Spectral analysis of storm surge in Hong Kong Victoria Harbour

Spectral analysis of storm surge in Hong Kong Victoria Harbour Stephen K. W. Tou Senior Research Engineer, Norton Christensen, Diamond Technology Center, 2532 South 3270 West. Salt Lake City, Utah 84119, USA K. Arumugam Head qfi Fluids Division, Department of Civil Engineering, The City University, Northampton Square, London ECI V OHB, England Based on a linear model the dynamic characteristics of Victoria Harbour (Hong Kong) is obtained by means of spectral analysis of the storm surge hydrographs. The results show that the harbour is an ideal one which has a small gain factor and a flat response in the frequency range from 0 to 6 x I0- s Hz. The results also show that the power spectra possess the narrow band features which indicates that the periodic components associated with tidal motions are predominant over the random components. The power spectrum corresponding to a frequency of 2.3 x 10- s Hz is likely to be associated with the astronomical tides. The peaks in the power spectra at zero frequency suggest that the pumping mode of oscillations is dominant in a storm surge. This mode of oscillations represents the temporal variations in mean sea level. To demonstrate the full potential of the present model, more case studies should be conducted when surge as well as non-surge data are available. Key Words: spectral analysis, spectrum, storm surges, harbour oscillations INTRODUCTION It has been suggested that more work should be done in the response of model and prototype of a harbour when subjected to various inputs in addition to uniform periodic incident waves 1. Hong Kong is frequently threatened by tropical cyclones. In this connection, further investigation into the problem associated with harbour oscillations should seek to extend the knowledge on the dynamic characteristics of Victoria Harbour and the behaviour of moored vessels and other floating or anchored structures under the action of surge excitation. One must be able to predict the expected surge amplitudes at a number of locations of immediate concern. DATA ACQUISITION AND H A R B O U R GEOMETRY Fig. 1 shows the geometry of the harbour and the locations of the two tide gauges from which the surge hydrographs are obtained. Since only two data sampling points were available and only the sea surface elevations were used in the analysis, the locations of these tide gauges were important. This is because the kinetic energy of the system in a form of fluid velocity head may not necessarily be reflected in the sea surface elevation. The kinetic energy and the potential energy can be exchanged due to local topographical configurations, bottom friction and other nonlinear effects. The location of the North Point tide gauge was nearly at the middle of the harbour to minimize Accepted October 1985. Discussion closes November 1986. 0309 170886/030179-0652.00 1986 Computational Mechanics Publications

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Adv. Water Resources, 1986, Volume 9, September

the nonlinear effects as well as the conversion of kinetic energy into potential energy by piling up of water along the coast lines. This location is likely correspondent to the antinode of standing waves in the harbour. The tide gauge at Wanglan Island was located at a distance sufficiently far away from the land boundary. The average depth of the harbour is about 14 metres. The narrowest entrance has an opening of about 1.5 km on the west side. The fundamental frequency of the harbour is calculated to be 1.4x 1 0 - 4 H z 2. SPECTRAL ANALYSIS T E C H N I Q U E The subject of the dynamic characteristics of harbour has been investigated by a number of workers t for basins with simple configuration and constant water depth, The treatment usually requires the assumption of irrotational motions and simplified boundary conditions. The numerical model has the advantage of being flexible and able to handle nonlinear problems easily. It gives valuable insight into harbour behaviour under surge excitation. However, it requires empirical coefficients or calibration and entails a time-consuming step by step integration. A great deal of interest has recently evolved in the use of power spectral technique for analysing off-shore structures. The power spectral technique has only been developed t~ the level for a linear model but may be sufficient f,,r practical engineering design/analysis purposes. This first order approximation method is ideally suited to the study of dynamic characteristics of harbour in the frequency domain.

Spectral analysis

of storm surge: S. Tou and K. Arumugam

0 Figure i .

Victoria Harbour and the Locatior of Tide Gauges ~

North

0

Kowloon

ca N'~rth'--~oint'~'Tide Gaugel --

o

,,,

C "l-

"%

"f7'01

La,, au

.o.g '~~ll [

Island

"8

Kon,

o

__J'! --,/Tide

~ll~i'll Fig. I.

Lama Island

$

Gauge

%

at Wanglan Islan(

Scale I.0 km

Victoria Harbour and the locations of tide gauges

The dynamic response of harbour depends upon the natural periods of the harbour with respect to the energy available in the storm surge in open sea. In this context, the distribution of energy content as a function of frequency is represented by a power spectrum. The power spectrum is obtained by analysing a given time series. It provides insight into the possible effect of the surge phenomenon as well as furnishes a means for estimating the response. The spectral analysis of storm surge presents a very efficient tool in the frequency domain to reveal the dominant frequencies that are not readily discernible in a time series. If there is forced resonance in the harbour, the spectrum will show an energy peak corresponding to the harbour's natural frequencies.

component

iRandom

R(t).Periodic

component P(t)

n(t)

T I M E SERIES ANALYSIS Storm surge is a transient phenomenon which can be regarded as a random oscillation composed of reflection, diffraction and attenuation of forced surface waves with respect to time. This special class of nonstationary random processes is usually of short duration and has a clearly defined beginning and an end. The surge hydrograph can be treated as a time series. In general, a time series can be considered as a sum of three components (Fig. 2): a periodic component P(t), a trend component H(t) and a random component R(t), i.e.,3:

~l(t)= P(t) + H(t) + R(t)

(1)

Time

t. Fig. 2. Time series demonstrating the components of parameter variability In the absence of any external disturbances except tide, the surge hydrograph may be represented adequately by a periodic function in terms of a trigonometric series. The periodicities of the hydrograph will appear as discrete lines in the power spectrum. Because the mean sea level changes as storm develops or decays in intensity over a

Adv. Water Resources, 1986, Volume 9, September

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Spectral analysis Of storm surge: S. Tou and K. Arumuyam time scale of hours and days, the transient behaviour of storm surge is a nonstationary process for which the average of the mean sea level is no longer a constant but a function of time. This temporal variation in mean sea level average is known as a trend. The trend component, when it exists in a series, becomes discernible only when the data record is long in duration. It will occur in the power spectrum as a peak at the pumping mode (i.e., zero frequency). A polynomial is usually used to describe a trend. The random component of a time series is the fluctuating component that has no deterministic pattern of behaviour and is described only by its statistical properties. If there are random components in a time series, they will be concentrated in certain frequency bands. Any one component or any combination of the three components may exist in a particular time series. In the case of storm surge, all three components are present because storm surge is a forced transient disturbance of progressive type having wave lengths comparable to those of the astronomical tides 4. An important characteristic of transient data, contrary to periodic and almost periodic data is that a continuous spectral representation for transient data is obtained rather than discrete spectral representation. Since the storm surge power spectrum describes the distribution of long period wave energy with respect to frequencies, it can be used to examine the strength of the three components individually. The dynamic characteristics of the harbour can then be determined accordingly. This is because, for a linear system, it is possible to work entirely in the frequency domain and link the response spectrum to the storm surge forcing spectrum via the harbour transfer function.

P R O C E D U R E O F SPECTRAL ANALYSIS

The autocovariance function is defined as: R(z)=~_~] 0

12)

(q(t)-~l(t))Ol(t+r)-il(t))dt

which describes the characteristics of the hydrograph in the lag time domain and is solely dependent on the time lag ~. For numerical integration, equation (2J is written as: 1

- q

R ( Z ) = N - q i~l (qlti)--IllOl[ti+')--l]l

13)

where r = q A t

¢I=Ni=12"tl([i) N = t o t a l number of sample data and q = 0 , 1, 2, 3 . . . M (the lag number) The number of lags, M, used in the autocovariance function is determined by the period to be resolved. Too many lags used in an analysis will reduce the accuracy of the spectral estimates, whereas too few will decrease the resolution of the spectral components. It has been recommended 5"~ that the optimum number of lags should be

M"O. IN Figs 3 and 4 show the autocovariance function for the surge induced by Typhoon Iris in the open sea (Wanglan Island) and Victoria Harbour (north point) respectively. It can be seen that R(r) remains positive for lags up to 6 h, indicating that the upsurge tends to persist for at least 6 h. For negative values of R(z), the upsurge will likely to be followed by a down-surge at some time : later. No

The procedure of spectral analysis follows that of a time series analysis for which the original hydrograph is manipulated in a digital form. The spectral structure of transients can be evaluated in a manner similar to the power spectrum of stationary random data (1). The method requires the determination of the appropriate autocovariance function from a given time series which is then Fourier transformed to yield the corresponding power spectrum.

.14

.12

.I0

The autocovariance fimction The autocovariance function is used in the study of the characteristics of surge hydrographs. It basically measures the degree of self-similarity and provides insight into the persistence underlying the surge phenomenon. To obtain such an autocovariance function, the time series of the hydrograph is first digitized at a sampling interval At which is carefully chosen to ensure that no significant energy exists at frequency above g/At, i.e., the Nyquist frequency. This sampling interval will resolve the aliasing problem and hence avoid folding energy into the lower frequency estimates. In the present study, the sampling interval is 0.5 h which will resolve one-hour period waves. This is because the high frequency components are filtered out by the tide gauge at North Point and because the fundamental frequency of the harbour is less than that of the one-hour period waves. This sampling interval will exclude the wind/gravity wave energy. The time history of the recorded data covers 85 h.

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Adv. Water Resources, 1986, Volume 9, September

R('C)

1112

.08

• 06

.04

.02

3.6

7.2

I0,8

14.4

18

21.<, 3

I

x

i0

se

Fut. 3. Autocovariance function (~f storm surges in the open sea

Spectral analysis of storm surge: S. Tou and K. Arumugam frequencies. Since the long period wave energy is proportional to the mean square of sea surface elevation, the subject density function then describes the distribution of long period wave energy between frequency components.

F. (~') m2

Smoothing of spectral estimates In order to improve the accuracy of the spectral estimates, it is necessary to apply a running average on adjacent estimates by some kind of digital filters. A digital filter is basically a spectral window which is represented by discrete coefficients to generate the smoothing weight function. Hamming window is the most simple and popular one and is adopted in this study to smooth out some of the distortions of the spectrum, i.e.:

.14 .12 .1 • ,38 .06 .04 .02

3 ~

iJ 8

18

)) S(fo) =:(S(fo)+S(f~ '

(8)

S(fk)=¼(S(fk_,)+2S(L)+S(L+,))

(9)

s(A,)=k(s(A,_ , )+ s(L,))

1 }

I

x I0

Fig. 4. Autocovariance function of storm surges in Victoria Harbour

correlation or persistence is indicated approaches a value of zero.

when

R(~)

Spectral density function The underlying principle of spectral analysis is the application of Fourier transform of the autocovariance function from the time lag domain to the frequency domain. Since surge is measured only at a single point, the one-dimensional, one-sided spectral density function is obtained as follows: S(f) = 4

R(v) cos(2n,h) dr

0 < f < ,x

(lO)

sec

(4)

Using the trapezoid rule to approximate the above integration one has: S ( f ) = 2At(R(0)

where k = 0 , l, 2, 3, 4 . . . M - I

Power spectrum Figs 5 and 6 show the power spectrum of the surge input from the open sea and the output response in Victoria Harbour for Typhoon Iris. Both power spectra consist of fairly continuous and narrow band of energy in the range of 0 to 0.8 x 10 -4 Hz. This continuous spectral representation is typical for transient phenomenon such as storm surge. The narrow band feature suggests that the periodic components are predominant over the random components in a storm surge. The power spectra corresponding to the frequency of 0.23 × 10 .4 Hz are likely to be associated with the astronomical tides. This tidal frequency is typically the semidiurnal tidal frequency in Hong Kong coastal waters. The peaks in the power spectra at zero frequency (or the pumping mode) indicate that the pumping mode of oscillations is dominant during a storm surge. These energy peaks also show that: (1) The surge hydrographs have trend components due to the temporal variations in mean sea level. O

m~

sec

.%1 - 1

+2 ~

R(q) cos(2~fqAt) + R(M) cos(2nfMAt))

q=l

(5)

$1( 0

sooo

\

The frequency of the spectrum corresponding to each lag is calculated as follows" 4000

f=--

k

(6)

2ArM

such that M-I

S(jkl=2At(R(O)+ 2 ~

3000

/

(nqk)

R(qAt)cos - ~ -

/

Semidiurnal

tidal

frequency

/

2000

+(- I)kR(M))

q=l

(7)

i000

for k=O, 1, 2, 3 . . . M The spectral density function may be interpreted as describing the manner in which the mean square of the sea surface elevation is distributed with respect to the

@

Y

2.3

4.6

6.9

i

IL5

13.8

I

1~.2

18.5 f

Fig. 5.

i

20.8

i

23

I

I

25

27.5

x

lO-5ttz

Power ,spectrum in the open ,sea

Adv. Water Resources, 1986, Volume 9, September

181

I

Spectral anah'sis of storm surge: S. Tou and K. Arumugam Z

where S 2 ( f ) = p o w e r spectral density of oscillation at a point (North Point) inside the harbour. $1 ( f ) = power spectral density of open sea surge at the boundary (Wanglan Island). H ( f ) = harbour transfer function.

sec

52(" ~:,: :,

/

4000

5e=i,{=urna_

tidJl

Equation (11) is not restricted to a single degree of freedom system, but also applicable to system with single output and single input provided that the relevant transfer function is used. The detailed form of the transfer function depends on the nature of the excitation and response variable as chosen to characterize the problem. Knowing the surge input and response spectra, one can proceed to determine the dynamic characteristics of the harbour [or the system gain factor) as a function of frequencies, i.e.:

frequenc,,

3000

23CC

IH(,f )lz = S2( f)/SI ( f )

The system gain factor for Victoria H a r b o u r is shown in Fig. 7. The results show that the harbour surge in response to the open sea surge much likes a highly damped springmass system as the harbour is well protected by the narrow entrances and there would not be sufficient time to increase the energy density in the harbour to a high level. Victoria H a r b o u r is therefore an ideal one which has a small gain factor and a flat response in the frequency range from 0 to 6 x 10 -s Hz.

12,2,2

:

Fig. 6.

~

~

6

~

1#

12

14

]O

18 f

20

I

I

I

x

2:'_ 5 i0

Hz

22

(12)

2O

Power spectrum in Victoria Harbour

(2) The low frequencies are so close to zero that they also appear as trends. In storm surges, it is most likely that the pumping mode of oscillations represents the temporal variations in mean sea level entirely. The lack of energy in the frequency range from 0.8 x 10 -4 Hz to 2.8 x 10 -4 Hz is because the storm surge does not generate wave energy in that frequency band. Instead, wave lengths comparable to those of astronomical tides are generated 4.

CONCLUSIONS The spectral analysis technique is shown to be effective to resolve a dynamic problem in the frequency domain. The

,l(,

In the scaling of oscillatory flows in the harbour, the effect ofcoriolis force is small as compared to that of inertia. The effect of coriolis force is therefore neglected. Nevertheless, the effect of convective m o m e n t u m transport and the effect of bottom friction are important especially when the harbour is small and shallow and the boundaries are complicated. These nonlinear effects are to be considered in most cases. In view of the situation that the power spectral technique has only been developed to the level for a linear model, the present analysis assumes the first order approximation by excluding these nonlinear effects from the governing Navier Stokes equation. Such analysis may be considered necessary and sufficient for small oscillations and for practical engineering applications with first order approximation. Based on the above considerations, the response of a harbour subjected to a storm surge forcing is now approximated by a linear model such that the mean square of harbour oscillation is computed as follows:
,

t,~lU ~ ,, l , r

o S2(,/')d/" .... u.

=

S,(flH(f)H*(f)df

(11)

~0

182

" i IL~.1*' Lnput / o u t l:~J

DYNAMIC CHARACTERISTICS OF VICTORIA HARBOUR

Adv. Water Resources, 1986, Volume 9, September

1 2.

I a.

b,

'~. i'rt.quenl~

Fig. 7.

System gain .factor jor Victoria Harbour

l × l~J ' }i

Spectral analysis of storm surge: S. Tou and K. Arumugam study shows that Victoria Harbour is an ideal one which has a small gain faetor and a flat response in the frequency range from 0 to 6 x 10- ~ Hz. The power spectra possess the narrow band feature which suggests that the periodic components associated with tidal motion are predominant over the random components during storm surges. The results show that the frequency range in the analysis is entirely within the tidal energy band. The power spectrum corresponding to a frequency of 0.23x 1 0 - 4 H z is likely to be associated with the astronomical tides. It would be interesting to Perform the same analysis during a nonstorm period and compare to the present results. The peaks in the power spectra occur at zero frequency indicating that the pumping mode of oscillations is dominant in storm surges. This mode of oscillations represents the temporal variations in the mean sea level. The nonlinear effects are difficult to grasp at present but should be investigated when such theory becomes available so that improvement over the linear model can be made.

NOTATION f k N

q,M T t At

~l(t}

frequency kth spectral estimate of the spectral density function total number of sampling points lag number time span of recorded data time coordinate sampling interval lag time surge height

REFERENCES I Wiegel,R. L. Oceanographical Engineering, Prentice-Hall, 1964 2 "Tou.S. K. W. and Arumugam,K. The Natural Modes ofHong Kong Victoria Harbour by the Finite Element Method, to be published 3 Rich,L. G. EnvironmentalSystemsEngineering,McGraw-Hill, 1973 4 Schenck,H,, Jr lntoduction to Ocean Engineering, McGraw-Hill, 1975 5 Bendat, J. S. and Piersol, H. G. Random Data: Analysis and Measurement Procedures, Willey-Interscience, 1971 6 Newland,D. E. An Introductionto RandomVibrations and Spectral Analysis, tongman, 1975

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