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Some results and comments on the Tidal Flow Forum exercise
Some results and c o m m e n t s on the Tidal Flow Forum exercise Bruno M. Jamart and J o s 6 0 z e r
M U M M (Management Unit o f the Mathematical Models o f the North Sea and the Scheldt Estuary), Gulledelle 100, 1200 Brussels, Belgium Two approaches are followed to estimate the fit between 'observations' and the results obtained with different numerical models within the frame of the Tidal Flow F o r u m exercise. The first approach (i.e. the computation of the RMS errors on the timeseries for a specific day) is shown to be insufficient because the RMS error computed over a short period of time is strongly dependent on the selected test period. Such RMS values are significant if and only if they are computed over a sufficiently long period of time, T. Hence, the model validation must be done on the basis of the results of long term simulations. The limit of the RMS values as T becomes arbitrarily large is related to the m a x i m u m error made on each constituent. Those m a x i m u m errors are easily calculated from the harmonic constants (computed versus observed) of the constituents. The results of the calculation of the m a x i m u m errors indicate that the ME, $2, N2 an d M4 components are the largest contributors to the overall error for the area under consideration. On the basis of m a x i m u m errors, the T F F results obtained so far are almost of the same order of accuracy. The errors affecting the TFF results are not much larger than those affecting most reported results of similar models used in similar applications 1. I N T R O D U C T I O N The Tidal Flow F o r u m (henceforth TFF) is an exercise of model assessment and comparison launched several years ago under the leadership of Gray, Le Provost, Lynch and V e r b o o m (in alphabetical order). The aim of the exercise was to set up a realistic, challenging and well-documented problem in tidal hydrodynamics which could serve as a benchmark or test problem for numerical models. The selected problem consists in computing the motion due to tides in the English Channel and in the southern part of the North Sea. This area has been extensively surveyed and modelled, and the tidal constituents are known to display here rather complex patterns (e.g., Charbert d'Hi~res and Le Provost2). A gridded bathymetric data set and the harmonic constants of 11 constituents of the vertical tide (i.e., sea surface elevation) along the two open boundaries constitute the first part of the database of the TFF. The second part of that database are the same harmonic constants at 11 coastal checkpoints and short timeseries of observed current velocities at various locations inside the domain of interest. The most recent version of the T F F database has been circulated by Werner and Lynch 2°. A detailed description of the reference computation, as well as a number of results, are given by Werner and Lynch x9, Gray et al. 5, and Waiters 17. By and large, the numerical models used to date on the TFF problem are two-dimensional (vertically integrated) models solving the classical shallow water wave equations. In a first round, all the models resorted
Accepted March 1989. Discussion closes June 1990.
to the finite element method (FEM) to handle the spatial discretization (see Advances in Water Resources 10). In the second round of discussions these FEM were extended*'l°'lS'2~; in addition, several entries based on the finite difference technique (FDM) both in time and space were also presented ~1'12A4'22'23. Most of the papers related to the TFF exercise describe the application of a specific model to the problem at hand (equations, numerical procedure, results) and conclude with a comparison between the model results and the sacrosanct 'observations'. The present contribution does not follow that outline because its purposes are: i) to compare the results of several different models and approaches; ii) to discuss the methods used to evaluate the results of numerical tidal models. Concerning our own model (briefly described in Ozer and J a m a r t 11'12) suffice to say that it solves the same governing equations as all other entries, using the same forcing and parameters, with an F D M / A D I (semiimplicit) techniquet. We have elected not to show the conventional sequence of timeseries o f elevation and velocity for the 17th of March, 1976: like those of most other models applied to the T F F problem, our results go up and down smoothly (these are tidal signals, for heavens' sake) and they agree 'reasonably well' with the 'observations'. In the next section, we present an overview of the various models included in our comparison, with an indication of the type of results available to us at the time of writing this. Section 3 is devoted to a discussion of the results in terms of RMS error, an estimator which we have used previously in a more restricted
© 1989 Computational Mechanics Publications
Adv. Water Resources, 1989, Volume 12, December 211
Results and comments on Tidal Flow Forum exercise: B. M. Jamart and J. Ozer
comparison 12 and which we now deem an insufficient and, to some extent, misleading indicator of the quality of a model. Another approach to the evaluation of the results, in the frequency domain, is discussed in section 4. Since the latter approach, which appears more suitable, can only be performed on spectral results, and since only short-term velocity observations are available within the TFF, the evaluation presented here is restricted to sea-surface elevation results. Conclusions are presented in section 5.
free of the deleterious effects linked to the 'cold start' procedure. Note that this simulation corresponds exactly to the prescriptions of the reference computation of the TFF exercise. Their second simulation 1° henceforth WEQN88, is performed over a period of time sufficiently long for the 11 TFF constituents to be retrievable by harmonic analysis of the resulting timeseries. When dealing with the results of WEQN88 for the 17th of March 1976, the model signal is reconstructed on the basis of the 11 TFF constituents and only those. A more exhaustive exposition of these results and attendant discussion appears in Werner and Lynch 21. The last four entries in Table 1 correspond to FDM computations. The three models use essentially the same finite difference grid, although some slight discrepancies have been noted between the various implementations. For example, the coastline geometry is not exactly the same in all three models and the interpolation procedure applied at the open boundaries also differs slightly. This is rather unfortunate, since it is then impossible to unequivocally ascribe discrepancies between model results to the differences in the numerical algorithms. We shall consider only those results of the W A Q U A and FADI models which have been obtained with the TFF parameters, and in both cases we have been provided with timeseries of the results for the 17th of March 1976. With MUMM's model, we have carried out both a short (MU87) and a long (MU89) integration. The results of the latter have been analyzed using the software package developed by Foreman 3. Since the inception of the TFF exercise, two somewhat different sets of bathymetric data have been used by the various groups. The reasons for dealing with an 'original' and a 'revised' bathymetry are discussed by Werner and Lynch 2°. These datasets are referred to as versions 1 and 2, respectively, in Table 1.
2. M O D E L RESULTS I N C L U D E D IN T H E COMPARISON We have at our disposal the results of 5 of the models involved in the TFF exercise. Table 1 contains a summary of the models and simulations considered herein, all of which use the TFF parameters. For some models, more than one set of results are available. In addition to the simulations listed in Table 1, several results have been reported in order to show that some amount of 'tuning' (i.e., modification of the basic TFF parameters) could improve the apparent fit of the model results to the TFF reference data. We do not include such 'calibrated' results in our comparison, for reasons which will be discussed later. The first three simulations listed in Table 1 are performed with models of the FEM-type. These two models use exactly the same computational grid but they differ with respect to the handling of the time variable. Walters 17 uses the spectral method pioneered by Pearson and Winter 13 and Le Provost and Poncet 7, wherein a spectral or harmonic decomposition is performed on the governing equations, yielding a system of coupled elliptic pde's which are solved iteratively. Waiters' computation involves the 11 constituents introduced as boundary forcing and only those. Werner and Lynch, on the other hand, apply a timestepping procedure to the wave equation resulting from manipulations of the momentum and continuity equations (see, e.g., Lynch and Gray, 9 Kinnmark6). In the first simulation performed by Werner and Lynch, referred to as WEQN87 in Table 1, the integration is carried out for 72 hours, starting with the sea at rest at 0:00 G M T on March 15, 1976. The results obtained for the last day of integration (March 17, 1976) are presumed
3. E V A L U A T I O N OF T H E MODELS IN T E R M S OF R M S E R R O R S 3.1. Results
In the original setup of the TFF exercise, the model responses are to be compared to observations for one specific period of time, namely the 17th of March 1976.
Table 1. Models and simulations considered by this paper Reference
Spatial discretization
Time discretization
Length of simulation
Bathymetry
Results available to us
Wemerand Lynch (1987)
FEM
Time-stepping ("wave equation")
3 days
VA
March 17,'1'976
Lynch and Wemer (1988) Wemer and Lynch (1988)
FEM
Time-stepping ("wave equation")
190 days
V.I+2
WALTE~S
w*Iters (19s7)
FEM
Spectral approach
DNA
V.I
WAQUA FADI
Praagman et'al. (1989)
FDM
ADI
3 days
Ya et al. (1989)
FDM
Falsified ADI
3 days
MU87
Ozer and Jamart (1987, 1988) this paper
V.2 V.2 V.l+2
Abbreviation
WEQN87
I
• WEQN88
11 TFF constituents 11 TFF constituents March 17, 1976 1,1
i
MU89
212
i
FDM
days ,
FDM
Adv. Water Resources, 1989, Volume 12, December
ADI
190days
,i ,
,,
March 17.' i976
March 17, 1976 ,,
V. 1+2
96 constituents
Results and comments on Tidal Flow Forum exercise." B. M. Jamart and J. Ozer Hence, most papers dealing with the T F F problem display timeseries of computed versus observed sea surface elevation at the coastal reference stations. Such graphs are not convenient for the purposes of comparing models between themselves. Therefore, in an attempt to present a general overview of the abilities of the models, we have introduced two presumably 'global' estimators easily derived from the one-day timeseries 11'1z. The first estimator is the m a x i m u m absolute error, i.e., the m a x i m u m difference between observed and computed elevations at each station for March 17, 1976. The second estimator is the root mean square error (RMS) over that 24 hour period. The first estimator, i.e., the m a x i m u m absolute error, shows that the discrepancies between observations and model results are not small (e.g., 113 cm at Boulogne for MU87; 115 cm at St. Malo for WEQN87; 84 and 96 cm at Christchurch, for WEQN88 and Walters, respectively). Our subjective opinion lz is that such errors are 'large', i.e., well above some as yet ill-defined standard that could be qualified as state-of-the-art, most likely in the order of 5 to 10 cm. A second remark concerning the m a x i m u m errors is that they are approximately twice as large as the RMS errors and provide no additional insight, so that the latter, viewed as a more general measure of fit, might be better suited for comparing model results. Table 2 contains the one-day RMS errors of the five models for which we have results calculated with the original bathymetry. Except for the MU89 model, these results have been published and discussed previously (Ozer and J a m a r t 12); they are included here for the sake of completeness. The main conclusion f r o m that earlier comparison, which is confirmed by the results of MU89, can be re-stated as follows: •
model results are compared to a so-called 'observed' or 'reference' signal which is in fact a harmonic recomposition based on a subset of the total tidal spectrum at each station; the constituents included in the reconstructed signal are the same as those used as boundary forcing, and only those; hence, the observations should be considered as filtered; • against the benchmark so-defined, the models for which a similarly filtered response is considered for the evaluation do p e r f o r m better (i.e., they have a smaller mean RMS error) than the models for which only a raw or unfiltered response is evaluated; note that at some stations, the unfiltered response (corresponding to what we called a 'time domain simulation') is actually better than the filtered result (referred to as 'spectral simulation' in our previous paper; this designation will not be used further to avoid confusion). This conclusion is further discussed by Werner and L y n c h 21.
The RMS errors for the 17th of March 1976, calculated for the models that have used the revised bathymetry, are listed in Table 3. These results support the conclusion that filtered model results compare generally better to filtered observations than do unfiltered simulation results. Table 3 shows that all the F D M models yield smaller RMS errors along the British coast (last 4 stations) than on the continental side, a point discussed by Yu et al. 23. The standard deviations
Table 2. Compar&on between model results and the TFF database: values o f the root mean square error, in cm, f o r the 17th o f March 1976. Model simulations performed using the original bathymetric data set
~rATION
Z~B~UC,aE HOEKVANHOLLAND LOWESrOrr
MIJ~ 20.4 13.8 44.2 61.4 61.8 36.6 21.2 12.1
WALTON
24.4
14.9
47.0
36.4
50.6
oovEn
Mean
31.4 52.2 34.5
40.9 37.4 26.8
33.6 56.5 36.0
15.6 48.2 22.2
22.5 56.6 27.5
Standard deviation
18.1
16.5
16.4
12.5
14.7
st. ~,~,o
CUEe,nOURG OtEt't'E nOUtX~NE CaZatS
CHRtSTCHURCH
MI.~9 WEQN$7 WEQNU WALTERS 24.9 47.1 15.0 19.4 6.8 12.7 13.3 13.7 16.2 54.1 17.3 25.1 63.4 41.6 15.6 32.5 39.4 46.1 37.7 34.7 18.3 30.3 9.4 21.3 15.6 14.8 21.0 12.8 17.6 12.8 15.1 13.8
Table 3. Comparison between model results and the TFF database." values o f the root mean square error, in cm, f o r the 17th o f March 1976. Model simulations performed using the revised bathymetric data set. (Unpublished results f o r WEQN88 courtesy o f F, Werner) STATION
MU~/
MU~
FADI
WAQUA
WF.~N~
~ . M,u,o
64.2 30.7
40.5 27.3
75.8 26.0
103.4 30.2
15.7 17.7
otem'~
69.6
46.2
80.0
63.3
22.8
SOULOGNt"
54.6
32.0
64.4
50.5
20.1
cm.ats ZEEnnOC,Gt~
63.6 46.3
43.3 26.3
57.4 51.5
57.2 40.2
36.6 14.7
HOEK VAN HOLLAND
29.4
20.5
26.6
17.8
16.1
LOW£STOFr
14.0
17.5
I 1.4
13.7
8.1
waLrou
26.5
14.0
20.3
27.5
36.1
DOV~
20.2
20.6
27.3
22.5
16.9
cnm~cnut¢ctt Mean
17.8
8.9
21,7
20.3
15.7
39.7
27.0
42.0
40.6
20.0
Standard deviation
19.5
11.7
23.2
25.4
8.8
CUEnSOttRC
on the mean of the local RMS errors are comparable to those of Table 2, i.e., about 50% of the mean value. The revised bathymetry was introduced with the hope that some of the problems affecting the original dataset might have contributed to the relative inaccuracies of the model results. Hence, better model results were expected. I f the RMS errors are used to evaluate the quality of the fit, Table 3 shows that the revised bathymetry does not yield better results, on the average, than the first TFF database. Note also that the results o f the FADI- and MU-models listed in Table 3 are obtained by solving the equations in Cartesian rather than spherical coordinates, although the bathymetry appears to be defined on a spherical grid. As remarked by Yu et al. 23, the RMS errors are larger when using the presumably more proper spherical code. A puzzling conclusion emerging from Table 3 is the fact that the F D M models, which all use basically the same grid and solve the same equations, yield results which appear to be substantially different. We feel that the comparison of those three models should be done through a more tightly defined procedure and that the remaining discrepancies, if any, would deserve further investigation. Similarly, it appears from Table 2 that comparable results of the FEM models (i.e., WEQN88 and Walters)
Adv. Water Resources, 1989, Volume 12, December
213
Results a n d c o m m e n t s on Tidal F l o w F o r u m exercise: B. M. Jamart and J. Ozer
can be locally quite different from each other. Some of those differences can be explained by the numerical handling of particular aspects of the equations and/or the boundary conditions (Waiters and Werner, pers. communication). Such explanations seem worth a detailed discussion.
the model result, A(t), is then given by M
A ( t ) = ~] fk Ak COS(c0gt+ [ V o + U ] k - - { 3 k ) where, for 1 ~ < k ~ < N = 1 1 , A k = { ( aok
COS
In retrospect, it is legitimate to ask whether the above conclusion that filtered model results compare globally better to filtered observations than unfiltered model results might not have been foregone. It is also legitimate to inquire whether that conclusion hinges on the evaluation method (RMS error for a specific day), and, more generally, whether the TFF exercise is adequately formulated. A tidal signal, either observed or calculated in a numerical model, can always be viewed as a sum of harmonic constituents. The number and the nature of such constituents that can be calculated from a given timeseries depend mostly on the length of the record and on the location (e.g., open ocean versus shallow coastal water) where the signal is recorded. Consider, for instance, a timeseries of the computed tidal elevation at one of the coastal checkpoints of the TFF problem. Let us assume that the series is long enough that all important constituents can be analyzed by a conventional harmonic decomposition, i.e., that the residual (difference between the original signal and its series representation) is small. Those 'important' constituents include those that are specified at the open boundaries of the model and, usually, additional harmonics resulting from the nonlinear dynamics of the system. The model result, r/c(t), can then be written as M
A~k cos(wkt- rbCk)
(1)
k=O
with Ack
fkack
=
q~c~ = ¢c~ -
[ Vo + U ] k
O~o= ¢bco = 0
where the ak's and ek's are the harmonic constants, the fck's are factors used to reduce the mean amplitudes to the year of analysis, and Vo + u is the value of the equilibrium argument of a particular constituent when t=0. In order to evaluate the quality of such a model result, one needs to have an equivalent series representation of the observed signal at the same location. In the TFF exercise, however, this is not the case: the reference provided by the TFF database can only be reconstructed on the basis of 11 = N < M constituents. Defining the sequence of the OJk'S in (1) in such a way that wl, ~oz.... ~ t correspond to the frequencies of the TFF constituents, the so-called 'observed' signal can be written as N
Oo(t) = ~
Aok COS(Wkt-- q~ok)
(2)
k=l
with Aok
=
fkaok
• ok = Oo~ - [ Vo + u / k
The difference between the truncated observation and
214
(9ok-- ack
COS
Ock ) 2
+ (aok sin Ook -- ack sin ~¢k) 2 } 1/z
3.2. Analysis o f the evaluation m e t h o d
rl¢(t) = ~
(3)
k=0
Adv. W a t e r Resources, 1989, Volume 12, December
&
atan
[ aok sin ¢ok - ack sin ¢ck ] / Laok cos ¢ o k - a~g cos ¢¢kJ /
and, for all other values of k, A k = -- ack t3k = Ock
The RMS error, calculated over some arbitrary period of time T starting at t = to, is defined as RMS(to, T) =
z~2(t) dt
(4)
to
and one can easily show (see Appendix) that the square of the RMS error consists of three terms: i) the first term is equal to M
1 Z f2 A2; 2 k=o it is independent of T, and it varies very slowly as a function of to (through the time variation of the nodal correction factors, fk); we shall refer to this term as the long term RMS error; ii) the second and third terms have a more complicated algebraic expression; they are functions of both to and T due to the fact that T is not a common multiple of the periods of all constituents; the sum of these terms will be called the time-dependent part of the RMS error. Note that as T becomes arbitrarily large, this contribution to the error vanishes. Two obvious conclusions follow from these equations. First, if equation (1) representing the full model result is replaced by a truncated series consisting only of the same constituents as those included in (2), the long term RMS error will be smaller. Hence, in a time-average sense, 'filtered' model results can only look better than 'unfiltered' ones when compared to the truncated observation. This artefact is a mere consequence of insufficient information on the prototype. Second, since the RMS error varies with both T and to, it appears unwise to evaluate the TFF model results on the basis of the fit caculated for one single day of simulation. To make these points clear, we have calculated the 24 hour RMS error on both the filtered and the unfiltered results of the MU89 simulation. Sample timeseries of such daily RMS errors are displayed in Figs 1 (St. Malo), 2 (Dover), and 3 (Zeebrugge). These figures show that the (time-) mean values of the daily RMS errors, i.e., the long term RMS errors defined above, are smaller for filtered than for unfiltered results. They also show that the time-dependent part of the RMS error, for T fixed and equal to 24 hours, is quite significant. At St. Malo, for example, there are times when the unfiltered model result looks better than the filtered one. More importantly, the subjective evaluation of the
Results and comments on Tidal Flow Forum exercise." B. M. Jamart and J. Ozer
ENGLISH Station
CHANNEL
AND
SOUTHERN
NORTH
SEA
PROBLEM
: Saint-Malo
cm)
RMS
70
60
50
40 ~
si
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Note
: starting
]. . . . . . . .
results
date
:
1'5/03/76
at
O h
. filtered
(day)
remu!tl
GMT.
Fig. 1. Timeseries of RMS errors (T= 24 hours) at Saint-Malo computed from the results of the MU89 (V.2) simulation. The solid line shows the RMS errors of the unfiltered results; the dashed line corresponds to the filtered results. The horizontal solid (dashed) line indicates the time-averaged value of the RMS for the unfiltered (filtered) results. quality of the various model results, with respect to some standard acceptable level of error, can obviously not be based on the RMS error over a single day: for these three sample stations, the results of MU89 look much better about 10 days after March 17, 1976, than they do on the arbitrarily selected quiz day of the TFF exercise. A corollary of the inadequacy of the RMS error calculated for a specific day as an estimator of model quality is that attempts to calibrate the TFF database on that basis are pointless. Both the evaluation and the calibration procedures have to he performed either in the frequency domain or on the basis of long simulations.
4. E V A L U A T I O N OF T H E M O D E L S IN T H E FREQUENCY DOMAIN
A general expression for the difference between observations and model results is given by equation (3). In the TFF exercise, the number of observed constituents, N, is smaller than the number M of harmonics retrieved from the long-term (180 days) simulations. Consider first the constituents which are both present in the calculated signal (equation (1)) and available from the TFF database (equation (2)). The reliability of the latter is somewhat questionable (see Ozer and
Jamart, 11'12) and will be further discussed in a subsequent paper. For the time being, we shall assume that the observed amplitudes and phases of the signal are heaven-sent, error-free constants. Equation (3) indicates that the contribution of the k-th constituent to the inaccuracy of the numerical result is fkAk COS(~0kt+ [Vo + U]k- ~k). The maximum error due to the k-th constituent is thus fkAe, a simple function of the observed and calculated amplitudes and phases, whose time-dependence is almost entirely contained in the fk factor. The term Ak is essentially space- and modedependent. A summary of the maximum error (more precisely, the Ak factor) for the 11 TFF modes at the eleven coastal checkpoints is presented in Figs 4 to 6 for various model simulations (Waiters, WEQN88, MU89). We conclude from the evaluation of the models on the basis of maximum error per constituent that:
i) the three models yield answers of comparable global (in)accuracy, although the details of the spatial distributions of those maximum errors are different between the models; ii) the largest absolute errors, in all models, are linked to the first three semi-diurnal constituents (M2, $2, Nz) and to the M4 component; hence, these are the constituents which need to be improved upon first in order to increase the overall quality of the simulation results.
Adv. Water Resources, 1989, Volume 12, December 215
ENGLISH
CHANNEL
AND
SOUTHERN
NORTH
SEA
PROBLEM
: Dover
Station RMS
(cm)
35-
30
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Fig. 2. Same as Fig. 1 at the station Dover
ENGLfSH Station RMS
CHANNEL AND S O U T H E R N NORTH SEA PROBLEM
: Zeebrugge
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(cm)
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date
.
I: - , - - -
.
: 15/03/76
at
0
Fig. 3. Same as Fig. 1 at the station Zeebrugge
216
Adv. Water Resources, 1989, Volume 12, December
h
. . ._ ._ ,.f .i l.t e.r.e.d.
GMT.
results
,
....
(day)
ENGLISH Waiters'
CHANNEL
AND
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NORTH
SEA
PROBLEM
model
Maximum
error
(cm)
50-
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Constituent
number
Fig. 4. M a x i m u m error (Ak) computed f o r the 11 TFF constituents at the 11 coastal stations f r o m the results o f Waiters" model. The 11 constituents are ordered as follows: 01, Kl, M2, $2, N2, 1£2, M4, MS4, MN4, M6, 2MS6
ENGLISH
CHANNEL
Lynch
Werner's
and
Maximum
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AND
SOUTHERN
NORTH
SEA
PROBLEM
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(ore)
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+ ÷
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Fig. 5. Same as Fig. 4 f o r the results o f the WEQN88 (V. 1 + V.2) simulations. The A k "s calculated f r o m the WEQN88 results with the original bathymetry are shifted to the left with respect to the constituent number; the values f o r WEQN88, revised bathymetry, are shifted to the right
Adv. Water Resources, 1989, Volume 12, December
21]"
Results and comments on Tidal Flow Forum exercise: B. M. Jamart and J. Ozer
ENGLISH
CHANNEL
AND
SOUTHERN
NORTH
SEA
PROBLEM
MU-rnodel Maximum 50
error
(cm)
40
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+
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number
u u .8. 9. . . . ( v a )
Fig. 6. Same as Fig. 5 for the MU89 (V.1 and V.2 bathymetries) simulations. It is also of interest to compare the errors obtained within the TFF exercise to those reported for similar models used in similar applications. We have performed such a comparison between various North Sea and/or English Channel numerical models, and we have considered only the dominant (M2) tidal constituent. The results of that comparison are summarized in Table 4, from which it appears that the errors affecting the TFF results are, in fact, not unreasonably large. It is important to note that the computations of the AM2'S listed in Table 4 are based, for each application, on the observed harmonic constants reported by each author. These constants can be author-dependent at a given location. A significant weakness of the TFF exercise, on the other hand, is related to constituents which are not included in the database (neither as forcing nor at the coastal checkpoints) and which appear important based on the results of the analysis of the long-term simulations. Our results fully support the conclusion of Werner and Lynch 21 that at least the 2MN2 and 2MS2 constituents should not be neglected for several reasons. First, since these internally generated harmonics contribute significantly to the total amplitude of the calculated signal at certain locations in the interior of the domain (see, e.g., spectra at Dieppe, Boulogne, Calais or Zeebrugge in Werner and LynchZX), their omission from the observed signal artificially increases the misfit of the unfiltered, time-dependent numerical results. Second, these non-negligible harmonics are certainly poorly simulated in the models since their values are de facto forced to vanish at the two open
218
Adv. Water Resources, 1989, Volume 12, December
boundaries. Third, these poorly represented harmonics are likely to interact with some of the other modes. This may affect the overall quality of the simulation in a way which is difficult to assess, although, as pointed out by an anonymous reviewer, higher harmonics will generally be more affected than lower ones, especially if their generation is not mainly due to well-represented constituents.
5. C O N C L U S I O N S The results of several vertically-integrated numerical models (and of various model simulations) obtained within the framework of the Tidal Flow F o r u m exercise have been compared to observations which are assumed to be correct for the present purposes. A first estimation of the fit between model results and observations is based on the computation of the RMS errors over the 24 hours timeseries obtained for the 17th of March 1976. Those RMS values seem to indicate that the best agreement is between model results and observations submitted to the same filtering procedure. However, the RMS value computed over such a short period of time (fixed T in equation (4)) varies significantly with time (to in equation(4)). The evaluation would be different had another test period been selected. Therefore, to be significant, the RMS values must be computed over a sufficiently long period of time.
R e s u l t s a n d c o m m e n t s on Tidal F l o w F o r u m exercise: B. M . J a m a r t a n d J. Ozer Table 4.
Space-averaged maximum error on the M2 constituent for 5 TFF entries and several other models Model area
Reference or abbreviation
Number of
Minimum value
Maximum value
Mean value
stations
(cm)
(cm)
(cm)
Standard deviation
(era)
English Channel and southern North Sea
Waiters (I 987)
11
6.9
44.6
i 8.4
11.7
Idem
WEQN88 (bath. V.1)
II
2.5
39.2
16.8
10.9
Idem
WEQN88 (bath. V.2)
11
2.3
30.zl
14.7
8.6
Idem
MU89 (bath. V.1 )
11
1.1
47.6
19.9
1i.3
ldem
MU89 (bath. V.2)
11
5.7
31.4
19.2
7.3
English Channel
Le Provost and Fomerino (1985) Ronday ('1976)
3.7
30.9
17.8
9.5
22
0.
37.4
16.2
11.5
North Sea North Sea
Y. Adam (pers. communication)
8
3.0
59.0
26.8
16.8
North European continental shelf
Beckers and Neves (1985)
31
0.
91.1
26.0
26.0
ldem
Verboom et al. (1987)
55
0.
31.0
7.7
7.2
T h e v a l i d a t i o n o f the m o d e l r e s p o n s e with respect to all i m p o r t a n t tidal c o n s t i t u e n t s m u s t be d o n e b y m e a n s o f s i m u l a t i o n s p e r f o r m e d over a p e r i o d which allows to retrieve t h o s e c o n s t i t u e n t s b y h a r m o n i c d e c o m p o s i t i o n . G i v e n the b o u n d a r y f o r c i n g o f the T F F p r o b l e m , s i m u l a t i o n s o f at least 180 d a y s are r e q u i r e d . F r o m the m o d e l ' s results, it a p p e a r s t h a t within t h o s e i m p o r t a n t c o n s t i t u e n t s , the 2 M N 2 a n d the 2 M S 2 tides, which are p r e s e n t l y neglected in the T F F p r o b l e m , are sufficiently i m p o r t a n t to be i n c l u d e d in the v a l i d a t i o n o f the m o d e l responses. T h e y s h o u l d thus be t a k e n into a c c o u n t in the ' r e f e r e n c e ' a n d in the f o r c i n g a l o n g the o p e n boundaries. B o t h the c o n s t a n t a n d the t i m e - d e p e n d e n t p a r t s o f the R M S are l i n k e d to the m a x i m u m errors m a d e o n each c o n s t i t u e n t . T h o s e m a x i m u m e r r o r s can be easily d e d u c ed f r o m the h a r m o n i c c o n s t a n t s ( c o m p u t e d versus o b s e r v e d a m p l i t u d e s a n d phases) o f the constituents. It is suggested t h a t t h o s e m a x i m u m errors might c o n s t i t u t e a m o r e general e s t i m a t i o n o f the fit b e t w e e n m o d e l results a n d o b s e r v a t i o n s . O n the basis o f the results a v a i l a b l e to us a n d a s s u m i n g t h a t the d a t a are reliable, it is s h o w n t h a t f o u r c o n s t i t u e n t s (Mz, $2, N2, M4) need to be i m p r o v e d u p o n first in o r d e r to increase the overall q u a l i t y o f the results. It a p p e a r s t h a t all the T F F results o b t a i n e d so far are a l m o s t o f the s a m e o r d e r o f a c c u r a c y . E v e n if one p a r ticular c o m p u t a t i o n a p p e a r s slightly better t h a n the others, o u r lack o f c o n f i d e n c e in the reliability o f the reference h a r m o n i c c o n s t a n t s a n d the l i m i t e d s a m p l e size keep us f r o m d r a w i n g definitive c o n c l u s i o n s . F r o m e q u i v a l e n t c a l c u l a t i o n s o n results r e p o r t e d for similar m o d e l s used in similar a p p l i c a t i o n s , we c o n c l u d e t h a t the errors affecting the T F F results are n o t u n r e a s o n a b l y large.
6. A C K N O W L E D G M E N T S
This w o r k was p a r t i a l l y s u p p o r t e d b y Det n o r s k e Veritas within the f r a m e o f an i n d u s t r y research a n d d e v e l o p m e n t p r o j e c t called the B o t t o m Stress E x p e r i m e n t (BSEX) f u n d e d b y F i n a E x p l o r a t i o n N o r w a y . It is also s u p p o r t e d b y the E u r o p e a n C o m m u n i t y C o m m i s s i o n within the f r a m e o f the O P E R A (Oil P o l l u t i o n : E n v i r o n m e n t a l Risk A s s e s s m e n t ) p r o j e c t . W e t h a n k all the p a r t i c i p a n t s in the T i d a l F l o w F o r u m for f r u i t f u l discussions a n d J. D i j k z e u l , R. A . W a i t e r s , F. E. W e r n e r , a n d C. S. Yu for the c o m m u n i c a t i o n o f their results.
REFERENCES
1 2 3 4 5. 6
7 8
Adv.
Beckers, P. M. and Neves, R. J. A semi-implicit tidal model of the North European Continental Shelf, Applied Mathematical Modelling, 1985, 9 (6), 395-402 Chabert D'Hi~res, M. M. G. and Le Provost, C. Atlas des composantes harmoniques de la mar6e dans la Manche, Annales Hydrographiques, 1979, 6, 5-36 Foreman, M. G. G., Manual for Tidal Heights Analysis and Prediction. Pacific Marine Science Report 77-10, Institute of Ocean Sciences, Patricia Bay, Sydney, B.C., 1977 Gray, W. G., A finite element study of tidal flow data for the North Sea and English Channel, Advances in Water Resources, 1989, 12 (3), 143-154 Gray, W. G., Drolet, J. and Kinnmark, I. P. E. A simulation of tidal flow in the south part of the North Sea and The English Channel, Advances in Water Resources, 1987, 10, 131-137 Kinnmark, I. P. E. The shallow water wave equations: formulation, analysis and application. Lecture Notes in Engineering, 15, Springer-Verlag, C. A. Brebbia and S. A. Orszag eds., 1985 Le Provost, C. and Poncet, A. Finite element method for spectral modelling of tides, International Journal for Numerical Methods in Engineering, 1978, 12, 853-871 Le Provost, C. and Fornerino, M. Tidal spectroscopy of the
W a t e r Resources, 1989, Volume 12, December
219
Results and comments on Tidal Flow Forum exercise: B. M. Jamart and J. Ozer
9
10
11
12
13
14
15
16
17
18
19
20
220
English Channel with a numerical model, Journal of Physical Oceanography, 1985, 15, 1009-1031 Lynch, D. R. and Gray, W. G. A wave equation model for finite element tidal computations, Computer and Fluids, 1979, 7, 207-228 Lynch, D. R. and Werner, F. E. Long-term simulation and harmonic analysis of North Sea]English Channel tides, Computational Methods in Water Resources, Volume I: Modeling Surface and Sub-Surface Flows, M. A. Celia et al., eds., Computational Mechanics Publications, Elsevier, 1988, 257-266 Ozer, J. and Jamart, B. M. Computation of tidal motion in the English Channel and southern North Sea: comparison of various model results, MUMM's contribution to BSEX, Technical Report No. TR07, 1987 Ozer, J. and Jamart, B. M. Tidal motion in the English Channel and southern North Sea: comparison of various observational and model results, Computational Methods in Water Resources, Volume I: Modeling Surface and Sub-Surface Flows, M. A. Celia et al., eds., Computational Mechanics Publications, Elsevier, 1988, 267-273 Pearson, C. E. and Winter, D. F. On the calculation of tidal currents in homogeneous estuaries, Journal of Physical Oceanography, 1977, 7 (4), 520-531 Praagman, N., Dijkzeul, J., van Dijk, R. and Plieger, R. A finite difference simulation model for tidal flow in the English Channel and the southern North Sea, Advances in Water Resources, 1989, 12 (3), 155-164 Ronday, F. C. Mod61es hydrodynamiques, Projet Met, Rapport Final, Services du Premier Ministre, Programmation de la Politique Scientifique, Bruxelles, Belgium, 1976 Verboom, G. K., van Dijk, R. P. and de Ronde, J. G. Een model van het Europese Kontinentale Plat voor windopzet en waterkwaliteitsberekeningen, Rijkswaterstaat-Dienst Getijdewateren Waterloopkundig Laboratorium, November 1987 Walters, R. A. A model for tides and currents in the English Channel and southern North Sea, Advances in Water Resources, 1987, 10, 138-148 Waiters, R. A. and Werner, F. E. A comparison of two finite element models using the North Sea data set, Advances in Water Resources, 1989, this issue Werner, F. E. and Lynch, D. R. Field verification of wave equation tidal dynamics in the English Channel and southern North Sea, Advances in Water Resources, 1987, 10, 115-130 Werner, F. E. and Lynch, D. R. Tides in the southern North Sea/English Channel: data files and procedure for reference computations, February 1988 distribution.
Adv. Water Resources, 1989, Volume 12, December
21
Werner, F. E. and Lynch, D. R. Harmonic structure of English Channel/Southern Bight tides from a wave equation simulation, Advances in Water Resources, 1989, 12 (3), 121-142 Yu, C. S., Fettweis, M. and Berlamont, J. A 2D model for tidal flow computations, Computational Methods in Water Resources, Volume I: Modeling Surface and Sub-Surface Flows, M. A. Celia etal. eds., Computational Mechanics Publications, Elsevier, 1988, 281-286 Yu, C. S., Fettweis, M., Hermans, I. and Berlamont, J. Tidal flow simulation in the English Channel and southern North Sea, Advances in Water Resources, 1989, this issue
22
23
7. A P P E N D I X
With A(t) given by equation (3), and defining o~k as [Vo + u ] k - fig, the integrand of equation (4) can be expanded as 1M 1M Az(t) = 2 k=o ~] f2 A~ + 5 k=0~f2 AkZ COS[2(wkt + c~k)] M
M
+ ~
~ f k f i Ak Ai COS(e0kt + ~k)COS(Wit + C~i) k=0 i=O i#1¢
Hence, the square of the RMS error is given by 1 M
RMS2=~ Z j2 A~ k=O
1
2 k=O
M
+ ~
M
T
I
Oto
1 [.to+T
~ fkfi&kZ~i --I
k=0 i=0
i#k
cos[2(wkt + ~k)] dt
lJto
COS(~okt+ o~k)COS(wit+ o~i) dt
The first term is independent of T and ctepends on to through the choice of fk. The second and third terms do depend on T, and they tend to 0 as T--, oo.
M U M M (Management Unit o f the Mathematical Models o f the North Sea and the Scheldt Estuary), Gulledelle 100, 1200 Brussels, Belgium Two approaches are followed to estimate the fit between 'observations' and the results obtained with different numerical models within the frame of the Tidal Flow F o r u m exercise. The first approach (i.e. the computation of the RMS errors on the timeseries for a specific day) is shown to be insufficient because the RMS error computed over a short period of time is strongly dependent on the selected test period. Such RMS values are significant if and only if they are computed over a sufficiently long period of time, T. Hence, the model validation must be done on the basis of the results of long term simulations. The limit of the RMS values as T becomes arbitrarily large is related to the m a x i m u m error made on each constituent. Those m a x i m u m errors are easily calculated from the harmonic constants (computed versus observed) of the constituents. The results of the calculation of the m a x i m u m errors indicate that the ME, $2, N2 an d M4 components are the largest contributors to the overall error for the area under consideration. On the basis of m a x i m u m errors, the T F F results obtained so far are almost of the same order of accuracy. The errors affecting the TFF results are not much larger than those affecting most reported results of similar models used in similar applications 1. I N T R O D U C T I O N The Tidal Flow F o r u m (henceforth TFF) is an exercise of model assessment and comparison launched several years ago under the leadership of Gray, Le Provost, Lynch and V e r b o o m (in alphabetical order). The aim of the exercise was to set up a realistic, challenging and well-documented problem in tidal hydrodynamics which could serve as a benchmark or test problem for numerical models. The selected problem consists in computing the motion due to tides in the English Channel and in the southern part of the North Sea. This area has been extensively surveyed and modelled, and the tidal constituents are known to display here rather complex patterns (e.g., Charbert d'Hi~res and Le Provost2). A gridded bathymetric data set and the harmonic constants of 11 constituents of the vertical tide (i.e., sea surface elevation) along the two open boundaries constitute the first part of the database of the TFF. The second part of that database are the same harmonic constants at 11 coastal checkpoints and short timeseries of observed current velocities at various locations inside the domain of interest. The most recent version of the T F F database has been circulated by Werner and Lynch 2°. A detailed description of the reference computation, as well as a number of results, are given by Werner and Lynch x9, Gray et al. 5, and Waiters 17. By and large, the numerical models used to date on the TFF problem are two-dimensional (vertically integrated) models solving the classical shallow water wave equations. In a first round, all the models resorted
Accepted March 1989. Discussion closes June 1990.
to the finite element method (FEM) to handle the spatial discretization (see Advances in Water Resources 10). In the second round of discussions these FEM were extended*'l°'lS'2~; in addition, several entries based on the finite difference technique (FDM) both in time and space were also presented ~1'12A4'22'23. Most of the papers related to the TFF exercise describe the application of a specific model to the problem at hand (equations, numerical procedure, results) and conclude with a comparison between the model results and the sacrosanct 'observations'. The present contribution does not follow that outline because its purposes are: i) to compare the results of several different models and approaches; ii) to discuss the methods used to evaluate the results of numerical tidal models. Concerning our own model (briefly described in Ozer and J a m a r t 11'12) suffice to say that it solves the same governing equations as all other entries, using the same forcing and parameters, with an F D M / A D I (semiimplicit) techniquet. We have elected not to show the conventional sequence of timeseries o f elevation and velocity for the 17th of March, 1976: like those of most other models applied to the T F F problem, our results go up and down smoothly (these are tidal signals, for heavens' sake) and they agree 'reasonably well' with the 'observations'. In the next section, we present an overview of the various models included in our comparison, with an indication of the type of results available to us at the time of writing this. Section 3 is devoted to a discussion of the results in terms of RMS error, an estimator which we have used previously in a more restricted
© 1989 Computational Mechanics Publications
Adv. Water Resources, 1989, Volume 12, December 211
Results and comments on Tidal Flow Forum exercise: B. M. Jamart and J. Ozer
comparison 12 and which we now deem an insufficient and, to some extent, misleading indicator of the quality of a model. Another approach to the evaluation of the results, in the frequency domain, is discussed in section 4. Since the latter approach, which appears more suitable, can only be performed on spectral results, and since only short-term velocity observations are available within the TFF, the evaluation presented here is restricted to sea-surface elevation results. Conclusions are presented in section 5.
free of the deleterious effects linked to the 'cold start' procedure. Note that this simulation corresponds exactly to the prescriptions of the reference computation of the TFF exercise. Their second simulation 1° henceforth WEQN88, is performed over a period of time sufficiently long for the 11 TFF constituents to be retrievable by harmonic analysis of the resulting timeseries. When dealing with the results of WEQN88 for the 17th of March 1976, the model signal is reconstructed on the basis of the 11 TFF constituents and only those. A more exhaustive exposition of these results and attendant discussion appears in Werner and Lynch 21. The last four entries in Table 1 correspond to FDM computations. The three models use essentially the same finite difference grid, although some slight discrepancies have been noted between the various implementations. For example, the coastline geometry is not exactly the same in all three models and the interpolation procedure applied at the open boundaries also differs slightly. This is rather unfortunate, since it is then impossible to unequivocally ascribe discrepancies between model results to the differences in the numerical algorithms. We shall consider only those results of the W A Q U A and FADI models which have been obtained with the TFF parameters, and in both cases we have been provided with timeseries of the results for the 17th of March 1976. With MUMM's model, we have carried out both a short (MU87) and a long (MU89) integration. The results of the latter have been analyzed using the software package developed by Foreman 3. Since the inception of the TFF exercise, two somewhat different sets of bathymetric data have been used by the various groups. The reasons for dealing with an 'original' and a 'revised' bathymetry are discussed by Werner and Lynch 2°. These datasets are referred to as versions 1 and 2, respectively, in Table 1.
2. M O D E L RESULTS I N C L U D E D IN T H E COMPARISON We have at our disposal the results of 5 of the models involved in the TFF exercise. Table 1 contains a summary of the models and simulations considered herein, all of which use the TFF parameters. For some models, more than one set of results are available. In addition to the simulations listed in Table 1, several results have been reported in order to show that some amount of 'tuning' (i.e., modification of the basic TFF parameters) could improve the apparent fit of the model results to the TFF reference data. We do not include such 'calibrated' results in our comparison, for reasons which will be discussed later. The first three simulations listed in Table 1 are performed with models of the FEM-type. These two models use exactly the same computational grid but they differ with respect to the handling of the time variable. Walters 17 uses the spectral method pioneered by Pearson and Winter 13 and Le Provost and Poncet 7, wherein a spectral or harmonic decomposition is performed on the governing equations, yielding a system of coupled elliptic pde's which are solved iteratively. Waiters' computation involves the 11 constituents introduced as boundary forcing and only those. Werner and Lynch, on the other hand, apply a timestepping procedure to the wave equation resulting from manipulations of the momentum and continuity equations (see, e.g., Lynch and Gray, 9 Kinnmark6). In the first simulation performed by Werner and Lynch, referred to as WEQN87 in Table 1, the integration is carried out for 72 hours, starting with the sea at rest at 0:00 G M T on March 15, 1976. The results obtained for the last day of integration (March 17, 1976) are presumed
3. E V A L U A T I O N OF T H E MODELS IN T E R M S OF R M S E R R O R S 3.1. Results
In the original setup of the TFF exercise, the model responses are to be compared to observations for one specific period of time, namely the 17th of March 1976.
Table 1. Models and simulations considered by this paper Reference
Spatial discretization
Time discretization
Length of simulation
Bathymetry
Results available to us
Wemerand Lynch (1987)
FEM
Time-stepping ("wave equation")
3 days
VA
March 17,'1'976
Lynch and Wemer (1988) Wemer and Lynch (1988)
FEM
Time-stepping ("wave equation")
190 days
V.I+2
WALTE~S
w*Iters (19s7)
FEM
Spectral approach
DNA
V.I
WAQUA FADI
Praagman et'al. (1989)
FDM
ADI
3 days
Ya et al. (1989)
FDM
Falsified ADI
3 days
MU87
Ozer and Jamart (1987, 1988) this paper
V.2 V.2 V.l+2
Abbreviation
WEQN87
I
• WEQN88
11 TFF constituents 11 TFF constituents March 17, 1976 1,1
i
MU89
212
i
FDM
days ,
FDM
Adv. Water Resources, 1989, Volume 12, December
ADI
190days
,i ,
,,
March 17.' i976
March 17, 1976 ,,
V. 1+2
96 constituents
Results and comments on Tidal Flow Forum exercise." B. M. Jamart and J. Ozer Hence, most papers dealing with the T F F problem display timeseries of computed versus observed sea surface elevation at the coastal reference stations. Such graphs are not convenient for the purposes of comparing models between themselves. Therefore, in an attempt to present a general overview of the abilities of the models, we have introduced two presumably 'global' estimators easily derived from the one-day timeseries 11'1z. The first estimator is the m a x i m u m absolute error, i.e., the m a x i m u m difference between observed and computed elevations at each station for March 17, 1976. The second estimator is the root mean square error (RMS) over that 24 hour period. The first estimator, i.e., the m a x i m u m absolute error, shows that the discrepancies between observations and model results are not small (e.g., 113 cm at Boulogne for MU87; 115 cm at St. Malo for WEQN87; 84 and 96 cm at Christchurch, for WEQN88 and Walters, respectively). Our subjective opinion lz is that such errors are 'large', i.e., well above some as yet ill-defined standard that could be qualified as state-of-the-art, most likely in the order of 5 to 10 cm. A second remark concerning the m a x i m u m errors is that they are approximately twice as large as the RMS errors and provide no additional insight, so that the latter, viewed as a more general measure of fit, might be better suited for comparing model results. Table 2 contains the one-day RMS errors of the five models for which we have results calculated with the original bathymetry. Except for the MU89 model, these results have been published and discussed previously (Ozer and J a m a r t 12); they are included here for the sake of completeness. The main conclusion f r o m that earlier comparison, which is confirmed by the results of MU89, can be re-stated as follows: •
model results are compared to a so-called 'observed' or 'reference' signal which is in fact a harmonic recomposition based on a subset of the total tidal spectrum at each station; the constituents included in the reconstructed signal are the same as those used as boundary forcing, and only those; hence, the observations should be considered as filtered; • against the benchmark so-defined, the models for which a similarly filtered response is considered for the evaluation do p e r f o r m better (i.e., they have a smaller mean RMS error) than the models for which only a raw or unfiltered response is evaluated; note that at some stations, the unfiltered response (corresponding to what we called a 'time domain simulation') is actually better than the filtered result (referred to as 'spectral simulation' in our previous paper; this designation will not be used further to avoid confusion). This conclusion is further discussed by Werner and L y n c h 21.
The RMS errors for the 17th of March 1976, calculated for the models that have used the revised bathymetry, are listed in Table 3. These results support the conclusion that filtered model results compare generally better to filtered observations than do unfiltered simulation results. Table 3 shows that all the F D M models yield smaller RMS errors along the British coast (last 4 stations) than on the continental side, a point discussed by Yu et al. 23. The standard deviations
Table 2. Compar&on between model results and the TFF database: values o f the root mean square error, in cm, f o r the 17th o f March 1976. Model simulations performed using the original bathymetric data set
~rATION
Z~B~UC,aE HOEKVANHOLLAND LOWESrOrr
MIJ~ 20.4 13.8 44.2 61.4 61.8 36.6 21.2 12.1
WALTON
24.4
14.9
47.0
36.4
50.6
oovEn
Mean
31.4 52.2 34.5
40.9 37.4 26.8
33.6 56.5 36.0
15.6 48.2 22.2
22.5 56.6 27.5
Standard deviation
18.1
16.5
16.4
12.5
14.7
st. ~,~,o
CUEe,nOURG OtEt't'E nOUtX~NE CaZatS
CHRtSTCHURCH
MI.~9 WEQN$7 WEQNU WALTERS 24.9 47.1 15.0 19.4 6.8 12.7 13.3 13.7 16.2 54.1 17.3 25.1 63.4 41.6 15.6 32.5 39.4 46.1 37.7 34.7 18.3 30.3 9.4 21.3 15.6 14.8 21.0 12.8 17.6 12.8 15.1 13.8
Table 3. Comparison between model results and the TFF database." values o f the root mean square error, in cm, f o r the 17th o f March 1976. Model simulations performed using the revised bathymetric data set. (Unpublished results f o r WEQN88 courtesy o f F, Werner) STATION
MU~/
MU~
FADI
WAQUA
WF.~N~
~ . M,u,o
64.2 30.7
40.5 27.3
75.8 26.0
103.4 30.2
15.7 17.7
otem'~
69.6
46.2
80.0
63.3
22.8
SOULOGNt"
54.6
32.0
64.4
50.5
20.1
cm.ats ZEEnnOC,Gt~
63.6 46.3
43.3 26.3
57.4 51.5
57.2 40.2
36.6 14.7
HOEK VAN HOLLAND
29.4
20.5
26.6
17.8
16.1
LOW£STOFr
14.0
17.5
I 1.4
13.7
8.1
waLrou
26.5
14.0
20.3
27.5
36.1
DOV~
20.2
20.6
27.3
22.5
16.9
cnm~cnut¢ctt Mean
17.8
8.9
21,7
20.3
15.7
39.7
27.0
42.0
40.6
20.0
Standard deviation
19.5
11.7
23.2
25.4
8.8
CUEnSOttRC
on the mean of the local RMS errors are comparable to those of Table 2, i.e., about 50% of the mean value. The revised bathymetry was introduced with the hope that some of the problems affecting the original dataset might have contributed to the relative inaccuracies of the model results. Hence, better model results were expected. I f the RMS errors are used to evaluate the quality of the fit, Table 3 shows that the revised bathymetry does not yield better results, on the average, than the first TFF database. Note also that the results o f the FADI- and MU-models listed in Table 3 are obtained by solving the equations in Cartesian rather than spherical coordinates, although the bathymetry appears to be defined on a spherical grid. As remarked by Yu et al. 23, the RMS errors are larger when using the presumably more proper spherical code. A puzzling conclusion emerging from Table 3 is the fact that the F D M models, which all use basically the same grid and solve the same equations, yield results which appear to be substantially different. We feel that the comparison of those three models should be done through a more tightly defined procedure and that the remaining discrepancies, if any, would deserve further investigation. Similarly, it appears from Table 2 that comparable results of the FEM models (i.e., WEQN88 and Walters)
Adv. Water Resources, 1989, Volume 12, December
213
Results a n d c o m m e n t s on Tidal F l o w F o r u m exercise: B. M. Jamart and J. Ozer
can be locally quite different from each other. Some of those differences can be explained by the numerical handling of particular aspects of the equations and/or the boundary conditions (Waiters and Werner, pers. communication). Such explanations seem worth a detailed discussion.
the model result, A(t), is then given by M
A ( t ) = ~] fk Ak COS(c0gt+ [ V o + U ] k - - { 3 k ) where, for 1 ~ < k ~ < N = 1 1 , A k = { ( aok
COS
In retrospect, it is legitimate to ask whether the above conclusion that filtered model results compare globally better to filtered observations than unfiltered model results might not have been foregone. It is also legitimate to inquire whether that conclusion hinges on the evaluation method (RMS error for a specific day), and, more generally, whether the TFF exercise is adequately formulated. A tidal signal, either observed or calculated in a numerical model, can always be viewed as a sum of harmonic constituents. The number and the nature of such constituents that can be calculated from a given timeseries depend mostly on the length of the record and on the location (e.g., open ocean versus shallow coastal water) where the signal is recorded. Consider, for instance, a timeseries of the computed tidal elevation at one of the coastal checkpoints of the TFF problem. Let us assume that the series is long enough that all important constituents can be analyzed by a conventional harmonic decomposition, i.e., that the residual (difference between the original signal and its series representation) is small. Those 'important' constituents include those that are specified at the open boundaries of the model and, usually, additional harmonics resulting from the nonlinear dynamics of the system. The model result, r/c(t), can then be written as M
A~k cos(wkt- rbCk)
(1)
k=O
with Ack
fkack
=
q~c~ = ¢c~ -
[ Vo + U ] k
O~o= ¢bco = 0
where the ak's and ek's are the harmonic constants, the fck's are factors used to reduce the mean amplitudes to the year of analysis, and Vo + u is the value of the equilibrium argument of a particular constituent when t=0. In order to evaluate the quality of such a model result, one needs to have an equivalent series representation of the observed signal at the same location. In the TFF exercise, however, this is not the case: the reference provided by the TFF database can only be reconstructed on the basis of 11 = N < M constituents. Defining the sequence of the OJk'S in (1) in such a way that wl, ~oz.... ~ t correspond to the frequencies of the TFF constituents, the so-called 'observed' signal can be written as N
Oo(t) = ~
Aok COS(Wkt-- q~ok)
(2)
k=l
with Aok
=
fkaok
• ok = Oo~ - [ Vo + u / k
The difference between the truncated observation and
214
(9ok-- ack
COS
Ock ) 2
+ (aok sin Ook -- ack sin ~¢k) 2 } 1/z
3.2. Analysis o f the evaluation m e t h o d
rl¢(t) = ~
(3)
k=0
Adv. W a t e r Resources, 1989, Volume 12, December
&
atan
[ aok sin ¢ok - ack sin ¢ck ] / Laok cos ¢ o k - a~g cos ¢¢kJ /
and, for all other values of k, A k = -- ack t3k = Ock
The RMS error, calculated over some arbitrary period of time T starting at t = to, is defined as RMS(to, T) =
z~2(t) dt
(4)
to
and one can easily show (see Appendix) that the square of the RMS error consists of three terms: i) the first term is equal to M
1 Z f2 A2; 2 k=o it is independent of T, and it varies very slowly as a function of to (through the time variation of the nodal correction factors, fk); we shall refer to this term as the long term RMS error; ii) the second and third terms have a more complicated algebraic expression; they are functions of both to and T due to the fact that T is not a common multiple of the periods of all constituents; the sum of these terms will be called the time-dependent part of the RMS error. Note that as T becomes arbitrarily large, this contribution to the error vanishes. Two obvious conclusions follow from these equations. First, if equation (1) representing the full model result is replaced by a truncated series consisting only of the same constituents as those included in (2), the long term RMS error will be smaller. Hence, in a time-average sense, 'filtered' model results can only look better than 'unfiltered' ones when compared to the truncated observation. This artefact is a mere consequence of insufficient information on the prototype. Second, since the RMS error varies with both T and to, it appears unwise to evaluate the TFF model results on the basis of the fit caculated for one single day of simulation. To make these points clear, we have calculated the 24 hour RMS error on both the filtered and the unfiltered results of the MU89 simulation. Sample timeseries of such daily RMS errors are displayed in Figs 1 (St. Malo), 2 (Dover), and 3 (Zeebrugge). These figures show that the (time-) mean values of the daily RMS errors, i.e., the long term RMS errors defined above, are smaller for filtered than for unfiltered results. They also show that the time-dependent part of the RMS error, for T fixed and equal to 24 hours, is quite significant. At St. Malo, for example, there are times when the unfiltered model result looks better than the filtered one. More importantly, the subjective evaluation of the
Results and comments on Tidal Flow Forum exercise." B. M. Jamart and J. Ozer
ENGLISH Station
CHANNEL
AND
SOUTHERN
NORTH
SEA
PROBLEM
: Saint-Malo
cm)
RMS
70
60
50
40 ~
si
I
\
I
I
I
tI
O0
20
',/ V!/ ,,
', /
' ,/
'4
'
'2
',l
I0
o
'
,6o Time
Unfiltered
Note
: starting
]. . . . . . . .
results
date
:
1'5/03/76
at
O h
. filtered
(day)
remu!tl
GMT.
Fig. 1. Timeseries of RMS errors (T= 24 hours) at Saint-Malo computed from the results of the MU89 (V.2) simulation. The solid line shows the RMS errors of the unfiltered results; the dashed line corresponds to the filtered results. The horizontal solid (dashed) line indicates the time-averaged value of the RMS for the unfiltered (filtered) results. quality of the various model results, with respect to some standard acceptable level of error, can obviously not be based on the RMS error over a single day: for these three sample stations, the results of MU89 look much better about 10 days after March 17, 1976, than they do on the arbitrarily selected quiz day of the TFF exercise. A corollary of the inadequacy of the RMS error calculated for a specific day as an estimator of model quality is that attempts to calibrate the TFF database on that basis are pointless. Both the evaluation and the calibration procedures have to he performed either in the frequency domain or on the basis of long simulations.
4. E V A L U A T I O N OF T H E M O D E L S IN T H E FREQUENCY DOMAIN
A general expression for the difference between observations and model results is given by equation (3). In the TFF exercise, the number of observed constituents, N, is smaller than the number M of harmonics retrieved from the long-term (180 days) simulations. Consider first the constituents which are both present in the calculated signal (equation (1)) and available from the TFF database (equation (2)). The reliability of the latter is somewhat questionable (see Ozer and
Jamart, 11'12) and will be further discussed in a subsequent paper. For the time being, we shall assume that the observed amplitudes and phases of the signal are heaven-sent, error-free constants. Equation (3) indicates that the contribution of the k-th constituent to the inaccuracy of the numerical result is fkAk COS(~0kt+ [Vo + U]k- ~k). The maximum error due to the k-th constituent is thus fkAe, a simple function of the observed and calculated amplitudes and phases, whose time-dependence is almost entirely contained in the fk factor. The term Ak is essentially space- and modedependent. A summary of the maximum error (more precisely, the Ak factor) for the 11 TFF modes at the eleven coastal checkpoints is presented in Figs 4 to 6 for various model simulations (Waiters, WEQN88, MU89). We conclude from the evaluation of the models on the basis of maximum error per constituent that:
i) the three models yield answers of comparable global (in)accuracy, although the details of the spatial distributions of those maximum errors are different between the models; ii) the largest absolute errors, in all models, are linked to the first three semi-diurnal constituents (M2, $2, Nz) and to the M4 component; hence, these are the constituents which need to be improved upon first in order to increase the overall quality of the simulation results.
Adv. Water Resources, 1989, Volume 12, December 215
ENGLISH
CHANNEL
AND
SOUTHERN
NORTH
SEA
PROBLEM
: Dover
Station RMS
(cm)
35-
30
25
A 20
" - : - .......
15 t
•I
it
t
a
'i ,'"' " '" ," i,,; '",,,// ~
w
\,
'~ I
~
'-:/
f' ,"
;
' ...... L
\s
10
''
/ tl
"'
,
. . . . . .
I
o
/
50
i
.
.
.
.
.
1 O0
.
.
,
I '
2O0
Time Unfiltered
Note
results
: starting
date
, ]........
: '1g/03/76
at
J
15o
O h
filtered
GMT.
"
(day)
results
"
Fig. 2. Same as Fig. 1 at the station Dover
ENGLfSH Station RMS
CHANNEL AND S O U T H E R N NORTH SEA PROBLEM
: Zeebrugge
......
(cm)
50
40
30
1.!
20
I
10
~/.
tt tL
,;
I I
LI
\
t
I
t
V
0
'l'
o
I
50
I'
1oo
'
200
15o
Time .
.
Note
.U n f i l.t e r e d.
: starting
r .e s u l t s.
date
.
I: - , - - -
.
: 15/03/76
at
0
Fig. 3. Same as Fig. 1 at the station Zeebrugge
216
Adv. Water Resources, 1989, Volume 12, December
h
. . ._ ._ ,.f .i l.t e.r.e.d.
GMT.
results
,
....
(day)
ENGLISH Waiters'
CHANNEL
AND
SOUTHERN
NORTH
SEA
PROBLEM
model
Maximum
error
(cm)
50-
40
30
20 o o
•
$
o o
0
0
°
10
e
S |
°
0
•
!
i 1
i +
g
"i
'l 3
2
i
o
.
0
I
0
o
I
4
0
0 o
o
8
~
°
°
9
I 10
ii
6
7
o
8 I'
I
12
11
Constituent
number
Fig. 4. M a x i m u m error (Ak) computed f o r the 11 TFF constituents at the 11 coastal stations f r o m the results o f Waiters" model. The 11 constituents are ordered as follows: 01, Kl, M2, $2, N2, 1£2, M4, MS4, MN4, M6, 2MS6
ENGLISH
CHANNEL
Lynch
Werner's
and
Maximum
error
AND
SOUTHERN
NORTH
SEA
PROBLEM
model
(ore)
50
40
30
20
* +
÷
. +.
10
÷
* ÷ .
0
~
i+
,: +:
1
2
"+
•
w
.
.*
"+ ,+
.+
4
+ +
,,~ * +
-", 6
;+
.
.+
J + + +
~.+
;+
4
7
8
9
Constituent
WLa9 ( v l )
+ ÷
÷ +
10
number
WL89 (v2)
Fig. 5. Same as Fig. 4 f o r the results o f the WEQN88 (V. 1 + V.2) simulations. The A k "s calculated f r o m the WEQN88 results with the original bathymetry are shifted to the left with respect to the constituent number; the values f o r WEQN88, revised bathymetry, are shifted to the right
Adv. Water Resources, 1989, Volume 12, December
21]"
Results and comments on Tidal Flow Forum exercise: B. M. Jamart and J. Ozer
ENGLISH
CHANNEL
AND
SOUTHERN
NORTH
SEA
PROBLEM
MU-rnodel Maximum 50
error
(cm)
40
30
+
*
+
20 ÷
.
10
+
÷
+
° +
*
•
,!
o
+
*
+
÷
i 0
t~*
*
*
'l
I
o
-÷
*
I
~l
I +
~
I
' I
1
(VI)
I
11
Constituent MU89
:i÷
.
.
.
.
.
.
.
.
l,
÷ ÷
*
number
u u .8. 9. . . . ( v a )
Fig. 6. Same as Fig. 5 for the MU89 (V.1 and V.2 bathymetries) simulations. It is also of interest to compare the errors obtained within the TFF exercise to those reported for similar models used in similar applications. We have performed such a comparison between various North Sea and/or English Channel numerical models, and we have considered only the dominant (M2) tidal constituent. The results of that comparison are summarized in Table 4, from which it appears that the errors affecting the TFF results are, in fact, not unreasonably large. It is important to note that the computations of the AM2'S listed in Table 4 are based, for each application, on the observed harmonic constants reported by each author. These constants can be author-dependent at a given location. A significant weakness of the TFF exercise, on the other hand, is related to constituents which are not included in the database (neither as forcing nor at the coastal checkpoints) and which appear important based on the results of the analysis of the long-term simulations. Our results fully support the conclusion of Werner and Lynch 21 that at least the 2MN2 and 2MS2 constituents should not be neglected for several reasons. First, since these internally generated harmonics contribute significantly to the total amplitude of the calculated signal at certain locations in the interior of the domain (see, e.g., spectra at Dieppe, Boulogne, Calais or Zeebrugge in Werner and LynchZX), their omission from the observed signal artificially increases the misfit of the unfiltered, time-dependent numerical results. Second, these non-negligible harmonics are certainly poorly simulated in the models since their values are de facto forced to vanish at the two open
218
Adv. Water Resources, 1989, Volume 12, December
boundaries. Third, these poorly represented harmonics are likely to interact with some of the other modes. This may affect the overall quality of the simulation in a way which is difficult to assess, although, as pointed out by an anonymous reviewer, higher harmonics will generally be more affected than lower ones, especially if their generation is not mainly due to well-represented constituents.
5. C O N C L U S I O N S The results of several vertically-integrated numerical models (and of various model simulations) obtained within the framework of the Tidal Flow F o r u m exercise have been compared to observations which are assumed to be correct for the present purposes. A first estimation of the fit between model results and observations is based on the computation of the RMS errors over the 24 hours timeseries obtained for the 17th of March 1976. Those RMS values seem to indicate that the best agreement is between model results and observations submitted to the same filtering procedure. However, the RMS value computed over such a short period of time (fixed T in equation (4)) varies significantly with time (to in equation(4)). The evaluation would be different had another test period been selected. Therefore, to be significant, the RMS values must be computed over a sufficiently long period of time.
R e s u l t s a n d c o m m e n t s on Tidal F l o w F o r u m exercise: B. M . J a m a r t a n d J. Ozer Table 4.
Space-averaged maximum error on the M2 constituent for 5 TFF entries and several other models Model area
Reference or abbreviation
Number of
Minimum value
Maximum value
Mean value
stations
(cm)
(cm)
(cm)
Standard deviation
(era)
English Channel and southern North Sea
Waiters (I 987)
11
6.9
44.6
i 8.4
11.7
Idem
WEQN88 (bath. V.1)
II
2.5
39.2
16.8
10.9
Idem
WEQN88 (bath. V.2)
11
2.3
30.zl
14.7
8.6
Idem
MU89 (bath. V.1 )
11
1.1
47.6
19.9
1i.3
ldem
MU89 (bath. V.2)
11
5.7
31.4
19.2
7.3
English Channel
Le Provost and Fomerino (1985) Ronday ('1976)
3.7
30.9
17.8
9.5
22
0.
37.4
16.2
11.5
North Sea North Sea
Y. Adam (pers. communication)
8
3.0
59.0
26.8
16.8
North European continental shelf
Beckers and Neves (1985)
31
0.
91.1
26.0
26.0
ldem
Verboom et al. (1987)
55
0.
31.0
7.7
7.2
T h e v a l i d a t i o n o f the m o d e l r e s p o n s e with respect to all i m p o r t a n t tidal c o n s t i t u e n t s m u s t be d o n e b y m e a n s o f s i m u l a t i o n s p e r f o r m e d over a p e r i o d which allows to retrieve t h o s e c o n s t i t u e n t s b y h a r m o n i c d e c o m p o s i t i o n . G i v e n the b o u n d a r y f o r c i n g o f the T F F p r o b l e m , s i m u l a t i o n s o f at least 180 d a y s are r e q u i r e d . F r o m the m o d e l ' s results, it a p p e a r s t h a t within t h o s e i m p o r t a n t c o n s t i t u e n t s , the 2 M N 2 a n d the 2 M S 2 tides, which are p r e s e n t l y neglected in the T F F p r o b l e m , are sufficiently i m p o r t a n t to be i n c l u d e d in the v a l i d a t i o n o f the m o d e l responses. T h e y s h o u l d thus be t a k e n into a c c o u n t in the ' r e f e r e n c e ' a n d in the f o r c i n g a l o n g the o p e n boundaries. B o t h the c o n s t a n t a n d the t i m e - d e p e n d e n t p a r t s o f the R M S are l i n k e d to the m a x i m u m errors m a d e o n each c o n s t i t u e n t . T h o s e m a x i m u m e r r o r s can be easily d e d u c ed f r o m the h a r m o n i c c o n s t a n t s ( c o m p u t e d versus o b s e r v e d a m p l i t u d e s a n d phases) o f the constituents. It is suggested t h a t t h o s e m a x i m u m errors might c o n s t i t u t e a m o r e general e s t i m a t i o n o f the fit b e t w e e n m o d e l results a n d o b s e r v a t i o n s . O n the basis o f the results a v a i l a b l e to us a n d a s s u m i n g t h a t the d a t a are reliable, it is s h o w n t h a t f o u r c o n s t i t u e n t s (Mz, $2, N2, M4) need to be i m p r o v e d u p o n first in o r d e r to increase the overall q u a l i t y o f the results. It a p p e a r s t h a t all the T F F results o b t a i n e d so far are a l m o s t o f the s a m e o r d e r o f a c c u r a c y . E v e n if one p a r ticular c o m p u t a t i o n a p p e a r s slightly better t h a n the others, o u r lack o f c o n f i d e n c e in the reliability o f the reference h a r m o n i c c o n s t a n t s a n d the l i m i t e d s a m p l e size keep us f r o m d r a w i n g definitive c o n c l u s i o n s . F r o m e q u i v a l e n t c a l c u l a t i o n s o n results r e p o r t e d for similar m o d e l s used in similar a p p l i c a t i o n s , we c o n c l u d e t h a t the errors affecting the T F F results are n o t u n r e a s o n a b l y large.
6. A C K N O W L E D G M E N T S
This w o r k was p a r t i a l l y s u p p o r t e d b y Det n o r s k e Veritas within the f r a m e o f an i n d u s t r y research a n d d e v e l o p m e n t p r o j e c t called the B o t t o m Stress E x p e r i m e n t (BSEX) f u n d e d b y F i n a E x p l o r a t i o n N o r w a y . It is also s u p p o r t e d b y the E u r o p e a n C o m m u n i t y C o m m i s s i o n within the f r a m e o f the O P E R A (Oil P o l l u t i o n : E n v i r o n m e n t a l Risk A s s e s s m e n t ) p r o j e c t . W e t h a n k all the p a r t i c i p a n t s in the T i d a l F l o w F o r u m for f r u i t f u l discussions a n d J. D i j k z e u l , R. A . W a i t e r s , F. E. W e r n e r , a n d C. S. Yu for the c o m m u n i c a t i o n o f their results.
REFERENCES
1 2 3 4 5. 6
7 8
Adv.
Beckers, P. M. and Neves, R. J. A semi-implicit tidal model of the North European Continental Shelf, Applied Mathematical Modelling, 1985, 9 (6), 395-402 Chabert D'Hi~res, M. M. G. and Le Provost, C. Atlas des composantes harmoniques de la mar6e dans la Manche, Annales Hydrographiques, 1979, 6, 5-36 Foreman, M. G. G., Manual for Tidal Heights Analysis and Prediction. Pacific Marine Science Report 77-10, Institute of Ocean Sciences, Patricia Bay, Sydney, B.C., 1977 Gray, W. G., A finite element study of tidal flow data for the North Sea and English Channel, Advances in Water Resources, 1989, 12 (3), 143-154 Gray, W. G., Drolet, J. and Kinnmark, I. P. E. A simulation of tidal flow in the south part of the North Sea and The English Channel, Advances in Water Resources, 1987, 10, 131-137 Kinnmark, I. P. E. The shallow water wave equations: formulation, analysis and application. Lecture Notes in Engineering, 15, Springer-Verlag, C. A. Brebbia and S. A. Orszag eds., 1985 Le Provost, C. and Poncet, A. Finite element method for spectral modelling of tides, International Journal for Numerical Methods in Engineering, 1978, 12, 853-871 Le Provost, C. and Fornerino, M. Tidal spectroscopy of the
W a t e r Resources, 1989, Volume 12, December
219
Results and comments on Tidal Flow Forum exercise: B. M. Jamart and J. Ozer
9
10
11
12
13
14
15
16
17
18
19
20
220
English Channel with a numerical model, Journal of Physical Oceanography, 1985, 15, 1009-1031 Lynch, D. R. and Gray, W. G. A wave equation model for finite element tidal computations, Computer and Fluids, 1979, 7, 207-228 Lynch, D. R. and Werner, F. E. Long-term simulation and harmonic analysis of North Sea]English Channel tides, Computational Methods in Water Resources, Volume I: Modeling Surface and Sub-Surface Flows, M. A. Celia et al., eds., Computational Mechanics Publications, Elsevier, 1988, 257-266 Ozer, J. and Jamart, B. M. Computation of tidal motion in the English Channel and southern North Sea: comparison of various model results, MUMM's contribution to BSEX, Technical Report No. TR07, 1987 Ozer, J. and Jamart, B. M. Tidal motion in the English Channel and southern North Sea: comparison of various observational and model results, Computational Methods in Water Resources, Volume I: Modeling Surface and Sub-Surface Flows, M. A. Celia et al., eds., Computational Mechanics Publications, Elsevier, 1988, 267-273 Pearson, C. E. and Winter, D. F. On the calculation of tidal currents in homogeneous estuaries, Journal of Physical Oceanography, 1977, 7 (4), 520-531 Praagman, N., Dijkzeul, J., van Dijk, R. and Plieger, R. A finite difference simulation model for tidal flow in the English Channel and the southern North Sea, Advances in Water Resources, 1989, 12 (3), 155-164 Ronday, F. C. Mod61es hydrodynamiques, Projet Met, Rapport Final, Services du Premier Ministre, Programmation de la Politique Scientifique, Bruxelles, Belgium, 1976 Verboom, G. K., van Dijk, R. P. and de Ronde, J. G. Een model van het Europese Kontinentale Plat voor windopzet en waterkwaliteitsberekeningen, Rijkswaterstaat-Dienst Getijdewateren Waterloopkundig Laboratorium, November 1987 Walters, R. A. A model for tides and currents in the English Channel and southern North Sea, Advances in Water Resources, 1987, 10, 138-148 Waiters, R. A. and Werner, F. E. A comparison of two finite element models using the North Sea data set, Advances in Water Resources, 1989, this issue Werner, F. E. and Lynch, D. R. Field verification of wave equation tidal dynamics in the English Channel and southern North Sea, Advances in Water Resources, 1987, 10, 115-130 Werner, F. E. and Lynch, D. R. Tides in the southern North Sea/English Channel: data files and procedure for reference computations, February 1988 distribution.
Adv. Water Resources, 1989, Volume 12, December
21
Werner, F. E. and Lynch, D. R. Harmonic structure of English Channel/Southern Bight tides from a wave equation simulation, Advances in Water Resources, 1989, 12 (3), 121-142 Yu, C. S., Fettweis, M. and Berlamont, J. A 2D model for tidal flow computations, Computational Methods in Water Resources, Volume I: Modeling Surface and Sub-Surface Flows, M. A. Celia etal. eds., Computational Mechanics Publications, Elsevier, 1988, 281-286 Yu, C. S., Fettweis, M., Hermans, I. and Berlamont, J. Tidal flow simulation in the English Channel and southern North Sea, Advances in Water Resources, 1989, this issue
22
23
7. A P P E N D I X
With A(t) given by equation (3), and defining o~k as [Vo + u ] k - fig, the integrand of equation (4) can be expanded as 1M 1M Az(t) = 2 k=o ~] f2 A~ + 5 k=0~f2 AkZ COS[2(wkt + c~k)] M
M
+ ~
~ f k f i Ak Ai COS(e0kt + ~k)COS(Wit + C~i) k=0 i=O i#1¢
Hence, the square of the RMS error is given by 1 M
RMS2=~ Z j2 A~ k=O
1
2 k=O
M
+ ~
M
T
I
Oto
1 [.to+T
~ fkfi&kZ~i --I
k=0 i=0
i#k
cos[2(wkt + ~k)] dt
lJto
COS(~okt+ o~k)COS(wit+ o~i) dt
The first term is independent of T and ctepends on to through the choice of fk. The second and third terms do depend on T, and they tend to 0 as T--, oo.