ~ .

%

Fig. 4. P1 amplitude (cm) contours based on a 60-day analysis of Flather's 18 model with advection

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The harmonic analysis of tidal model time series: M. G. G. Foreman and R. F. Henry

128

5 0

0 v

I.U

J

..-I

Vancouver

P A C I F I C

O C E A N

4 7 I I I

(a) 128 LONGITUDE (ow)

1 2 3

I

123

50

47

O,)

Fig. 5. (a) Freeland's 21 CODE-II current meter sites. Barotropic K 1 (b), and M 2 (c) current ellipses at the CODE-II sites. The largest semi-major axis is 15.3 cm/s for M 2. Lines within each ellipse denote g. Arrows denote the direction of rotation around the ellipse. Observed - - , model - - -

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The harmonic analysis of tidal model time series: M. G. G. Foreman and R. F. Henry

(c)

i t t )

the beginning of the analysis period, and the analysis period itself is long enough to resolve constituents such as Px and K 1, then including all constituents in the analysis is the best strategy. However, it may either be difficult to determine the time required for this decay, or it may not be economically feasible to run a model this long. So the noise may not be as small as one would hope, and inference may be required to resolve closely separated constituents. The numerical tests whose results are shown in Table 3 suggest that neighbouring constituents can be adequately separated with R ~> 1/6 in equation (1) when the noise level is less than 0.5 × 10 - 4 for each observation. This translates to a 30-day period for separating P1 and K 1. The Flather ~8 model tests with no advection confirm this value.

If computer storage is at a premium during a model run, the information to be retained for a subsequent harmonic analysis can be reduced significantly if at each model grid point, the right hand sides for the normal equations, rather than hourly time series, are stored. For example, if a model with 500 grid points is run for 60 days, storage of all hourly time series requires 720 000 words per variable. If on the other hand, the mean and 15 constituents are to be included in the harmonic analysis, storage of the right hand sides requires only 15 500 words. (Each right hand side entry is the summation of a series of products of a sinusoid with a grid point value. See Foreman 9 for further details.) However a major disadvantage with storing right hand sides is that the constituents to be included in the analysis must be chosen prior to running the model. In other words, the harmonic analysis cannot be repeated with additional constituents unless the model run is also repeated. So when calculating right hand sides during a model execution, one must anticipate all the constituents that could be present in the model time series.

In comparing model time series with observations there are several points of caution. Obviously it is important to ensure that the two time series contain the same constituents. As demonstrated by Lynch and Werner 17, the presence of shallow water constituents generated in

the model interior but not included in the forcing can disrupt a comparison. A safer technique is to compare amplitudes and phases for individual constituents.

Special mention is also warranted for comparing model and observational currents. Obviously if the model is two dimensional (i.e. barotropic), the current observations should also be averaged in some sense. This requires simultaneous observations at several depths. If baroclinic processes are suspected in the observations, some attempt should be made to remove them before comparing the observations with barotropic model results. For example, using baroclinic theory such as presented by Phillips 19, it is sometimes possible to fit the observations with the barotropic and one or more baroclinic mode shape functions.

As with one dimensional tidal signals, it is generally preferable with tidal currents to compare model results and observations on a constituent by constituent basis. Figures 5b and 5c show K 1 and M z current ellipse comparisons for an array of six current meter sites in the CODE-II experiment off the west coast of Vancouver Island. (Figure 5a shows the locations of these sites.) The vertically-averaged model currents were obtained with the Walters 2° finite element method and the barotropic currents were calculated by forming weighted averages of observations (courtesy of Freeland 2~) at a minimum of two depths. The lines within each ellipse are the phase lag g, measured in the same manner as shown in Fig. 2, and the arrows denote the direction of rotation around the ellipse. The K~ model ellipses are too thin because the barotropic model cannot accurately reproduce the clockwise rotary component of the current that is largely due to a baroclinic shelf wave. Although the model ellipses for M 2 are more accurate, the discrepancies are still not acceptable. As the moorings are near the edge of a canyon, it is likely that the model grid requires a refinement in order to more accurately represent the topography and the correct tidal behaviour. (See Foreman z2 for further details.)

When comparing model results and observations on a constituent by constituent basis, one may either compare

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The harmonic cnalysis o f tidal model time series: M. G. G. Foreman and R. F. Henry

Table 5. Combined standard deviations (cm) for the harmonic analysis results with and without inference. Types A, B, C and test numbers 2 --* 4 and 6 --* 8 are as in Table 3

teat

2

3

4

6

7

8

houra type Zo 01 K1 M2 $2

60 A .000 .002 .002 .003 .003

60 B .05 3.65 4.18

60 C .05 .22 .31 .14 .07

60 A .24 1.14 1.02 .89 .88 /

60 B .25 3.82 4.23

I 60 C .25 .40 .56 .26 .12

amplitudes with amplitudes and phases with phases (as is illustrated in Table 3), or form a complex-valued ampli tude and compare both simultaneously. Each approach has its advantages. Whereas the first approach permits separate measures of the magni tude and timing accuracy, the second approach condenses the error to one value, the distance between two complex numbers. When average amplitudes and phases are computed from a series of analyses (as in Table 3), it is safer to do the calculation in complex form. This avoids any potential difficulties due to the phase discontinuity at 360 °. Table 5 shows the complex ampli tude s tandard deviations associated with the harmonic analysis results shown in Table 3.

As confidence intervals or ellipses can be computed during harmonic analyses (e.g., Fig. 1), it is logical not only to calculate the difference between model results and observations, hut also to determine whether or not each model result lies within a part icular confidence region a round the observation. As discussed previously, due to a combina t ion of high background noise and a short observat ion period, these confidence regions may be large. So even though a model prediction may not seem to be close to an observation, it may in fact be well within an acceptable confidence region.

Finally, it should also be ment ioned that when using Fo reman ' s p rograms for analysing model time series, the input routines should be changed to acept more than 3 or 4 digits for each observation. Round ing to this number of digits creates a slight non-t idal noise and may thus affect the length of record required to separate close constituents. Rounding will also affect the accuracy of the analysis result. Fo r example, an analysis of the time series at forced bounda ry points will not return exactly the same amplitudes and phases as were used in the forcing. This is illustrated by the Test 2 results in Table 3.

SUMMARY

This paper has briefly reviewed the deve lopment of the harmonic analysis of tides and described the particular harmonic analysis technique in t roduced by God in 8 and circulated by Fo reman 9. The related issues of consti tuent selection, time series length, confidence regions, and inference were discussed. It was also shown that since numerical model time series have a much lower non-t idal content than observational time series, constituents can be separated with shorter records. Finally several guidelines for the analysis of tidal model time series were discussed.

A C K N O W L E D G E M E N T S

We thank Gabriel God in for helpful discussions; Roger Flather and Roy Walters for the use of their model codes; Patricia Kimber for drafting the figures; and the reviewers for helpful comments .

REFERENCES

1 Newton, I. Philosophia Naturalis Principia Mathematica, 1687 2 Laplace, P. S. Recherches sur plusieurs points du syst6me du

monde, M~moires de rAcadkmie royale des Sciences, 1775, 88, 75-182

3 Laplace, P. S. Recherches sur plusieurs points du syst6me du monde, Mdmoires de rAcadbmie royale des Sciences, 1776, 89, 177-267

4 Doodson, A. T. and Warburg, H. D. Admiralty Manual of Tides, His Majesty's Stationery Office, London, 1941

5 Darwin, G. H. Reports of a Committee ,for the Harmonic Analysis of Tides, British Association for the Advancement of Science, 1883 86

6 Harris, R. A. Manual of Tides, Appendices to Reports of the U.S. Coast and Geodetic Survey, 1897-1907

7 Doodson, A. T. The Harmonic Development of the Tide- Generating Potential, Proc. Roy. Soc. Series A, 1921, 100, 306 323. Re-issued in the International Hydrographic Review, May 1954

8 Godin, G. The Analyis of Tides, University of Toronto Press, 1972

9 Foreman, M. G. G. Manual Jot Tidal Heiyhts Analysis and Prediction, Pacific Marine Science Report 77-10, Institute of Ocean Sciences, Patricia Bay, Sidney, B.C., 1977, Unpublished manuscript

10 Ortega, J. M. Numerical Analysis A Seeond Course, Academic Press, New York, 1972, 193 pp.

11 Draper, N. R. and Smith, H. Applied Re,qression Analysis, John Wiley & Sons, New York, 1966

12 Schureman, P. Manual of Harmonic Analysis and Prediction of Tides, U.S. Department of Commerce Special Publication No. 98, Washington, 1958

13 Munk, W. H. and Hasselmann, K. Super-resolution of Tides, Studies on Oceanography (Hikada volume), Tokyo, 1964, 339-344

14 Munk, W. H. and Bullard, E. C. Patching the long-wave spectrum across the tides, Journal qf Geophysical Research, 1963, 68(12), 3627-3634

15 Crawford, W. R. and Thomson, R. E. Diurnal Period Shelf Waves along Vancouver Island: A Comparison of Observations with Theoretical Models, Journal of Physical Oceanoyraphy, 1984, 14(10), 1629 1646

16 Foreman, M. G. G. Manual jor Tidal Currents Analysis and Prediction, Pacific Marine Science Report 78-6, Institute of Ocean Sciences, Patricia Bay, Sidney, B.C., 1978, Unpublished manuscript

17 Lynch, D. R. and Werner, F. E. Long term simulation and harmonic analysis of North Sea/English Channel tides. In: Developments in Water Sciences, 35. Computational Methods in Water Resources. Volume I: Modeling Surface and Sub-Surface Flows. Proceedings of the Vllth International Conference, MIT, Boston, USA, June 1988. M. A. Celia et al., editors, Computational Mechanics Publications, Elsevier, 1988, 257 266

18 Flather, R. A. A Numerical Model Investigation of Tides and Diurnal-Period Continental Shelf Waves along Vancouver Island, Journal of Physical Oceanography, 1988, 18(1), 115-139

19 Phillips, O. M. The Dynamics of the Upper Ocean, 2nd ed., Cambridge University Press, London, 1977

20 Waiters, R. A. A Model for Tides and Currents in the English Channel and Southern North Sea, Advances in Water Resources, 1987, 10, 138-148

21 Freeland, H. J. Derived Lagrangian Statistics on the Vancouver Island Continental Shelf and Implications for Salmon Migration, Atmosphere-Ocean, 1988, 26(2), 267 281

22 Foreman, M. G. G. Two Finite Element Tidal Models for the Southwest Coast of Vancouver Island, submitted to Proceedings of the International Conference on Tidal Hydrodynamics, Gaithersburg, Maryland, Nov. 14~18, 1988

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