Sea level characteristics in the Bay of Fundy/Gulf of Maine

ENVIRONMETRICS

Environmetrics 2004; 15: 745–756

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/env.664

Sea level characteristics in the Bay of Fundy/Gulf of Maine

Bruce Smith*;y

Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5

SUMMARY

This article reports on the analysis of sea level records from the Bay of Fundy/Gulf of Maine. Descriptivestatistical methods are used to illustrate the increase in mean sea level, seasonal variance and nodal variation.Frequency domain time series methods are used to model increasing tidal amplitude, and to characterize featuresof sea level spectra. Copyright # 2004 John Wiley & Sons, Ltd.

key words: sea level; nodal variation; spectral analysis; seasonality; complex demodulation; Gaussian fitting

1. INTRODUCTION

The purpose of this article is to report on several features of sea level records from the Bay of Fundy/

Gulf of Maine. The specific data which will be examined consist of hourly sea level measurements at

St. John, New Brunswick and Portland, Maine. The Portland record includes nearly a century of hourly

sea level data, while prior to 1941 there are numerous gaps in the St. John record, and beyond 1978 the

St. John data may be unreliable due to silting in the vicinity of the tide guage. Therefore, long term

features will be studied using the more extensive Portland record. Frequency domain analyses will

focus on the St. John record, which shows a number of distinctive spectral features.

A typical stretch of data is illustrated in the upper panel of Figure 1. which shows hourly

observations at St. John in March, 1990. The ordinate is in metres above chart datum. The middle

panel of Figure 1 is a tidal prediction, and the lower panel is the difference between observed sea level

and tidal prediction, which will be referred to as the surge. The tidal prediction consists of predicted

values from an harmonic regression, the details of which are provided in Section 3. Tidal forces are

gravitational, and the tide varies with the motions of the planets. The surge residual is assumed to be

the result of meteorological forcing.

The semi-dirunal nature of the tide is evident, with two highs and two lows daily, and the spring-

neap tidal cycle is also clear. The surge shows several persistent elevated levels with duration of a few

days. For example, there are several extended excursions above 100 mm. In addition, there is some

cyclic variation in the surge with period near 12 h. In particular, there is a stretch from about 300 to

360 h with four local maxima and four local minima.

Received 21 November 2003

Copyright # 2004 John Wiley & Sons, Ltd. Revised 25 January 2004

*Correspondence to: B. Smith, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, CanadaB3H 3J5.yE-mail: [email protected]

Figure 1. Sea level (top), tidal prediction (middle) and surge (bottom). Sea level data are hourly measurements at St. John,

beginning March 1, 1990. Sea level and tidal predictions are in metres above chart datum. Surge is the difference between sea

level and tidal prediction

746 B. SMITH

Copyright # 2004 John Wiley & Sons, Ltd. Environmetrics 2004; 15: 745–756

The remainder of the article is as follows. Section 2 presents some descriptive analyses which

illustrate seasonality and long term trends. Section 3 describes a standard tidal analysis, and examines

the increasing trend in tidal amplitude. Section 4 presents some frequency domain time series

analyses, and is followed by a discussion.

2. DESCRIPTIVE STATISTICS

Figure 2 shows yearly mean sea level at Portland, Maine. Each point is the average of all hourly

measurements for the year. In addition to several years where no are data available, there are several

outlying data points, which are typically from years in which there was a substantial proportion of

missing data. The well known trend in mean sea level is apparent, although there is some indication of

a break in the trend around 1970.

Figure 3 shows yearly standard deviation of Portland sea level and tidal prediction. The sea level

standard deviation is calculated using all available sea level measurements for the year. The standard

deviation of the tide is based on all hourly tidal predictions made for the year. The cyclic variation is

referred to as the ‘nodal’ cycle. It is driven by changes in the moon’s declination, which varies by

approximately 5� over a period of about 18.61 years. A general increase in variability over time is

clear, both in the observed record and in the tidal predictions. Outliers typically result from poor

Figure 2. Yearly mean sea level (metres above chart datum) at Portland, Maine

SEA LEVEL CHARACTERISTICS 747


quality data. For example, only winter data were available for Portland in 1972, and this led to poor

tidal predictions for the year.

Figure 4 shows the daily standard deviation of the surge residual, which is the standard deviation for

a fixed calendar day, over all years. For example, the observation at Julian day 0 is the standard

deviation of all January 1 surge data, regardless of year. The pattern represents seasonality in

meterological forcing. In particular, standard deviations are relatively large in winter due to an

increased frequency of large storm surges in winter, as compared to summer. By way of example, the

extended excursions above 100 mm in Figure 1 are likely due to weather systems passing through the

area. The systems move relatively slowly, and so there is persistence in the surge. Further analyses

show similar patterns of seasonality in low order auto-correlations, with the surge showing greater

short term dependence in winter than in summer.

In summary, the available data show long term trends in mean, and both long and short term non-

stationarity of covariance structures. Similar temporal variation is found at other locations within the

Bay of Fundy/Gulf of Maine.

3. TIDAL ANALYSIS

Tidal analysis is used to remove astronomically forced components of the sea level record. The usual

sea level model is of the form

Yt ¼ �t þ �t þ �t ð1Þ

Figure 3. The standard deviation (metres) of sea level (s), and predicted tide (t), at Portland, Maine, by year

748 B. SMITH


where �t represents mean sea level, �t denotes the tide and �t is the surge, at time t. The tide is

generally modeled as a sum of harmonic components

�t ¼XKk¼1

Ak cosð!kt þ �kÞ ð2Þ

where the frequencies !k are assumed know based on knowledge of astronomical motions.

From a statistical viewpoint, this is a harmonic regression model. It is linearized by a change to

Cartesian co-ordinates, and so parameters can be estimated by least squares. The tidal predictions in

the center panel of Figure 1 were produced by a tidal analysis incorporating 69 known frequencies, and

1 year of hourly data. The frequencies chosen and other details of the tidal analysis can be found in

Foreman (1996). Godin (1988) describes the underlying physics.

The principal tidal constituent is known as M2 (M for moon, 2 for 2 cycles per day), and has

frequency of 0.0805114 cycles per hour, corresponding to a period of about 12.4 h. This constituent is

driven by the gravitational attraction of the moon, and its period results from a combination of the

periods of the earth’s rotation and period of the moon’s orbit about the earth. M2 accounts for the

majority of tidal variation at most locations around the globe.

Figure 4. The estimated surge standard deviation (metres), by day



Godin (1988) reported increasing M2 amplitude at St. John. Figure 5 shows a time varying estimate

of the M2 amplitude at Portland, Maine, calculated by complex demodulation (Tukey, 1961). This

procedure provides local estimates of amplitude and phase for the model

AðtÞcosð2�!t þ �ðtÞÞ

Here we have taken ! ¼ 0:0805 rad h�1 in order to extract the amplitude and phase of M2. In addition

to the clear nodal cycle of 18.61 years, an increasing trend in M2 is apparent. The trend and seasonality

in M2 is largely responsible for the trend and seasonality in sea level variation observed in Figure 3.

Similar results are found at other locations in the Bay of Fundy/Gulf of Maine, with the typical

increase in M2 amplitude being about 0.5 mm/year. This increase is not compensated by a

corresponding decrease in amplitude of other tidal constituents, and so it is an important factor,

together with the increase in mean sea level, in assessing increasing flood risk within the Bay of

Fundy/Gulf of Maine.

4. SPECTRAL ANALYSIS OF THE SURGE

When fitting the sea level model (1), the usual assumption is that the surge �t is a process with

continuous spectrum, while the tide (2) is a discrete spectrum process that can be filtered using least

Figure 5. The estimated amplitude (metres) of the M2 tidal constituent at Portland, Maine, calculated by complex

demodulation

750 B. SMITH


squares. As is clear in Figure 4, the surge variance is non-stationary, and low order autocorrelations

show similar seasonality. For this reason, spectral analyses of the surge has been restricted to fairly

short stretches of data - 2048 observations, or just under 3 months of hourly data.

The statistical methods used are described in the Appendix, and particular details of the

implementation are as in Smith and Miyaoka (1999).

Figure 6 shows log periodograms of data from St. John, together with superimposed spectral

estimates. In the left hand panels, the spectral estimates are local averages of 21 periodogram

ordinates. The predominant peaks are at integer multiples of 0.0805, the M2 frequency.

In the right hand panel, the estimates were obtained by Gaussian fitting (Whittle, 1953); a

procedure sets down an approximate likelihood for a parametric spectral model. Details are given

in the Appendix. The particular form chosen for the surge spectrum is

fXXð�j�Þ ¼XJj¼1

�j

ð j2!2 � �2Þ2 þ �2Jþ1�

2

þ �Jþ2

1 �XJþKþ2

k¼Jþ3�k expð�ik�Þ

�� 2ð3Þ

where ! ¼ 2�� 0:0805. The model has J þ K þ 2 parameters.

The first part of this expression is used to model the broad spectral peaks in the neighborhood of

multiples of M2. If the sea level in the bay is a tidally forced oscillation, then under certain simplifying

assumptions on the underlying dynamics, the response to forcing with period T will be linear

combinations of the sinusoidal motions

cos2�ð2mþ 1Þt

T

� �m ¼ 0;�1;�2; . . . ð4Þ

See Smith and Miyaoka (1999) for details. Here we take T�1 ¼ 0:0805, which implies forcing by the

M2 tidal constituent. This sinusoidal motion has a discrete spectrum with infinite peaks at odd integer

multiples of 0.0805 cycles per hour. By introducing non-linearity into the underlying equations of

continuity and motion, the response may also include energy at even multiples of M2 (Smith and

Miyaoka).

This explains the positioning of the spectral peaks, but not their broad nature. To explain the latter,

consider the following equation of an idealized, frictionally damped, forced, harmonic motion

@2YðtÞ@t2

þ �1

@YðtÞ@t

þ �2YðtÞ ¼ XðtÞ

where �1 is the frictional damping coefficient, andffiffiffiffiffi�2

pis the resonant frequency of an undamped,

resonant system. If the forcing function XðtÞ has a spectrum fXXð�Þ, then the response YðtÞ has

spectrum

fYYð�Þ ¼fXXð�Þ

ð�2 � �2Þ2 þ �21�

2ð5Þ



752 B. SMITH


The resonant frequency of the Bay of Fundy is near the M2 frequency (see Proudman, 1953, p. 232),

which means that frictional damping would spread out the spectrum of the forcing function in the

vicinity of M2. A tidal analysis at M2 would therefore give a surge residual with remaining energy

near this frequency.

The first term of (3) is an attempt to model the effects of friction, and includes a number of terms

with denominators of the form ð�2 � �2Þ2 þ �21�

2. The frequencies of the harmonic motions are

integer multiples of M2, and a single parameter �Jþ1 is used as a common frictional damping factor at

all harmonics of M2.

Based on observations of the degree of persistence in surge data such as those in the lower panel of

Figure 1, the atmospherically driven component of sea level is assumed to be a low order

autoregressive process, whose spectrum is given by the second part of (3).

Several models, consisting of different numbers of harmonic components and different autore-

gressive orders were fit to each dataset, and model selection was carried out by minimizing the Bayes

Information Critierion,

BIC ¼ �2 log Lð�Þ þ p log ðTÞ

where p ¼ J þ K þ 2 is the number of model parameters, T is the number of observations and Lð�Þ is

Whittle’s likelihood. The log spectra of the best fitting models (chosen by BIC) are plotted in the right

hand panels of Figure 6. The model for the 1950 data consisted of a first order autoregressive

(meteorologically driven component), plus three frictionally attenuated tidally forced components (at

frequencies M2, 3M2 and 5M2). The model for 1965 data consists of a second order autoregressive

and four tidally forced components (M2, 2M2, 3M2 and 4M2), and the model at 1990 consists of a first

order autoregressive and seven harmonic components (M2, 2M2, 3M2, . . . , 7M2). The seventh

harmonic lies beyond the Nyquist frequency, and has been folded back into the interval [0,0.5]. In each

case the parametric model appears to have captured the salient features of the non-parametric spectral

estimate. Models without frictional damping did not fit nearly as well.

As pointed out above, spectral energy at odd harmonics of the fundamental frequency is compatible

with a linear, Gaussian model, while even harmonics may arise from non-linearity or non-Gaussianity.

Higher order spectra are sometimes useful in assessing the nature of non-linear and/or non-Gaussian

structure. Figure 7 shows estimates of the bicoherency, a quantity related to the bispectrum, but with

simplified sampling properties (see the Appendix for details). The dashed lines on the plot are at those

multiples of M2 which were identified as model components by the BIC model selection process. The

contours are drawn at the 90th percentile of the null distribution of the bicoherency, after Bonferroni

correction for the number of estimates constructed.

For the 1990 data, there is evidence that much of the significant portion of the bicoherency, and

hence the bispectrum, occurs when one frequency is near a multiple of M2, and this is true to a lesser

extent in 1965. Following the discussion in Smith and Miyaoka (1999), this type of structure in the

bicoherency is compatible with an M2 tidally forced oscillation, and non-linear dynamics.

3——————————————————————————————————Figure 6. Spectral estimates for hourly data collected at St. John, New Brunswick; October-December 1950 (top); January-

March 1965 (middle); and March-May 1990 (bottom). Ordinate is log power, and abcissa is frequency (cycles/h). Plotted points

are periodogram ordinates. The superimposed curves in the left-hand panels are spectral estimates obtained by averaging 21

periodogram ordinates. Right-hand panel shows parametric estimates obtained by minimizing a BIC criterion based on Whittle’s

likelihood



5. DISCUSSION

An analysis of sea level records in the Bay of Fundy/Gulf of Maine indicates several forms of non-

stationarity. These include increasing mean sea level, and both long and short term seasonality of

variance. Additional findings, not presented here, showed that other second order features (estimated

autocovariance function and spectrum) also have marked seasonality. Parametric modelling of the sea

level data should therefore be restricted to short records, or must incorporate the various sources of

non-stationarity.

This work corroborates findings of Godin (1988), who reported increasing M2 amplitude at St.

John. Our estimates suggest that the increase in M2 amplitude in the Bay of Fundy/Gulf of Maine is

about 10% of the magnitude of the increase in global mean sea level, and it is therefore vital that the

increase in M2 be accounted for in making flood risk assessments.

An analysis of the St. John record from January to March 1991 was carried out by Brillinger (1994),

who pointed out the unexpected breadth of the spectral peaks. We have attributed this to frictional

attenuation, which is in keeping with Ku et al. (1985), who suggested that features of the Bay of Fundy

record may be the result of frictional forces.

The models fit here, and the Gaussian fitting method in general, are capable of identifying only

linear, Gaussian, structure. However, the bicoherency estimates suggest the presence of a non-linear or

non-Gaussian structure. Further work will focus on the identification and fitting of non-linear models

using the bispectral fitting method (Brillinger, 1972).

The longer term goal of this work concerns the assessment of flood risk in the Bay of Fundy/Gulf of

Maine. In this direction, future work will focus on modeling the point processes of exceedances of

high levels, with the parametric form of the intensity function to include features identified here,

including increasing mean sea level and seasonality.

Figure 7. Estimated bicoherency for the St. John data; October-December 1950 (top); January-March 1965 (middle); and

March-May 1990 (bottom). The axes are frequency in cycles per hour. Dashed lines indicate multiples of the M2 frequency at

which spectral components were identified using BIC. The contour level is the 90th percentile of the null distribtion, Bonferroni

corrected for the number of estimates plotted

754 B. SMITH


ACKNOWLEDGEMENTS

This research was partially supported by a grant from the National Sciences and Engineering Research Council ofCanada. The author thanks David Greenberg and Keith Thompson for helpful discussions.

APPENDIX

Let Xt be a stationary process having spectral representation Xt ¼Ð 2�

0ei�tdZXð�Þ. The spectrum fXXð�Þ

and bispectrum fXXXð�; �Þ of the process are given by

CumðdZXð�Þ; dZXð�ÞÞ ¼ �ð�þ �Þ fXXð�Þd�d�

CumðdZXð�Þ; dZXð�Þ; dZXð!ÞÞ ¼ �ð�þ �þ !Þ fXXXð�; �Þd�d�d!

where �ð�þ �Þ is a 2� periodic extension of the Dirac function.

If T observations X0; X1; . . . ;XT�1 are made at regular time intervals, the discrete Fourier

transform, periodogram and biperiodogram of the data are defined respectively as

dðTÞX ð�Þ ¼

XT�1

t¼0

Xtei�t

IðTÞXX ð�Þ ¼

1

2�TdðTÞX ð�Þ

�� 2

and

IðTÞXXXð�; !Þ ¼

1

ð2�Þ2TdðTÞX ð�ÞdðTÞX ð!ÞdðTÞX ð�� !Þ

These can be locally smoothed to give estimates of the spectrum and bispectrum as:

fðTÞXX ð�Þ ¼

Xj

Wð1ÞT ð�� jÞIðTÞXX ð�jÞ

fðTÞXXXð�; !Þ ¼

Xj;k

Wð2ÞT ð�� j; !� !kÞIðTÞXXXð�j; !kÞ

where the weight functions Wð1ÞT and W

ð2ÞT are concentrated in the neighbourhood of the origin. Details

can be found in Brillinger and Rosenblatt (1967).

Because of its simplified sampling properties as compared to the bispectrum, the sample

bicoherency

hðTÞXXXð�; !Þ ¼

fðTÞXXXð�; !Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

fðTÞXX ð�Þ f ðTÞXX ð!Þ f ðTÞXX ð�þ !Þ

q

is often considered. It is an estimator of the bicoherency

hXXXð�; !Þ ¼fXXXð�; !Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

fXXð�Þ fXXð!Þ fXXð�þ !Þp



In particular, if spectral and bispectral estimators are constructed by averaging L periodogram or

biperiodogram ordinates and hXXXð�; !Þ ¼ 0, then the bicoherence jhðTÞXXXð�; !Þj2

is approximately

exponentially distributed with mean T=ð4�LÞ, which allows for straightforward testing of zero

bicoherency or bispectrum.

Parametric spectral models can be fit by Gaussian fitting, a technique introduced by Whittle (1953).

The method sets down an approximate negative log likelihood as

�log Lð�Þ ¼Xj

ðlogð fXXð�j j �ÞÞ þ IðTÞXX ð�jÞ=fXXð�j j �Þ ð6Þ

where the spectrum fXXð�j j �Þ is parameterized by �.The form of the likelihood is motivated by the asymptotic normality of the discrete Fourier

transform, and minimization of (6) leads to an estimate of �. See Brillinger (1985) for details on this

and other statistics based on the discrete Fourier transform.

REFERENCES

Brillinger DR. 1985. Fourier inference: some methods for the analysis of array and non-Gaussian series data. Water Res. Bull.21(5): 743–756.

Brillinger DR. 1994. Comment on ‘Report on statistics and physical oceanography’. Statistical Science 9(2): 201–202.Brillinger DR, Rosenblatt M. 1967. Computation and interpretation of k’th order spectra. In Spectral Analysis of Time Series,

Harris B (ed.). Wiley: New York; 153–188.Foreman MGG. 1996. Manual for tidal heights analysis and prediction. Pacific Marine Science Report 77-10, Sydney, BC.Godin G. 1988. Tides. Anadyomene Edition: Ottawa.Ku LF, Greenberg D, Garrett CJR, Dobson FW. 1985. Nodal modulation of the lunar semi-diurnal tide in the Bay of Fundy and

Gulf of Maine. Science 230: 69–71.Proudman J. 1953. Dynamical Oceanography. Wiley: New York.Smith B, Miyaoka E. 1999. Frequency domain identification of harbour seiches. Environmetrics 10: 575–587.Tukey JW. 1961. Discussion, emphasizing the connection between analysis of variance and spectrum analysis. Technometrics3: 1–29.

Whittle P. 1953. The analysis of multiple stationary time series. J. R. Statist. Soc. B 15: 125–139.

756 B. SMITH


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Sea level characteristics in the Bay of Fundy/Gulf of Maine