AUTOMATED TIDAL REDUCTION OF

SOUNDINGS

E. G. OKENWA

November 1978

TECHNICAL REPORT NO. 55

PREFACE

In order to make our extensive series of technical reports more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text.

AUTOMATED TIDAL REDUCTION OF SOUNDINGS

E.G. Okenwa

Department of Surveying Engineering University of New Brunswick

P.O. Box 4400 Fredericton, N.B.

Canada E3B5A3

November 1978

Latest Reprinting January 1993

ABSTRACT

In Hydrographic Surveying, soundings are reduced to a

chart datum established at a reference gauge station from

a long period of tidal observations. Unfortunately, due to

the variations in tidal characteristics from place to place,

soundings can only be reduced to the chart datum within the

v icin_i ty or the gauge station. As we move away from the

gauge station, 'it becomes necessary to obtain new information

on the tidal characteristics and apply necessary corrections

to the chart datum to obtain an appropriate sounding datum

for reducing the soundings.

To reduce soundings means to subtract the heights of

tide, at the sounding locations and at the times of

soundings, from the depths sounded to obtain the depths

rPfcrenced to the chosen datum. hlanual reduction of sound­

ings is a tedious aspect of the field hydrographer's list

of ehorcs. There have been some attempts to automate the

tidal reductions using digitized cotidal charts.

The objective of this work has l;een to develop

alternative approaches to automated tidal reductions, namely,

using analytical cotidal models. The range ratio aHd time

lag fields have been approximated by surfacesdescribed by

two dimensional algebraic polynomials (Pn(~.A)). ThP

observed time series at a referenc.P station has he~''ll

--"'roxima ted by one dimensional trigonometric polynomial

Wi til tiH· coe f ficj en ts or t.iH•sc Pu lynorni a 1 s s t.o rr·d in

the computer, the range ratio and 1he time lag at auy point

(¢., >..)in the area can readily bE' predicted and the height 1 1

of tide at the point and at time t can be predicted from

the predicted height of tide at thP reference station.

Test computations, using data from the 'Canadian Tides

and Current Tables, 197R' for the Bay of Fundy have been

done. It has been shown that the water level (h) at a loca-

tion (<Pi' >.i) can be predicted with a standard deviation

(c!hi) of 0.5 m or better.

iii

TABLE OF CONTENTS

Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i i

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i x

I.

II.

INTRODUCTION ................................. . 1

1.0 Chart Datum and OthAr Water Levels . :.. . .. 1 1.1 Sounding Datum........................... 6 1.2 Reduction of Soundings . . . .. . . ... .. .. .. .. . 13

ANALYSIS AND PREDICTION OF TIDES ............. . 16

2. 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1 Theory of Tide Generation ... .... ..... .... 16

2.1.1 The Movement of the Moon (Real) and the Sun (Apparent) . .. . . . . . . .. . 19

2.1.2 The Tide Generating Forces and Potentials . . . . . . . . . . . . . . . . . . . . 21

2.1.3 Development According to Darwin ( 1886) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.1.4 Development According to Doodson (1921) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Least Squares Harmonic Analysis and · Prediction of Tides . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Tidal Analysis and Prediction by Hesponse Method . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2 . 3 . l General . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 2.3.2 Brief Outline of the Theory of

the Response Method............... 47

III. COTIDAL CHARTS AND THEIR USES . .. .... .. .. .. .. . . 49

3.0 Introduction . . . . . . . . . .. . . .. . . . . . . . . . . . . . . 49 3.1 Types of Cotidal Charts . .. .. .. .. .... .. .. . 49

3.1.1 Range/Time Cotida1 Charts . .. .. ... . 49 3.1.2 Amplitude/Phase Cotida1 Charts . . . . 51

3.2 Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . . . 54 3. 3 Uses of Cotidal Charts . . . . . . . . . . . . . . . . . . . 60

iv

Page

IV. THE PROPOSED ANALYTICAL SCHEME 65

4. 0 In traduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.1 Models ...................................... 67 4.2 Least Squares Solution of the Models ........ 70 4.3 Data Requirements and

Reduction Aligorithms ....................... 76 4.3.1 Amplitude/Phase Cotidal Models ........ 76 4.3.2 Range/Time Cotidal Models ............. 78

V. TEST COMPUTAIONS AND RESULTS ..................... 82

5.0 Data ........................................ 82 5. 1 Computations and Results . . . . . . . . . . . . . . . . . . . . 8 7

5.1.1 Determination of the Coefficients of the Approximating Polynomials ...... 87

5.1.2 Least Squares Polynomial Approximation of Observed Time Series at the Reference Station ....... 100

5 .1. 3 Tidal Reductions ..................... 101

VI. CONCLUSIONS AND RECOMMENDATIONS .................. 107

REFERENCES. . ...................................... l 09

APPEND! CES ....................................... 114

I. Outline of the Least Squares Approximation Theory .................... 115

II. Brief Description of the C?mputer Programs Used .................. 122

III. Canadian Definitions of Chart and Sounding Datums ................................. lS6

v

Figure 1-1

Figure 1-2

Figure 1-3

Figure 1-4

Figure 1-5

Figure 2-1

Figure 2-2

Figure 2-3

Figure 2-4

Figure 2-5

Figure 2-6

Figure 3-1

Figure 3-2

Figure 3-3

Figure 4-1

Figure 5-1

Figure 5-2

Figure 5-3

Figure 5-4

LIST OF FIGURES

Page

Relationship Between Various Water Levels

Bay of Fundy - Variation & Tidal Ranges and Sounding Datum Along the Northern Coast

Bay of Fundy - Variation of Tidal Ranges and Sounding Datum Along the Southern Coast

Variation of Sounding Datum in an Estuary or a River

Tidal Reduction Curve

Line Spectrum of Function h(t)

Belationship Between the Orbital Motions of the Moon and the Sun

Effects of the Gravitiational Attraction of a Heavenly Body M on the Earth

Di~tribntion of Tidal Forces

Orbital Parameters

Time Relationships

Range/Time Cotidal Char~

Amplitude/Phase Cotidal Chart

Digital Cotidal Chart - Hudson Bay 1976

4

8

9

ll

14

18

20

23

26

31

50

53

63

Tile Proposed Analytical Scheme- Flow Chart66

Bay of Fundy - Tido Gauge LocR~io~s 85

Bay of Fundy - Grid Nun,bering !::.15

Bay of Fundy - Range/Time Cotidal Curves Using the Proposed Analytical Cotidal Models 96

The Average Progression of the Semi-diurnal Tides in the Bay C' r Fundy 90

Figure ll-1

Figure II-2

Figure II-3

Figure Ill-1

Program for the Polynomial Approximation of Cotidal Curves - Flow Chart

Program for the Polynomial Approximation of Observed Time Series - Flow Chart

Program for the Tidal Reduction -Flow Chart

Reference Surfaces and Water Level Variations

\" i i

Page

123

141

151

158

Table 2-1

Table 5-l

Table 5-2

Table 5-3

Table 5-4

Table 5-5

Table 5-6

Table 5-7

Table 5-8

Table 5-9

LIST OF TABLES

Principal Tidal Constituents as Derived by Doodson

Bay of Fundy - Tidal Information on Secondary Ports

Tide Observations at the Port of Saint John

A Posterori Variance Factors for Various Degrees of the Polynomials

:Fb:urier Coefficients After Discarding ~hbse of Them Greater Than Their Standard Deviations

Original Coefficients of the Polynomials and Their Associated Standard Deviations

Predicted Range Ratios and Time Lags at

Page

35

83

86

89

93

94

fhe Grid Intersections 97

Coefficients of the Polynomial for the Time Series at the Reference Station 102

Tidal Reductions 105

Difference Between Predicted and Observed Values 106

viii.

ACKNOWLEDGEMENTS

I wish to express my deep gratitude to my supervisor

Dr. D.B. Thomson who suggested this topic and gave me

valuable guidance throughout the research.

My gratitude also goes ~o Dr. P. Vanicek for his use­

ful suggestions and advice, Mr. D.L. De Wolfe of the Bedford

Institute of Oceanography for the useful discussions I had

with him, and my fellow graduate students for useful exchange

of ideas and for their helpful attitudes.

I would also like to express my gratitude to the

University o~ Nigeria, Nsukka for offering me the opportunity

to undertake this research.

My thanks goes to Mrs. Rao for her excellent typing.

Finally, I wish to dedicate this work to my wife,

Chinyere, in appreciation of her support and encouragement

and for takjng care of our baby.

ix

I. INTRODUCTION

1.0 Chart Datum and Other Water Levels

The hydrographic surveyor must refer all his depth and

height measurements to a reference datum. This reference

datum, generally called chart datum, is a low water datum

which by international agreement is so low that water level

will seldom fall below it. The chart datum is, for purposes

of integration and consistency, normally tied to the

Geodetic datum which is usually defined by the mean sea

level. For bxample, over a period of some years, tide

gauges in Canada have been tied to the Geodetic Survey of

Canada Datum (G.S.C.D.) [Atlantic Tidal Power Engineering

and Management Committee Report, 1969]. This geodetic datum

is based on the value of the mean sea level prior to 1910

as determined from a period ot observations at tide gauge

stations at Halifax and Yarmouth, Nova Scotia and Father

Poj_nt, Quebec on the East Coast, and at Prince Rupert,

Vancouver and Victoria on the Paci fie. Mean Sea Lend

(M.S.L.), as its name implies. is the mean level taken up

by the sea. It is determined at a tide gauge station from

a long period of tide observations The geoid, which is

supposed to be the datum for the lwights, is defined as

2

"that equipotential surface which on the average coincides

with the mean sea level" [Thomson, 1974]. It therefore

leaves the problem of mean sea level determination to be

solved in order to define a height datum.

It is not easy to determine mean sea level since the

actual level of the sea is continuously changing.

Wemelsfelder [1970] , in his paper titled, 'Mean Sea Level

as a Fact and as an Illusion', outlined two concepts of mean

sea level: the Physical concept and the Emperical concept.

The Physical concept according to him 'is that of a common

parlance', it is the concept used in the verbal description,

"the height of the mountains above sea level". This concept

has the intent to overlook every motion of the sea, it

intends to say, no waves, no tides, no storm surges, no

wind influences, no seasonal changes, no density anomalies,

no temperature anomalies. The mean sea level is rather

conceptualized as, 'a physical object existing primarily

in space, the way in which the ocean spans the earth.'

The emperical concept tries to quantify the mean sea

level as the mean observed water levels at a tide gauge

station over a period of time. This mean level even on the

same sea varies from one tide gauge location to another and

varies also with different time epochs. Wemelsfelder, [1970],

enumerated 33 factors influencing the variations in the mean

sea level and grouped them under global, regional, 1< cal

3

and instrumental influences. Bamford, [1971], observed that

apart from tidal forces wpose mean effect over a long period

should be zero. other forces cause the mean sea level to

depart appreciably from a.n exact level (equipotential)

surface. Thomson [ 1974] , further noted that, 'the problem of

determining the true physical surface of the oceans is

analogous to that of using Stoke's formula for·geoid deter-

mination - we would require an infinite number of tide

gauges, atmospheric sensors, sea temperature and density

determinatioris'. It appears then that mean sea level, thus

the geoid, cannot be easily determined.

The various other water levels*that can be used as a

datum, or that will be relevant to the subject matter of

this work, will now be briefly defined and each is

illustrated in Figure l-1.

The average of recorded values of all the high and low

waters over a period is called the Mean Tide Level (M.T.L.).

It is obtained more easily than mean sea level and as such

is sometimes used in calculations instead of the M.S.L.

The average throughout the ye~r of heights of high

waters during the spring tides is termed Mean High Water

Springs (M.H.W.S.). The average throughout the year of the

heights of low water during tht~ spring tides is called Mean

Low Water Springs (M.L.W.S.).

Mean High Water Neaps (M.:i.W.Il.) is the average

*see Appendix Ill for further details reg;~rding definitions used in Canada.

......... M .. H.W.N.- •

Fi§ure 1-1

---

at Time. 't'

1'1

g ~----~------oil!( C)

......__ ......... _

\

............. .--. ......

AW~wtion~hfp S.tween Various Water Levels

5

throughout the year of heights of high waters during the

neap tides and the average throughout the year of heights

of low water during the neap tides is called Mean Low Water

Neaps (M.L.W.N.).

The highest tide which can be predicted to occur under

average meterological conditions and under any combination

of astronomical conditions is termed Highest Astronomical

Tide (H.A.T.), while the lowest predictable tide is called

the Lowest Astronomical Tide (L.A.T.).

Chart datum, as previously stated, is a low water

level. It is the datum to which all soundings on published

charts are reduced and to which tidal predictions and tide

readings are'referenced. Ideally, Lowest Astronomical Tide

level should be taken as chart datum. But, since we cannot

accurately define it, we choose chart datum arbitrarily as

close to L.A.T. as possible such that, (i) tides will

seldom fall below it, (ii) j_t is not so low as to give

unduely shallow depths.

6

1.1 Sounding Datum

When a chart datum is chosen; it can only be used with­

in the vicinity of the gauge location [Atlantic Tidal Power

Engineering and Management Committee, 1969]. Depending on

the variation of tidal characteristics, it is not advisable

to reduce depth measurements to this chart datum if the

reference tide gauge is more than 8 km away [Admiralty

.Manual of Hydrographic Surveying, 1969]. This leads to the

neyessity of establishing a local sounding datum. In the

Admiralty Manual of Hydrographic Surveying, 1969 , the

following rules are given as a guide to the choice of

sounding datum:

(i) if possible, a sounding datum should agree with

the chart datum.

(ii) changes in a sounding datum within the area of

interest must be made whenever the nature and

range of tides alter appreciably. It is difficult

to lay down precise figures, but a difference in

range of about one metre between two places would

normally indicate the necessity for a change of

datum somewhere between them.

(iii) the time difference between tides experienced at

two places will not have any effect on the

difference of sounding d~' tum between two points.

It may however have a considerable effect un the

7

value of the reduction required to reduce soundings

to datum. Therefore, it is important, even if the

sounding datum does not alter, to obtain time

differences between tidal stations so that time

differences may be interpolated and applied ·to

observed heights of tide used for the reduction of

soundings.

(iv) If there is any doubt in the surveyor's mind

concerning the behaviour of the tide, he. should set

up another tide gauge to find out what is happening.

Figures l-2 and l-3 show how the tidal ranges change

along the southern and northern coasts of the Bay of Fundy.

At Yarmouth~ the range at the spring tides is about 4.9

metres (16 feet). The range increases to the east and at

Burnt Coat Head, a distance of about 290 km away, the range

reaches about 16.7 metres (55 feet). Along the northern

coast, the range is about 8.5 mteres (28 feet) at Eastport,

Me. and increases going eastward, and at Joggins Wharf,

the range is about 12.2 metres (40 feet).

If a datum was established at Yarmouth or Eastport, Me.

for the reduction of soundings, as the soundings progressed

eastwards, the sounding datum should be altered. The ideal

thing is to alter a sounding datum in a series of steps.

Figures 1-2 and 1-3 depict the altPration of a sounding

datum in steps of 0.6 m (2 feet). The correction to be

+30 ----.. - ---·-..,.

./ ..............

~ ........... -

~ ' M.ki~ I

+25

+20

i-15 I

+1.0 ! I

+5 I M.T •. L.. iG.S.C. Oat"'m(l'ti.S.L.)

~ !· -0

-5 I

I i T -M

M.L.W. I ~ ' . " _shaft Datum

t.::S..- - - - - T - - - ---.-.--15

·~ .i ~~ i.

~ so~:i:: Oatum (!;: : -20

-25

-30 -,., i ~ Ji: I» ~ !'

1/) 0 .... ... ~ il n1 ~~ ··~ ("')~·

3:0 ;:....

0~ -~~ i -· ~ ., !" <IQ ~

., ~ ..... ~~ -en ~ tt It --~

Figure 1-2

Bay of Fundy; Vari'ltion.of Tidal Ranges and Sounding Datum Along the Northern

Coast

':/:)

l ~ ___j

r i .. Ul ...

------- -tlt'

+25'

.. ao' +15'

N\.1-i.W. --- ·----·· +'It'

----· + 5'

-·~M·-- --- G.$.C. Qatum(f't1.$.L.} ,, ~:.Is. ·---- ~

M.T.L. • •• 4. .,

- 5'

-tfl' (!;)

-15'

-20'

um -25'

-3&' -< 0 ~ l r f 1». =· ~ ~~ i ~ Ill ""'' i li .... Ill

s. b! 8" 0 .., a ..... ! :r t

Figure 1-3

Bay of Fundy; Variation of Tida.l Ranges and Sounding Datum Along

the Southern Coast

10

applied to a chart datum (established datum at the reference

station) to obtain the sounding datum is given by [Admiralty

Manual of Hydrographic Surveying, 1969],

d = h - H-r-R ' ( 1.1)

~here h is the height of the M.S.L. above the zero of

the new reference gauge, H is the height of the M.S.L. above

the established chart datum, r is the range of tide at the

new reference station and R is the range of tide at the

established reference station. It means that when lal > 0.6 m

(2 feet) the sounding datum is changed by 0.6 m (2 feet).

Figure l-4 illustrates how a sounding datum could change

in an estuary or a river. The configuration of the land

and the slope of the sea bed will influence the tidal

characteristics and hence the tidal ranges. The range of

the tide increases at first proceeding up a river and then

starts to decrease until it reaches zero at a point inland

where the river ceases to be tidal.

It is not possible to establish one sounding datum tor

a hydrographic survey which covers a long stretch of coast-

line and where tidal conditions are unknown. Tidal informa-

tion in the area must be built up and a sounding datum

transferred gradually along the coast as the survey

progresses. A hydrographic surveyor on a sounding mission

could be met with any of the following situations regarding

sounding datum:

l l

8m

ESTUARY

I I I

----;;l~~~~-:":h;-~-~~::~;:··::~;=~~==~~---·~--......_ Geodetic Datum (M.S.L.)- - - .._ · · coiw .._ ____ _

-4

Satmdlng Datum __ _.....

-6

--8

100 80 60 40 20 0 20 40km

DISTANCE

Ffgufe 1-4

Variation of Sounding Datum in an Estuary or a River

(adaphtd from Admiralty Manua1, 1969)

12

(i) a chart datum has already been established

within the sounding area,

(ii) a chart datum has been established near the

sounding area,

(iii) a chart datum has not previously been es­

tablished anywhere nearby.

The actions correspondi_!lg to the above situations are:

(i) the surveyor should recover the established

chart datum and use it,

(ii) the surveyor should transfer the datum to the

survey area; in other words, he should obtain

a sounding datum for the area to be surveyed

referenced to the established chart datum,

(iii) the surveyor should aim at establishing a

chart datum.

13

1.2 Reduction of Soundings

FigurP l-1 illustrates the realtionship between a

sounding at a time t and the chart datum. The height of tide

at time t must be subtracted from the depth sounded to yield

a reduced sounding. Manual reducttons of soundings in tidal

waters is a tedious aspect of the field hydrographer's tasks.

It requires that a tide gauge be set up in the survey area

and the rise and fall of tides observed while the sounding

is performed. From the observed heights, it is possible to

plot a curve showing the variations in the water levels and

to reduce the soundings to a suitable reference plane.

Figure 1-5 illustrates a typical reduction curve [Admiralty "\

Manual of Hydrographic Surveying, 1969]. It has been drawn

from the height observations at half hourly intervals with

additional readings on either side of the high water. The

reductions are scaled in steps of one metre and noted in the

form of a table. For example, the reduction is 5 m from

1247 hrs to 1342 hrs, 6 m from 1343 to 1446 hrs.

For inshore surveys, it is usually convenient to set up

a tide gauge and observe the tides while sounding is

proceeding. If we are sounding offshore, the problem

becomes complicated. It may be possible to use drying banks,

islets or temporary structures such as drilling rigs as sites

for tide gauges. Another possibility in the near future

will be the use of automatic sea bed tide gauges[DeWolfe,l977J.

8m

I 1 r

I i

I 0 I u 6 ).. w ! > i oC(

1/1 IX w 5 1-w :1 z 1-

4 l :I: ~

w :I:

3

1t00

/' R~duction Table

PERIOD IN HRS. REDUCTION

- 1246 = 4m 1247 - 1342 = 5m 13.43 - 1441 = 6m 1447 - t$at = lm 1537 - 1639 = 8m 1131 - 1715 = 1m 171&- 1818 = &m 1819 - 1918 = 5m 1t10 - = 4m ·~

1 _L _ _ ___ !___ 1 _ __L______~ _ _ __ .t t _ ____ -~

1300 1400 1500 1600 1700

TtME

Figure 1-5

Tidal Reduction Curve

1100 1t00 2000 hrs

_. ~

15

In the absence of the above alternatives, tidal observations

could be made from an anchored survey vessel using an echo

sounder.

If the cotidal charts for the area of interest are

available or could be constructed, the necessary tidal

information for the reduction of the sounding can be recovered

from them. The objective of this report is to offer an

automated analytical alternative to the manual task of tidal

reduction of soundings through the use of tidal observations

or cotidal chart information or a combination of the two.

Before describing the proposed scheme, an understanding

of tidal theories ~nd phenomena, analysis and prediction of

tides, and fhe types and construction of cotidal charts are

pertinent. Chapter II covers the theory of tide generation,

harmonic analysis and prediction of tides. Chapter III is

devoted to the types, construction and uses of cotidal

charts.

II ANALYSIS AND PREDICTION OF TIDES

2.0 Introduction

When the water levels h(t) have been observed at

times t relative to a chosen datum at a tide gauge

station, we have obtained a record distributed in time

space (time series) and defined at the discrete time

intervals. There is a trigonometric polynomial, Pn(t),

of the form

n b(t) = I (a.cos w.t + b.sin ~.t),

. 0 1 1 1 1 1=

(2.1)

which can predict this time series at any time t in the

interval. The analysis of this time series means the

determination of tile real numbers a., b., and w.. If we 1 ' 1 1

seek a least squares solution to this problem, we would

llave a system of normal equat]()ns that would be nonlinear.

The presence of the non-linear trigonometric terms as

unknowns lends to a serious problem which may or may not

have a solution [Vani~ek and Wells, 1972]. If, however,

the frequencies w. are known, the coefficients a. and b. 1 1 1

can be determined using least squares harmonic analysis.

The first and basic problem of harmonic tidal analysis.

therefore, is the determination of the constttuent fre-

quencies w.. This is the first step in the complete de­l

composition of the observed time series into individual

trigonometric terms. The first practical attempt at the

determination of the constituent frequencies was made by

16

17

Darwin in 1886 using the orbital theories of the moon

and the sun. In 1921, Doodson improved on the method

by making a more complete expansion of the tidal potential

using the modern luni-solar orbital theories.

The careful analysis of the tides at Honolulu and

Newlyn by Munk and Cartwright [1966], indicated that the

spectrum of a tidal record is a continuous function of

frequency w over the low frequency band, but that it

approximates closely a line spectrum over the other fre-

quencies - 'the constituent lines emerge from the noise

background as trees from grass' [Godin, 1972]. As long as

we do not work with the low frequency band, (as is

generally t~e case.in Hydrographic Surveying), it is

reasonable to assume that to a good order of approxima-

tion the spectrum of a tidal record is a line spectrum.

We can therefore treat the observed heights as a problem

of spectral analysis of a time series. Letting

and

equation 2.1 can be rewritten as

00

h(t) = y Hkcos(wkt + ak), k=O

(2.2)

where Hk is the amplitude of the constituent frequency wk'

ak is the phase of the constituent at time t = 0. If the

function is defined on the finite set M = {0, ±1, ±2, ±3,

... ±!}, the frequency wk is ~iven by

lS

(2.3)

Hk is obviously a non negative real number that d0scribes

the magnitude of the constituent frequency wk. By plotting

the amplitude against integer frequencies, a visual inter-

pretation of the contributions of the individual constituent

frequencies (Figure 2-1) can be made. This represents the

discrete transformation of the function from time space into

frequency space [Vanicek and Wells, 1972].

H H1 n2 H3 ~--··---'---=--- --'---=---·-'-::-=----1 0 nfl 2nfl 3nfl

Figure 2-l

Line Spectrum of Function h(t)

Munk and Cartwright, [ 1966] introduced an en tin:' 1 y

different method of tidal analysis which they called the

response method. In this method, the potential is generated

as a time series V(t) and an a~tempt is made at the

prediction of height of the tjde at a time t as the weighted

sum of the past and present values of the potential

h(t) =I W(s)V(t- Ts). s

(2.4)

The weights W(s) are determined such that the prediction

error h(t) - h(t) is a minimum in the least square sense.

In this chapter, the theory of tidal generation and

the traditional harmonic analysis and prediction of tides

are described. The thinking behind the response analysis

and prediction is briefly outlined.

2.1 Theory of Tide Generation

2.1.1 The Movements of the Moon (Real) and the Sun (Apparent)

The·moon and the sun are the principal tide generating

agents. Other heavenly bodies are either too distant

or have too little mass to exert any significant force on

the earth's surface. Figure 2-2 shows the rela tiC>nship be-

tween the orbit of the moon and the apparent orbit of the

sun. The sun moves in an apparent path around the earth on

a plane called the ecliptic once every 365.25 solar days.

Fur our present purposes, this movement can be regarded as

uniform and inclined at an angle of 23° 27' (obliqui~y of

the ecliptic) to the celestial equator. The point where

the ecliptic crosses the celestial equator from south to

north (D in Figure 2-2) is called the Vernal equinox or

the first point of Aries T.

The moon moves eastward around the earth in an orbit

N

s

Figure 2,;..2

The Retatlonflifttf' Betvweett the Orbital Mo11ons

of tM Moo~ and the Sun

21

inclined at about 5° 9' [Admiralty Manual of Hydrographic

Surveying, 1969] to the ecliptic and crosses the ecliptic

at the nodes. It takes approximately 27.2122 mean solar

days for the moon to travel from the ascending node F to

the ascending node K (Figure 2-2). As indicated in

Figure 2-2, the lunar orbit does not cross the ecliptic

at the same place consecutively. The nodes continually

move westward along the ecliptic and this nodal movement

or regression, as it is often called, has a period of 18.61

tropical years (one tropical year= 365.2422 mean solar days).

Due to the nodal regression, the obliquity of the lunar

orbit with respect to the celestial equator varies pro-

gressively between a maximum and a minimum, namely, '

~ax. = 23° 27' + 5° 9' = 28° 36'

Min. = 23° 27'

2.1.2 The Tide Generating Forces and Potentials

To derive the mathematical expression for the tide

generating forces of the moon and tl.e sun, the principal

factors to be taken into consideration are:

(i) the revolution of the moon around the earth in

an orbit inclined to the equator,

(ii) the motion of the earth around the sun along

the ecliptic which is also inclined to th~

equatorial plane,

(iii) the rotation of the earth around its axis.

22

The tide generating forces at the earth's surface

result from a combination of two basic forces; (i) the

force of gravitation exerted by the moon (and sun) upon

the earth, and (ii) centrifugal forces produced by the

revolutions of the earth and the moon (and the earth and

the sun) around their common centre of mass known as the

barycentre.

The magnitude of centrifugal force produced by the

revolution of the earth-moon system around barycentre

(which lies approximately 1709 km beneath the earth's

surface on the side towards the moon and along the line

connecting centres of mass of the earth and of the moon)

is the same at any point on or beneath the earth's sur-

face [National Ocean Survey, 1977].

[Godin, 1972]

2 = KM/po ,

--

Its magnitude is

where p0 is the distance between the centres of mass

of the earth and of the moon (Figure 2-3), K is the

universal gravitational constant, and M is the mass

of the moon.* The gravitational force exerted by the

moon is different at different positions on or beneath

the earth's surface because the force of attraction

*Note: The earth-moon system is used here to develop the equations for tidal potential. The same develo1ment resulting in similar equations can be used for the sun or any other heavenly body.

23

V'

Ftgure 2--3

Etf*ta- 6ffflie &t·.wlttltional Attractf6ft

Of a- ••••"'IV ~Y M, on ttte Emt.

24

between two bodies is a function of the distance between

them. This gravitational force at 0 (Figure 2-3) is

and at X is

2 Fg0 = KM/P 0 ~

2 Fg = KM/px ~

(2.6)

(2,7)

where Px is the distance between the centre of mass of the

moon and point X on the earth's surface. The tide generat-

ing force due to the moon M at point X (Figure 2-3) on the

earth's surface is defined as the difference between the

gravitational force at X and that at the resultant centre

of mass of the earth-moon system where the gravitational

and centrifugal forces are in equilibrium [Dronkers, 1972].

In terms of potentials, the attra~ting potential at

X and at time t is

(2.8)

and the potential of the constant vector field of the

centrifugal force is

2 fc = KM a cos ~mx/Po , (2.9)

where ~mx is the zenith distance as shown in Figure 2-3,

and a is the mean radius of the earth. From equations

2.8 and 2.9 and making use of the definition of the tide

generating force given above, the tide generating potential

(Vm) due to the moon at X and at Lime t is [Dronkers, 1972].

= KM[l_ p

X

25

(2.10)

Figure 2-4 shows the distribution on the earth of tide

forces of lunar origin. At point A nearest to the moon, th~

force of attraction is greater than the centrifugal force.

The resultant is the tidal force (Ft) towards the moon. At

C, the centre of the earth, both centrifugal and the gra-

vitational forces are equal. The tidal force at tbe centre

consequently is zero. At B farthest from the moon where the

centrifugal force is greater than the attractive force, the

tidal force is directed away from the moon.

We can express px (equations 2.10) in terms_ of Po and

~mx using th~ cosine formula of plane trigonometry given by

Equation 2.11 can be rewritten as

1 = - [L -p

cos <P mx-2

( ~ ) J Po

(2.11)

(2.12)

When 1 Px

is expanded in powers of the parallax afpo by means

of a Taylor series, expansion in zonal harmonics is obtained

and equation 2.10 is given as [Godin, 1972]

The first term of the expansion

,0 / MOON

/ /

/ /

/ /

/

F;tgure 2-4

Distribution of Tidal Force

27

(2.13a)

can be overlooked because it is a constant and hence has no

physical significance.

The second term

2 v1 = KM/p0 a cos ~mx . (2.13b)

is the lunar gravitational force at the centre which is

equivalent to the centrifugal force.

The third term is

2 1 2 v 2 = KM a /Po 2 (3 cos ci>mx - 1) . (2.13c)

This is the significant term as far as tidal potential

is concerned. The fourth terrr1 is

= KM a ~4 ~(5 cos3 ct> - 3 COS ~mx). Po .... mx

(2.13d)

For practical purposes, the fourth term is of little

significancr. It must be considered when we are required

to determine the potential with a nigher degree of accuracy.

Henceforth in this report, v2 is the tidal potential. It

is decomposed into constituent frequencies and this, as

has been mentioned, is the first step in the harmonic

analysis of tidal records.

We can rewrite equation 2.13 as

(2.14)

28

The principal variable in the tide generating potential

defined by equation 2.14 is the zenith distance~ mx

quantity changes due to two effects [Dronkers, 1964],

namely,

(i) the daily rotation of the earth about its

axis (24 hours) combined with the motion

of the moon in its orbit (50 minutes per

day) giving a total periodicity of 24 hours,

50 minutes,

(ii) effects due to moon's motion in its orbit

during a lunar month which results in a mean

monthly periodicity of its declination o of

27.3 mean solar days.

This

The other variable in the potential that must be accounted

for is p0 , the mean distance of the moon to the earth which

varies due to the irregular elliptical nature of lunar

orbit.

The expression of the potential as a function of time

dependant variables and as a function of position on the

earth surface is achieved by transforming our Horizon

co-ordinate system to the Hour Angle system using [Smart,

1971]

cos ~ = sin ¢ sin o + cos o cos ~ cos t , mx (2.15)

where ¢ is the geodetic latitude, c is the declination and

t is the hour angle. We can evaluate cos2~mx in terms of

¢, o and t which after some manipulation yields

29

V G(a, p) [eos 2 .p cos2 o cos 2t + sin 2¢ sin 2o m

t 3( . 2 l) ( . 2 .l' cos + S1n ~ - 3 Sln u (2.16)

in which G(a, p) is defined as the Doodson constant. namely

G( a, p) = ~ KM. rf;c3( c is the mean semi-axis of the orbital

ellipse of the moon).

Equation 2.16 contains the variables p, o, t which are

dependant on time. Th~ first term of the equation con-

taining cos 2t includes the semi-diurnal constituents with

periods approximating half a lunar day. The second term

containing cos t determines the diurnal constituents with

periods approximating a lunar day. The third term is

independent of t and hence contains the long period con-

stituents. It is only subject to variations in declina-

tion o and distance p of the celestial body. We have now

been able to decompose the tidal potential into 3 frequency

bands

0 - for lon~ period constituents,

l- for diurnal constituents,

2 - for semi-diurnal constituents.

This is only a step towards the complete decomposition of

the tidal potential into the numerous periodic constituents.

For the complete decomposition, thr' work of Darwin and

Doodson are important. Darwin's d•,eomposi tion provides

readily the most important constituents and their relative

importance while Doodson's method is more suitable for

rigorous developments and pro"i.des a greater number of

30

constituents.

2.1.3 Development According to Darwin

This development is based on deriving relations for

sin o and cos o cos t, which occur in equation 2.16 in

terms of

t - the local solar time,

s - the longitude of the moon referred to

the equator,

h - the mean ecliptic longitude of the sun.

Darwin used the old lunar theory and all quantities were

given with respect to the moon's orbit projected onto the

celestial equator. He considered

p - the ecliptic

perigee,

n - the ecliptic

nodes,

Ps - the ecliptic

perigee,

as constant over one year.

longitude of the moon's

longitude of the moon's

longitude of the sun's

Referring to Figure 2-5, the relations are derived

from right spherical triangles MAM' and MX'M' and the

oblique triangle MAX'. A is a point of intersection of

the lunar orbit and the equator, X' and M' are the

projections of X and M onto the equator [Dronkers, 1964

Page 59]. From triangle MAM' and ~IX'M', the sine rule of

spherical trigonometry yields

31

figa·,_ 2~5

OfMt*l Pltrcrmetwrs

J2

sin !) sin s j_ n ( s - \_I + k ) .

cos 6 cos t - cos x.

(2. 17)

(2.18)

where I is the angle bctwP.en the orbit of the moon and the

celestial equator, s is thP. longitude of the mean moon on

the equator, v is the distance between the referred equinox

y' and the intersection of the lunar orbit with the equator

at A, X is thP. arc MX' and arc AM = s - v + k. k is the

difference between the true longitude of the moon (s') mea-

sured from -,' ( y '_M) and the longi t 11de of the mean moon in the

equator s. From oblique triangle MAX' and using the cosine

formula we have that

cos X = cos( 15"tx + h - v) cos(s - \_) + k)

+ sin ( 15 o t x + ll - v) sin ( s - v + k) cos I

' (2.19)

in wi1icii h is tlle mean eel ipti c 1 ongi tude of the sun and

v is the right ascension of A, 15° of arc is equal to one

hour in time. 2 2 The terms sin 6, sin 26 cost and cos 6 cos 2t

whie~1 are contained in the potential formula (equation 2. lG),

can be determined from equations 2.17, 2.18 and 2.19 in

terms of the orbital elements tx, s, h and v. When these

are substituted back into equation 2.16, we obtain a series

of harmonic terms of which the arguments depend on the

rotation of the earth (l5°tX), the mean motion of the moon

in its orbit (s) and the mean motion of the earth in orbit

(h) namely.

2 [ 4 I Vm = G(a, p){cos $cos ~ cos(30°tx- 2s- 2h - 2v- 2v- 2k)

+ ~ sin2 I cos(30°tx + 2h - 2v)

+ sin4 ~ cos(30°tx + 2s + 2h- 2v- 2v + 2k)J

+ sin 2¢[sin I cos2 ~ cos(l5 tx - 2s + h + 2v - v

- 2k - 90°) + ~ sin 2I cos(l5 tx + h - v + 90°)

+sin I sin2 ~ cos(l5°tx + 2s + h- 2v- v- 2k + 9Qo)]

(2.20)

In the development for solar constituents, the terms v

and v will vanish and angle I will change toE·

\

2.1.4 Development According to Doodson

Doodson's method principally involves the use of a

rigorous expansion of the ecliptic longitude and latitude

of the moon. For the development of sin o and coso cos t,

he introduced the ecliptic longitude Am and latitude Bm of

the moon and the local siderreal time 8 of the point X

(Figure 2-3) on the earth's surfacP. The equations are

sin o = sin E sin Am cos Bm + cos t sin Bm , (2.21)

cos 6 cos t = cos sm cos Am cos 8 + (cos E cos Bm sin Am -

- sin E sin Sm)sin 8. (2.22)

where E is the obliquity of the ee I ip tic.

Finally the potential Vm is developed as the sum of ~eriodic

functions of six variables, namely, tx, s, h, P, n and Ps.

31

Doodson obtained 400 periodic constituents from his

devP1opment of whic.h the principal ones are listed in Table

2-l [Vanicck, 1973].

The constituent frequencies can be described in mathe-

mat ica l terms using Doodson nun1bers and the ast ronomi ca 1

variables, namely

wk = ~1 = k 1 f 1 + k 2 f 2 + k 3 f 3 ~ k 4 f 4 + k 5 f 5 + k6 f 6 ,

(2.23) Ckx = o ± 1 ± 2).

f is a six dimensional vector whose components are the basic

frequencies of the motions of the earth, the moon and the

sun, namely

fi 1 is the period of the earth's rotation TX (1 day),

-l £2 is the period of moon's orbital m9tion ~ (1 month),

f-l is tiH· pt~riod of earth'~ orbital motion li. ( 1 year), . :J

f~ 1 is the period of lunar perivee P (8.R5 ypars),

-l r 5 is the period of regression <)f lunar nodes N (J8.61 years),

-1 . £6 is the period of solar perig,~e Ps (21000 years).

f 6 is usually omitted because it is insiginificant. kx = 0,

l, 2 refers to the tidal species, 0 for long period, J for

diurnal and 2 for semi-diurnal. (k 1 , k 2 ) is called the group

number. (k 1 . k 2 , k 3 ) is called the constituent number.

With the constituent freq~lPnc j es determined, whic ·l

are the same anywhere on the e:1 rth' '> surface, the first step

in the harmonic analysis is DO\\ co11:pleted. In the next

Symbol Velocity per hour

M 0

s 0

s a

s sa

M m

\fil

<Pl

0°,000000

0°,000000

0°,041067

0°,082137

0°,544375

1° 1 098033

13°,398661

13°,943036

14°,496694

14°,958931

15°,000002

15°,041069

15°,041069

15°,082135

15°,123206

15°,585443

16°,139102

27°,895355

35

Origin (L, lunar; S, solar

Long period components

+ 50458 L constant flattening

+ 23411 S constant flattening

+ 1176 S elliptic wave

+ 7287 S declinational wave

+ 8254 L elliptic wave

+ 15642 L declinational wave

Diurnal components

+ 7216

+ 37689

- 2964

+ 1029

+ 17554

- 423

- 36233

- 16817

- 423

- 756

- 2964

- 1623

L elliptic wave of o1

L principal lunar wave

m L elliptic wave of K1

s elliptic wave of P1

S solar principal wave

s S elliptic wave of K1

L declinational wave

S declinational wave

s S elliptic wave of K1

S declinational wave

m L elliptic wave of K1

L declinational wave

Semi-diurnal components

+ 2301 L elliptic wave of M~ "'·

Table 2-1 Principal Tidal Constituents As Derived by Doodson.

:3e

Table 2-1 -colltinued •

Symbol Velocity Amr- :. i tude 105 Origin per hour (L' lunar; S, solar)

112 27°,968208 + 2777 L variation wave

N2 28°,439730 + 17387 L major elliptic wave of M2

v2 28°,512583 + 3303 L evection wave

r-12 2f! 0 1 984104 + 90812 L principal wave

>..2 29°,455625 - 670 L evection wave

L2 29°,528479 - 2567 L minor elliptic wave of M2

T2 29°,958933 + 2479 s major elliptic wave of s 2

52 30°,000000 + 42286 s principal wave

R2 30°,041067 - 354 s minor elliptic wave of s 2

~2] 30°,082137 + 7858 L declinational wave

SK 30°,082137 + 3648 s declinational wave 2

Ter-diurnal component I

I M3 43°,476156

I - 1188

I L principal wave

37

section, the least squares harmonic analysis of observed

tidal records, to determine the tidal constants Hk and gk'

where Hk js the amplitude of the eonstituent k and gk the

phase lag of the constituent k at the observed station,

is described.

2.2 Least Squares Harmonic Analysis and Prediction of Tides

The height of tide h(t) :;~t any place and at any time

t can be expressed as the sum of harmonic terms [Dronkers,

1972] 00

(2.24)

where s 0 is the height of mean water level above the datum

in use, wk is the constituent frequency, Hk is the amplitude

of the constituent k and ak is the initial phase of the

constituent. The number of constituents included will

depend on the accuracy required for prediction. For ordinary

hydrographic works, the constituents M2 , s2 , N2 , o1 , K1 , P1

are sufficient to yield an accuracy of 0.2 m in a prediction.

ak depends on the varying mean longitudes of the moon's

perigee and sun's perigee with periods of approximately 8.61

and 21000 years respectively and the ecliptic longitude of

the moon's ascending node with a period of 18.61 tropical

years. To take these effects into account, f 5 and f 6 con­

stituents are eliminated and a node factor fk and a correc­

tion for equilibrium argument Uk are introduced.

Equation 2.24 is rewritten as N

h(t) == s 0 + kil fkHk c.os(tJ.Jkt +(Vk + Vk) - Xk), (2.25)

3S

in which (Vk + Uk) is the value of the equilibrium argument

of the constituent k when t = 0, generally called the

astronomical argument, Xk is the phase lag of the tidal

constituent behind the phase of the corresponding equilibrium

constituent at Greenwich, N is the number of constituents in use.

All tide observations are made on local standard time,

often referred to as zone time and denoted as ZT. Equation

2.25 therefore has to be modified so that allowance is made

both for the zone time and the local longitude since the

meridian of the observing station and the meridian defining

zone time are usually not coincident (Figure 2-6).

If (Vk - Uk) is the phase of the equilibrium constituent

k at the Greenwich, P(= 0, l, 2) is the tide species number,

0 for long period, l for diurnal and 2 for semi-diurnal and

Ax is the geodetic longitude of the point, say x2 (Figure 2-6)

west of the Greenwich, then (Vk + Uk) - PAx is the phase

expressed in Greenwich mean time of the equilibrium consti­

tuent k of the tide species P at the point x2 west of

Greenwich. This is now transformed into the zone time of

the place. If the correction for zone time is AT (where AT

is negative west of Greenwich and positive east of Greenwich)

and the frequency of the constituent is wk' we must subtract

wk.AT from the phase of the equilibrium tide. Thus with

respect to the point X2 west of Greenwich, Vk + Uk - PA +

wk.AT is the phase of the equilibrium tide expressed in the

local zone time.

If gk is the phase lag Xk corrected for longitude and

39

f\r~Ute 2-6

T\me Re\ati'Ortsbtps

40

zone time, then we have that

(2.26)

The determination of Hk in equation 2.25 and gk in equation

2.26 are the objectives in the harmonic analysis of tides.

They are determined from a series of observed tides at a tide

gauge station and are called the harmonic constants for that

station. The estimation of these constants for a station

is improved when more observations are available.

From equation 2.25, using trigonometric relations for

compound angles

• fkHk cos[wk.t + (Vk + Uk) - Xk)] = fkHk cos((Vk + Uk)

sin(wk. t).

If we let

equation 2.25 is rewritten as

N h(t) = s0 + I Ak cos(wk.t) +

k=l

N I Bksin(wk. t) •

k=l

(2.27)

(2.28)

(2.29)

(2.30)

Equation 2.30 is a trigonometric polynomial that can

predict the observed time series h(t) at time t in the

given interval of time. Least squares approximatior metho-

dology [Vanicek and Wells, 1972; Moritz, 1977; Appendix I]

can be used to determine the coefficients s0 , Ak' Bk

4]

(k = 1, 2, 3, ... N). The number of coefficients to be

solved is

U = 2N + 1, (2.31)

where N is the number of constituent frequencies used.

We can choose our base functions as

~ = {1, cos w1t, sin w1 t ...... cos wNt, sin wnt}.

(2.32)

The Vandermonde's design matrix A is

1' cos wltl, sin wltl, cos wNtl' sin wNtl

A 1' cos wlt2 sin wlt2, cos wNt2, sin wNt2 MxU

1 ' cos w1 tm, sin wltm ... cos wNtm, sin wNtm

(2.33)

in which m equals the number of measurements h(t) that have

been made. For weights, we can consider each observation

as having been made independently with equal amount of

reliability. The error in observations (OX~· can be taken

to be equal to the resolution of the tide gauge used so that

LL = diag[ o~1 • 2 aq. OL 2 mxm

and the corresponding weight matrix is

p = I~l = diag[ l 2 , 1

!~J (2.34) 2 , mxm OL oL

1 2

2 in which o 0 (the a priori variance factor) is taken as unity.

The solution for the vector of coefficients is given

as

c (2.35)

42

T in which 2 = [s0 . A1 . B1 , A2 . B2 .. Ak, Bk]

The solution for the residua] vector is

,., V = AC - F

where F is a vector of observed heights.

(2.36)

The associated variance covariance matrix of the vector of

coefficients is

(2.37)

where a~ is the estimated variance factor given by

(2.38)

df represents the degree of freedom given in this case by

the number of observations minus the number of coefficients

( df = m - u).

With the coefficients s0 , Ak, Bk determined, equations

2.26, 2.28 and 2.29 yield the harmonic constants Ilk and gk.

Note that if however it is not intended to pretiict the tides

in the past or in the future, the constants need not be

computed. The tide at any time t in the time interval can

be predicted using the polynomial.

From 2.28 and 2.29

fkHk sin((Vk - Uk) - Xk) tan((Vk + Uk) X )

Bk = - =

fkHk cos((Vk + Uk) Xk) k Ak '

or

(2.39)

and

43

(2.40)

or

( 2. 40.-::.)

To completely solve our problem, we have to determine the

astronomical argument (Vk + Uk) and the nodal factor (fk).

The values are usually tabulated in tide tables (eF,. Admiralty

Tide Tables), or they can be computed.

The astronomical argument is given as [Godin, 1972;

pp. 171-178]

A A

Vk(t) = k 1 ~ + k 2S + k 3h + k 4P + k 5N + k6Ps (2.41)

A

where T, S, b, P, Nand Ps are the values of the astronomical

variables at the instant of time t from the origin of time

and are given as

s so + t.t s , h ho + t.t h,

p = Po + t.t P,

N No + llt I~ ,

Ps Ps0 + t.t Ps, A A

T = 0.0416 (hh mm) + h s

s 0 , h 0 , P0 , N0 and Ps0 are the values of the astronomical

variables at the time t = 0, hh mm represents the hours

and minutes of the day, S, 6, ~. ~. ~s are the ~ates of

change of the astronomical variables in cycles per muan lunar

day. Uk is the phase of the astronomical argument (Vk) at

time t = 0.

The nodal (modulation) fae.tor is given by [Godin, 1972] no

fk = 1 + I jrk.jexp[2n i(6k4 (j)P + 6k5 (j)N + k6 (j)Ps)J, j=l J

(2.42)

in which rkj is a complex number which depends on L\k 4 ,

6k5 and 6k6 . The j's inside the differences in Doodson

numbers indicate that they depend on a specific constituent

within a cluster.

It is important to note that in the discussion so far,

there was no mention of removing the noise part of the ob-

served series before the analysis is made. The harmonic

constants obtained are therefore likely to include other

effects beside those of the astronomic forces and are conse-

quently in a certain measure variable. The harmonic ana-

lysis should be based on a series of very selective

filterings so as to permit isolation of an oscillation

having a maxi~um tide/noise ratio. Godin [1972] has given

several filters that could be used to eliminate the noise

part or suppress certain frequencies.

Vanicek [1970] pointed out that there is an obvious

danger in removing the noise part of a series when the

magnitudes are not known. On the other hand, it is

usually equally deterimental 1o leave these constituents

unattended because they may distort the spectral image of

the series to a considerable degree. He described a method

of least squares spectral analysis that could be used to

analyse a time series and locate the frequencies accurately

45

without first removing the noise part.

Mosetti and Manca [1972] described a number of methods

for separating a certain number of tidal constituents by

means of successive approximations and thus to completely

extract astronomic tide from the tidal records. The fre­

quency interval in which the tidal constituents occur are

divided into a number of wave groups, the periods within

each group being very close to each other but sufficiently

distinct from the periods of constituents in all other

groups. By drawing the graph of oscillations in each group,

it is easy to see that the modulations are perturbed to

some extent due to interference phenomena from waves within

the group. ~f we are dealing with series extending over a

fairly long period, it is possible to evaluate the intervals

on the record that are least perturbed and where the ampli­

tudes vary with regularity dictated by astronomic laws.

The harmonic constants can then by computed for those

intervals.

2.3 Tidal Analysis and Prediction by Response Method

2.3.1 General

Munk and Cartwright [1966] presented an entirely

different method of tidal analysis and prediction. They

applied the theory of time series to the tidal observations

at a gauge station to determine certain coefficients which

replaced the amplitudes Hk and the phase lags gk of the

tidal constituents as in the harmonic analysis. Even

46

though the theory of this method is more involved than the

harmonic method, the authors claim that the response method

gives a simpler and physically more meaningful representation

of tides than the harmonic method. Unlike the traditional

harmonic method which attempts to express the tides as the

sum of harmonic functions of time, the response method

expresses tide as the weighted sum of the past, present

and future values of a relatively small number of time

varying input functions.

Dronkers [1972] described the method as a more

empirical modification of the equilibrium tide based on

the theory of time series. He added that the principal

advantage of the response method is that the total number

of coefficients is less than the number of constituents

used for the harmonic prediction of comparable accuracy.

ln the response method we deal with complete potential

instead of a set of discrete frequencies as in the harmonic

method.

Lambert [ 1974] noted that the principal advantage of

response method over the harmonic method lies in the fact

that separate admittance functions (Fourier transform of

response weights) can be calculated for sufficiently dis­

tinct uncorrelated inputs, thus making the method

adaptable for earth tide analysis.

The response method of tide analysis and prediction

as developed by Munk and Cartwright [1966] is applied to

various observed series to obtain frequency dependent

47

admittances that describe the tidal characteristics in a

similar sense to what can be deduced from the traditional

harmonic constants. To bridge the gap between the response

and harmonic methods, Zetler, Cartwright and Munk [1969 J have

described procedures for deriving harmonic constants from

the response admittances. They showed that the harmonic

constants Hk and gk of a tide constituent k can be deter­

mined for a place using response analysis and the result is

compatible with the conventional harmonic analysis.

2.3.2 Brief Outline of the Theory of Response Method

The tidal potential can be generated as a time series

V(t) and an attempt can be made at predicting the height of

tide for a time t as the weighted sum of the past and

present values of the potential~ ~

h(t); I W(s) V(t- TS). (2.43)

The weights W(s) are determined such that the prediction

-error h(t) - h(t) is a minimum in the least squares sense,

TS is the time lag used in the argument of the potential.

The weights represent the sea level response at the place

of interest to a unit impulse

V(t) ; o(t).

In the response approach of Munk and Cartwright, V(t) is

expressed in spherical harmonies as

n n V(~, A, t) ; g l I [a~(t)U~(~, A) +ib~(t)V~(~. A)].

n;Q m;Q (2.44)

48

t• um + . vm nere 1 n n are a set of compl\:~x spherical harmonics

of order m and degree n, a(t), b(t) are the amplitudes of

the real and imaginary parts of the spherical harmonics and

can be computed for any desired time interval for any

location.

The prediction formalism becomes [Munk and Cartweight,

1966]

-hCt> = I

Letting

W~(s)

and

mn

U~(s) + i V~(s) .

C~(t- s) = a~(t- TS) - i b~(t- TS),

equation 2.45 is rewritten as

-h( t) = L I Wm(s) Cm(t- Ts).

mn s n n

(2.45)

(2.46)

The weights Wm(s) define the relation between the linear n

part of the tide and the equilibrium tide, thus the

determination of Wm(s) is the essential point in the n

response method.

III COTIDAL CHARTS AND THEIR USES

3.0 Introduction

In Chapter II we have seen how the tidal constituent

frequencies are obtained from the decomposition of tidal

potentials and how the tidal characteristic for a location,

that is, the tidal constants (amplitude Hk and phase lag

gk for any constituent k) for major constituents can be

determined using the harmonic or response methodsof tidal

analysis. In this chapter, the types and methods of

constructing cotidal charts and their uses, are discussed.

3.1

3.1.1

Types of Cotidal Charts

Range/Time Cotidal Charts

Most often, a range/time cotidal chart is constructed

by graphical means. On it, two sets of curves connect

points having equal range differences (or range ratios)

and points having simultaneous high and low waters

[Admiralty Manual of Hydrographic Surveying, 1969]. All

cotidal curves indicate a relationship to the tides at the

reference gauge station. Figure 3-1 illustrates a typical

range/time cotidal chart. The range curves (shown by

pecked lines) indicate the range ratios of the tide at

the reference station A. At B for example, the tidal

range is 0.65 times the range at A. The time curves (shown

by full lines) indicate time lags or corrections wh~ch must

be applied to the times of high or low waters at the

reference gauge station to obtain the times of high or

4!1

5l'

FiiJufe 3-1

Rattge/ Time Co-Tidal

Chart

51

low waters at a place of interest.

To construct this type of cotidal chart, simultaneous

tide observations are made at the reference station and at

other well distributed temporary tide stations such as at

points B, C, D and E in Figure 3-1. From mean high waters

and mean low waters, the mean range is obtained for each

station. The range ratios are determined from the relation:

mean range at a gauge stationjmean range at the reference

station. The mean time lag for each station is determined

by finding the mean time differences between the occurrence

of high and low waters at the reference station and at

other gauge stations. Both sets of cotidal curves are

interpolated in between stations as contours are inter­

polated in between spot heights for a topographic map.

3.1.2 Amplitude/Phase Cotidal Charts

This type of cotidal chart is referred to as being

semi-graphical. It is more difficult to produce and

more complicated to use than a range/time cotidal chart

but, could be more reliable and more versatile. The number

of such charts needed for an area would be equal to the num­

ber of constituent frequencies being taken into account

for our tidal predictions. For ordinary practical purposes

in hydrographic surveying, four major constituents are

considered, namely M2 , s2 . KJ. and o1 [Admiralty Manual of

Hydrographic Surveying, 1969]. This means that four

cotidal charts would be needed each containing two sets of

curves. Figure 3-2 illustratt~s on•· such eotidal chart of

an area for the M2 constituent. The full lines connect

points having equal values of phas<' 1 a.g g in degrc~~s and m

the pecked lines connect points having equal amplitudes

H . m

To produce the amplitude/phase cotidal charts, tide

gauges are set up at well distributed locations in the area

such that tidal characteristics should as much as possible

vary linearly from one gauge station to another. This

means that there should be no major physical features or

structures which may influence the propagation of tidal

waves between any two tide stations. (For example, Larsen

[1977] in his study of the tides in the Pacific Ocean near

the Hawaiian Islands, observed that the phase lag of the

M2 semi-diurnal tide differs by 46° between the nearby

tide stations at Mokuoloe and Honolulu that are on the

opposite sides of the Hawaiian ridge but differs by only 15°

between Mokuoloe and a distant station at Hilo that are

on the same side of the ridge. Also for the K1 diurnal

tide, the differences are found to be 8° and 3° respectively).

Tides are observed at the stations for a minimum period of

29 days. The tidal records are then analysed using the

harmonic or the response method to determine the harmonic

constants Hk and gk for each constituent frequency at each

gauge station. The amplitude and phase lag curves ar~ then

interpolated as contours are interpolated for a topographic

map.

. -. ''

/ /

/

/ /

/ /

/

e.a _. /

Fflme 3-2

Atnpf'tt~ CtF--TiAf Chart fbr ~

-

The amplitude/phase coticlal cltart cannot be used to

directly convert tide readings mad•c' at the reference sta-

tion to those observable at auy other place as is the case

with the range/time cotidal charts. With it however. tidP

at any point of interest in the area covered by the chart

can be predicted at any time t using equation 2.25.

Interpolating between gauge stations has been the

classical method of producing amplitude/phase cotidal

charts. Presently a more meaningful method of producing

this type of cotidal chart is through the solution of

numerical schemes. Luther and Wunsh [1974] however used

350 sets of constants, obtained partly from the publications

of the International Hydrographic Bureau (IHB) and partly

from other investigators, to produce .the cotidal charts

for the central Pacific ocean which they claim are comparable

with the numerical charts of Pekeris and Accad [1969] and

Hendershott [1972].

3.2 Numerical Schemes

The various numerical schemes for the production of

cotidal charts stem from various solutions of the Laplace

tidal equations [Bye and Heath, 1975; Hendershott and Munk,

1970]

au fv g act, - ~ )_ ( 3. 1) TI - a cos <P CIA.

av + fu = ~ acs - 0 (3.2) at a a¢

5 + l ['uQ avQ ·] 0 (3.3) at a cos <P ax-+~ cos '

55

where •· A are the geodetic latitude and longitude respecti­

vely,

u, v are the latitudinal and longitudinal components

of the fluid velocity,

a is the earth mean radius,

f(= 2n sin $) is the Coriolis parameter in which Q

is the angular velocity of the earth,

Q is the undisturbed depth of the ocean,

~ is the elevation of the sea surface above the

undisturbed level, and

t(= Vfg) is the equilibrium tide.

The Laplace tidal equations representing equations of

motion, thou~h they look simplified, are difficult to solve

even in the case of uniform depth covering the globe. The

early solutions were given by Lord Kelvin in 1845 and

Hough in 1897 who replaced the Laplace power series in sine

with an expansion in spherical harmonics thus regarding

the earth's rotation as very small. In 1898, Lord

Kelvin introduced the concept of f-plane approximation

in which he considered the oscillations of the horizontal

sheet of fluid of uniform depth rotating about its normal

and this reduces the Laplace tidal equations to [Hendershott

and Munk, 1970]

au fv a (E; - D (3.4) - = - g. at ax

av fu a(~ - ~) (3.5) + = - g. at aY

l_S_ + Q(au at ax

+ av) (ly

0 .

in which x, y are the Cartesian coordinates in the plane

( 3. 6)

of the fluid. Larsen [1977] used the f-plane solution to

produce the cotidal charts for the Pacific ocean near the

Hawaiian Islands. He approximated the Island as an ellip-

tically shaped cylinder with the plane ocean taken to be

tanga1t to the earth at the coordinates <f>o = 20.7 °N and

Ao = l56.8°W which corresponds to the coordinates of the

centre of the elliptically shaped Island. On the plane

ocean, he took the rectangular coordinate system with the

X-axis eastwards and normal to the axis of the ridge formed

by the island and the Y-axis northwards and parallel to the

ridge axis and with the origin at the tangent point (<f>o•

The boundary condition assumes that the velocity nor-

mal to the coast vanishes and free tide solutions are

added in order to fit the observed tide at the boundary.

The cotidal charts for the various constituents are con-

structed by mapping the amplitude and phase of the total

tide, that is the resultant of the equilibrium tide, forced

tide and free tide, as a function of the elliptic coordi-

nates. The author evaluated the accuracy of the cotidal

chart by comparing the observed tjdes at some locations

with the values of tides predicted by the model. He

observed that the plane wave model of the tides connect

the tidal observations together in a simple way and thus

57

allows the tide to be interpolated between gauge stations

and extrapolated into the ocean beyond the tidal sites.

Rossby in 1939 introduced the beta-plane approxima-

tion. In this, the Laplace tidal equations are written as

in f-plane approximation but with the coriolis parameter

made a linear function of y, namely

The variation of f with y corresponds to an expansion of

the coriolis parameter about the latitude <Po

2n sin <I> = 2n sin <Po + c 2~)a(<jl - <t>o)cos <Po·

in which B is of the order 2n a

When 8 = 0, we then have f-plane approximation.

With the advent of large computers, the application

(3.7)

(3.8)

of the method of finite differences to the tidal probl~ms

become popular. Freeman and Murty [1976] studied the

cooscillating and independent tides in Hudson Bay and

James Bay by applying the finite differences to solve the

Laplace tidal equations. They linearised the equations

of motion in spherical polar coordinates and vertically

integrated retaining variable coriolis, pressure gradient,

bottom stress and direct tidal potential terms. The

equations thus solved in the model are

au 2n sin <I> -~- .Q.n TB>. F>. (3.8) at = v --- + a cos <I> a>. p

av gh .ill TB Fct> - 2n u sin cj> - --~ + (3.9) at . a 3<1> 0

__ l __ ~~~ + ~~ cos ¢} ' a COS¢ 'Oil O'f'

( 3. 10)

where TB is the bottom stress, ~\' ~¢ are the horizontal

components of the tide generating force, n is the deviation

of the water level from the mean tide level, h is the water

depth and ~ is the density of water.

The cooscillating tide is modeled by setting the tide

generating force terms to zero and specifying the free

surface elevation across the mouth of the Hudson Bay by

(3.11)

l where nk belongs to the constituent k at the open mouth

boundary location and is referred to the mean tide level.

The independent tide is modeled by setting the normal

velocity on the open mouth boundary to zero and specifying

the tide generating force. For example, for M2

- 48 · 8 gh cos~ sin(w t + 2A + w T), a · ·-v m m ( 3. 12)

~2¢ - !8 · 8 .gh cos cp sin ¢ cos(t•Jmt + 2\ + 11\mT),

and for K1

- 28.5 h a .g

- 28.5 h( . 2 . g Sln q a

(3.13)

(3.14)

2 cos ¢)cos(wkt +A+ wkT) .

(3.15)

Here T is the number of hours from the Greenwich mean

time to the local zone time. The linear form of bottom

friction due to Heaps is used and is given as

TS <P = !_Jh~ V . (3.16)

The authors used a rectangular grid of 15 and 10 minutes

of arc in longitudinal and latitudinal directions respecti-

vely. The grids are drawn so that the fluid velocity

components (U, V) are defined on the closed boundary locations

and the water levels (n) at the open boundary at the mouth

of the Bay. In the formulation of the numerical scheme,

central finite differences are used in both space and time.

Using a leap-frog scheme, water levels (n) are computed

at even time steps (i.e. i = 2, 4, 6, 8 .. ) and the hori-

zontal flow components (U, V) computed at odd time steps

(i.e. i = 1, 3, 5, 7 ... ).

The numerical scheme is thus given by [Freeman and

Murth, 1976]

ui+l _ ui-1 kj k,j

2L1t i ghkj oj[nLl,j

)

at vkj sin \~ .-ll!-l,jJ J a cos

1 Tf3 i-1 -i (3.17) Ak,j + FA.k . ' p ,J

- 2~ ui sin <P.-k' j J

ghk 'j l( i i l a llk,j+l - llk,j-1

1 Ti -i (3.18) f3 <P . . + Fcp . p l,J k,J i ui .

i-1 1 uk+l · -i+1 k-l,J llk . - Ilk . = - 'J

'J 'J a cos <Pj 2f:.?t

cos (3.19)

and the output of the computati.ons are U, V and n as func­

tionstime. From these parameters, the current ellipses

and the co-phase and co-amplitude lines are constructed.

In numerical schemes, the problem generally posed is to

solve the Laplace tidal equations in their primitive form

or after elimination of one or two dependent variables with

prescribed boundary conditions. For example

(i) Vanishing normal velocity at coast lines

[Pekeries and Accad, 1967],

(ii) Specified or observed values of the consti­

tuents at the coastal stations only [Hendershott,

1966] .

(iii) Specified or observed values of the consti­

tuents at selected coastal and island sta­

tions plus vanishing normal velocity at the

remaining coastal boundary points [Larsen,

1977].

3.3 Uses of Cotidal Charts

Cotidal charts are found useful in many situations.

They are useful in the study of the impact of large

engineering structures on the tidal regime, for example,

the proposed tidal power project on the Bay of Fundy in

Eastern Canada [Atlantic Tidal Power Engineering and

Management Comn1ittee Report, 1969; Garrett and Greenberg,

1976].

They are indispensable in navigation especially

when deep draught ships have t.• navigate through a complex

61

estuary where drying sand banks alternate with deeps such

as that obtained in the port of London [White, 1971]. Here

deep draught tankers navigate to Thameshaven and Coryton

to evacuate oil from the principal oil refineries. In such

a situation the pilot and the captain of the vessel would

want the information on

(i) the critical depths in the channel at

chart datum,

(ii) the points along the track where these

critical depths occur,

(iii) the times the tidal heights at.these

points would be sufficient for safe

passage of a vessel with a particular

draught,

(iv) the latest times along the route that

the passage depths are available.

If the underkeel clearance is not so critical, this infor­

mation can easily be obtained using cotidal charts and

appropriate up to date navigation charts and tide tables.

If the underkeel clearance is critical, the use of cotidal

chartsis supplimented by several radio linked tide gauges.

The application of prime concern here is the use of

cotidal charts for the reduction of sounding dat~. As was

shown previously, all depth measurements are reduced to the

chart datum; therefore the height of tide at time t must be

subtracted from the depth sounded at the time t. This

implies that we should observe tides at the same time we

62

take our soundings. If we are working on the coast or on

the inland tidal waters, it is possible to establish tide

gauges close to the sounding area and observe tides at t.he

same time. If we are involved with extensive sounding

offshore, the possibilities of observing tides close to the

sounding area are remote. It becomes more feasible to do

the tidal reductions using predicted tides, and when this

is the case, the use of cotidal charts become convenient.

Rangejtime cotidal charts can be used in which case

we only need to observe or predict tides at the reference

station and then obtain the equivalent at the desired loca­

tions, or, we can use amplitude/phase cotidal charts and

predict the tides at the desired locations independent of

a reference station. Finally, a combination of the two

approaches can be used.

The Canadian Hydrographic Service has done some auto­

mated tidal reductions using digitized rangejtime cotidal

charts of the Hudson Bay and the Lower St. Lawrence River

[Tinney, 1977]. In these schemes, the cotidal charts were

digitized by breaking the survey area into equal size blocks

based on lines of latitude and longitude and approximating

the boundaries of the cotidal zones with the edges of those

blocks. Those digitizations were coded and stored in the

computer. To locate a particular block and retrieve the

cotidal values, the geodetic coordinates (¢. A) of the

position of the sounding were used.

The choice of the size of the blocks would obviously

55'

l

0 N

-TIDAL CHART DIGITAL CO IS76 Hudson Bay

I

l

63

i. 85

Figure 3-3

64

depend on the amount of computer space available and the

accuracy requirements. With smaller size blocks, the zone

boundaries would be better approximated but more computer

space would be required. Figure 3-3 shows the digital break­

down of the cotidal chart used for the Uuds<Jn ~ay. Ulock

sizes of 5' latitude and 10' longitude were used giving a

total of 13,986 blocks dividing the Bay into 93 reduction

zones. The tide station at Churchill served as the reference

station for the cotidal chart and during the survey, the

predicted heights from the reference station were used

instead of the observed heights. However, in the survey

of the Lower St. Lawrence River with Pointe-an-Pere as the

reference station, observed tides were used.

IV THE PROPOSED ANALYTICAL SCHEME

4.0 Introduction

The proposed analytical scheme is aimed at achieving

automated tidal reductions using little computer space and time

and with advantageous accuracy and flexibility. Figure 4-l

illustrates the proposed scheme in a flow-chart. It shows

that we can work with amplitude/phase cotidal model or

range/time cotidal model. The same objective is achieved

using either model but it does not necessarily mean that

the same degree of accuracy and flexibility is attained.

Basically the data requirements for either are the same

except that with amplitude/phase cotidal model, the ampli-

tude Hk and the phase lag gk for each constituent k we wish

to take into account and at each observation station are

required. With the range/time cotidal model, we require

the mean range ratios and the mean differences of the

times of occurrence of high and low waters between each tide

gauge station to be considered and a reference gauge

station. With the range/time cotidal model, we have the

option of carrying out the tidal reduction based on the

observed tides or on the predicted tides at the reference

station.

In each case, the aim is to produce an analytical

cotidal model using observed data or existing cotidal

charts. The analytical model could then be stored con-

veniently in a computer so that when observed sounding

65

66

Figure 4-1 fhe Proposed Scheme - Flow Chart

Amplitude/Phase Co-tidal Chart -1JATJ\

¢0, A0: Ref. Stn

¢., t\ : Obs. Stns. 1 1

Hk·· gk. 1 1

OR Existing Coti da 1 Charts

DATA ¢0 , A0: Ref. Stn.

¢;• ";= Obs. stns. t-R.,l.lT.{orr.

1 1 1 (rat i o) , l:l T 1) h0(t) at Ref.Stn. . OR

Exi s ti nq~oti da 1 charts+~0 (t) at

L.S.Poly­nom1al App­roximations for: -ffi.pl itudes

-Phase lags

. Eqns. 4.14, 4.15.

L.S.Polynomial Approxir;lation of Obs. time series at the Ref. Stn. Eqn. 2.30

L.S. Poly­nomi a 1 APP roxi mations for:

-range ratio

-time differ-ence

Eqns. 4. 14,

4. 15.

Predict: I

-Amplitudes! -fJhase lags at any point

P(¢, >.)

Eqns. 4.1,4.2.

rangerati time diff. at any point Pl¢,>·) Eqns. 4.1, 4.2.

Range/Time Co-tidal Chart

Predict: height of tide at the point

!-'[-¢, A) Eqn. 2.25

redict: eight of

tide at the point P(¢, A)

Eqn. 2.30

67

data are input, the output would be reduced soundings.

The theory and mathematical models for the two approaches

are basically the same. In Section 4.1 of this ehapter,

the mathematical models are discussed, and in Section 4.2

the data requirements are explained explicitly.

4.1 Models

Earlier, it was shown that the tides are functions of

time and position on the surface of the earth and that the

tidal characteristics, that is, the amplitude Hk and the

phase lag gk for the constituent k are constant for a place.

These constants can be estimated by performing harmonic

or response analysis of a long period tidal records.

Knowing the ~stimated tidal constants for a place, the

tide at the place can be predicted at any time t.

Now suppose we consider a section of a bo~~ of tidal

water, not so extensive in area and where the constants

Hk and gk are defined at a reference station whose geodetic

coordinates are (~0 , A0 ), and at several other points

P/~j' Aj) within the area. We can define mathematically

surfaces that can describe the distribution of those con-

stants with reference to the primary station. The aim is

to approximate, in the Least Squares sense, the amplitude

and phase lag fields by surfaces described by two dimen­

sional algebraic polynomials. The coefficients of these

polynomials are determined in such a way as to fit the

observed data in the Least Squares sense. Using this

6P.

technique, the amplitudes Hk z<nd tl1e phase lag gk ean be

predicted at any point of interest ~i(~i' li) within the

area by the polynomials

.t H I C. ~1-(X., yi). j~O J J l

(4.1)

.t \ c~ olo ( ) L '~' j xi· Y i ' . 0 J (4.2)

J=

where ~Hk(xi, Yi) and ~gk(xi, yi) are the predicted

differences in amplitude and phase lag respectively for

the constituent k between the reference station and the

point i, C~ and C~ are the coefficients of the polynomials,

lji(xi, yi) are base functions (two dimensional) of the

approximating polynomials, and .t is the number of base

functions. The selection of the prescribed functions ljJ

can be, from the theoretical point of view, purely arbitrary.

The sufficient and necessary condition for the prescribed

functions ljJ ~ {¢ 1 , ¢ 2 ... ~.e_} to create a base is that

they are linearly independent on the functional space (G). m

If and only i~ ljJ is a base can the coefficient~ of the be8t

fitting polynomial be uniquely determined [Vani~ek and Wells,

1972].

Even though the position of a point may be expressed

in terms of geodetic coordinates (¢i.li)' it is more con­

venient to work with local orthogonal coordinates (xi, y 1 ).

The relationship between the two systems is defined as

(4.3)

(4.4)

69

where R0 is the mean radius of curvature of the earth

computed at the reference station and is given by [Krakiwsky

and Wells, 1971]

(4.5)

in which·

(4.6)

and

N /(1 2 . 2 )1/2 0 = a - e s1n ~ 0 (4.7)

The first eccentricity squared is

2 2 2 2 e = (a - b )/a , (4.7b)

and for the Cla.Ike 1866 ellipsoid, the semi-major axis

a= 6378.2064 km, while the semi-minor axis b = 6356.5838 km.

Regarding the choice of base functions, we can use

mixed algebraic functions which are particularly simple to

Jeal with [Nassar and Vanicek, 1975], namely,

(4.8)

where n is the degree of the polynomial. Equation 4.1 and

4.2 can now be rewritten as

l;Hk( xi' Yi) f CH. xi. j (4.9) = Y. J i 1

1., j=O

~gk( xi' y i) = ~ cg. X£. y~ (4.10) J "i 1

.t, j=O CH. The problem is to solve for the coefficients and cg.

J J

of the polynomials. The number of coefficients U to be

solved for is determined from the relation

U = (n + l)d. (4.11)

70

where n is the degree of the polynomial and d is the dimen-

sionality of the base functions.

4.2.1 Least Squares Solution bf the Models

To determine the unknown coefficients CH. and Cg. of J J

the models represented by equations 4.9 and 4.10, obser-

vation equations can be written for each data point i where

the amplitude difference ~Hk and the phase lag difference

~gk referred to a reference station are known. The equations

are

~Hk(Xi' Yi) + v~ = ~Hk(X.i, y i). (4.12) 1-Iki

~gk(Xi' Yi) + v gki

= ~gk(Xi' y i). (4.13)

where 6Hk and ~gk are the predicted values, v Hki

and v gki

are the residuals of observations, and the terms on the

right hand side (~Hk and ~gk) are the known or observed

~alues. Substituting equations 4.9 and 4.10 into equations

4.12 and 4.13 yields

n ~~ l .J· I C,,e. V L X.Y. + H.· =

l,j=O J 1 1 ki (4.1-1)

n ,e . I c~. x.y~ + Vgk· = ~g(x., Y.).

l,j=O ~J l l 1 1 1 (4.15)

Putting equations 4.14 and 4.15 in matrix form we have

A en + VH = Lu (4.16) mxu ux1 mxl mx1

A cg + Ve- = Lg (4.17) mxu ux1 mxl mxl

It is pertinent to note here that equations 4.16 and 4.17 are

the same as the observation equations for a parametric case

71

in the least squares adjustmen1:s. The parametric least

squares adjustment differs only in purpose and notations

from the least squares approximation of a function (F)

defined on a discrete or compact domain (M) [Vanicek and

Wells, 1972]. The purpose of the least squares approxima-

tion is to find an approximating polynomial (Pn) for a

given function or for a given set of functional values.

The purpose of the least squares adjustment is to find the

least squares statistical estimates of unknown parameters

which are related to the observed values by linear (or

linearized) mathematical models.

The matrix A is known as Vandermonde's design matrix

and is given ,by

0 0 J.ll o< xl y 1)'

0 l J.ll1 ( xl Y 1)'

0 2 J.ll2(xlyl),

n n J.llu(xlyl)

0 0 0 1 0 2 J.llu(x~y~) A -1Vo<x2y2)' J.ll1 ( x2y2)' tJ;2(x Y ),

mxu

(4.18)

H C and Cg are the vectors of coefficients. VH and Vg are

the vectors of residuals of the observations LH and Lg

~ and Lg are the vectors of observed values (or the

functional values) at the discrete points i. The solution

of the system of equations given by 4.16 and 4.17 for the

coefficients, using least squares approximation methodology

[Vani~ek and Wells, 1972; Christodoulidis, 1973; Balogun,

1977; Appendix I] is given by

72

where N is the Gram's matrix defined by

N uxu

and

T - [A PA J _

T U =A PL = <L, ~ 1.> ux l

(4.19)

(4.20)

(4.21)

The sign < > indicates a scalar product [Appendix I].

Since our prescribed functions form a base, the Gram's

determinant must be different from zero and must have an

inverse.

The solution for the residual vector is given by

V = AC - L (4.22)

The associated variance covariance matrix for the coeffi-

cients is given by

\ ~ = ~2 -1 L a N , c

(4.23)

where &2 is the a posteriori variance factor given by

(4.24)

in which df represents the degree of freedom given by

df = m - u .

P is the weight matrix

p = I -1 L

= Diag[-1-2 1 OL1 ' OL2 2

(4.25)

1 ] OL 2

m (4.26)

73

where oL is the standard error of the observables. The

weight matrix is diagonal when we are dealing with statis-

tically independent observables, that is, the observations

are assumed uncorrelated.

For statistical reasons, we may wish to work with

orthogonal bases, and usually the base ~ is not an ortho-

gonal one. Schmidt's orthogonalization process [Appendix I]

may be applied to obtain an orthogonal base ~*· Using

an orthogonal base, the normal equation is

(4.27)

Again setting

A*TPA* = N* '

.\

and

A*TPL =· U* ' WE' have that

C* = N*-lU* (4.28)

A* is the Vandermonde's design matrix obtained using

the orthogonal base. t* is a vector of Fourier coefficients,

N* is the Gram's matrix, this time diagonal because we

are dealing with orthogonal base functions and is given by

<1/J~ 1/1~ > 0 0

0 * * 0 N* <1/Jl WI> (4.29) -

uxu

0 0 * * <1/J u 1/1 u'

and U* is given by

71

* <\IJo L>

* L> U* - <\jJl (4.30) uxl

* L> <wu

The associated variances are given by

\ "2 N*-1 Lc* = a * • (4.31)

where

(4.32)

The solution of normal equation becomes trivial as the normal

equation matrix N* (Grams matrix) is diagonal and each Fourier

coefficient can be solved for independently.

We subject the Fourier coefficients to statistical

screening by comparing each coefficient against j times

its standard error. (Christodoulidis, i973], that is, if

j a"* c. l

(4.33)

then a~ is st2tistically insignificant at that level and l

is discarded. j takes the values 1, 2 or 3 depending upon

what level of significance of their standard deviations we

wish to test the coefficients. The discarded Fourier co-

efficients are set equal to zero. Once the appropriate

Fourier coefficients are discarded, the residuals, the

variance factor and the variances are recomputed using only

the accepted coefficients. The residuals are given by

A 1 A

V* = A*C* - L (4.34)

75

The a posteriori variance factor is recomputed by

(4.35)

where

l df = m - u + d ,

in which d represents the number of Fourier coefficients

discarded. The new variances are

(4.36)

Using the transformation matrix (see Appendix I)

l 812 813 81u

B - 0 l 823 82u (4.37) u X U

0 0 0 l '·

and the remaining statistically significant Fourier

coefficients, the original coefficients are computed by

A A

C = BC* (4.38)

The correct number of original coefficients are obtained

even though we are solving for them using fewer number of

Fourier coefficients. If, however, the last Fourier

coefficients are the ones discarded, a fewer number of

original coefficients will be recovered. The variance-

covariance matrix of the original coefficients can be

computed using the variance-covariance law, namely,

(4.39)

where Lc*l is given by equation 4.36.

76

Once we have computed the co(~fficients of the

original polynomials and their variance-covariance matrix

from the statistically significant Fourier coefficients,

statistically significant surfaces which describe the

distributions of amplitudes and phase lags (or range ratios

and time differences) in the area of interest have been

obtained. Analytical cotidal models for amplitudes and

phase lags (or range ratios and time differences) have

thus been obtained. With the analytical models, the

values of the amplitude and phase lags (or range ratio

and time lags) can be predicted for any point Pi(~iAi)

in the area using equations 4.1 and 4.2. The prediction

variance covariance matrix is given by

(4.40)

in which J is Jacobian of transformation defined by A matrix.

4.3 Data Requirements and Reduction Algorithms

4.3.1 Amplitude/Phase Cotidal Model

As previously noted, to produce cotidal models for

amplitudes and phase lags, we need to define the ampli­

tudes and the phase lags of each constituent frequency

at a reference station and at several other observation

stations adequately distributed in an area of interest.

Working with four major consti tue::ts, eight analytical

models are needed to describe the tidal characteristics

of the area. For a fair estimate of the amplitudes and

the phase lags, the tidal analysis must be made from

77

369 days of tidal records, and for a barely acceptable

estimate, observation should cover a period of 29 days.

The more observations added in the analysis, the better

will be the estimate of the harmonic constants.

It may not be easy to adequately distribute observing

stations and obtain sufficient data to enable the produc-

tion of a desired analytical model. An alternative is

to use cotidal charts,produced from the numerical schemes

such as those described in III, Section 3.2, as a source

of data. The cotidal charts are digitized as mentioned in

section 3.3 and the digitized values are used in the least

squares polynomial approximations to produce the analytical

cotidal models. As a check on the compatibility of the

analytical models and the original chart, the area is

grided at close intervals and the amplitudes and phase lags

predicted at the grid intersections using equations 4.1

and 4.2. The co-amplitude and co-phase curves can then

be easily drawn in.

If the amplitude/phase cotidal models are being used

for the reduction of soundings, the reduction algorithms

can be summarized in steps as follows:

(i) At each sounding location i, the depth

(Di), the time (t) and the geodetic co-

ordinates (~i' Ai) are observed.

(ii) With the observed geodetic coordinates

(~., A.), the amplitudes and phase lags l l

78

of the constituents being used can be

predicted using the analytical models.

(iii) Using the tide prediction approach as

described in II Section 2.2 and the

predicted amplitudes and phase lags

from (ii) above, the height of tide

h.(t) at the sounding location above 1

chart datum are predicted.

(iv) The sounding reduced to the chart

datum is d. =D. - h.(t) 1 1 1

4.3.2 Range/Time Cotidal Model

(4.41)

Some assumptions must be made at the outset for this

model. Considering a body of water of relatively small

extent, such that one can safely assume that the meteoro-

logical variables in the area are not remarkably different

from place to place, it can be further assumed that given

any two points A(~, A) and B(¢, A) in the area, the tides

at A bear constant relationships with the tides at B.

Those relationships will change when there are marked

topographical changes due, for example, to errosion,

engineering structures, which tend to change the pattern

of the propagation of tidal waves. If we establish the

relationship existing between a reference station and any

other point, it is possible to preclict with some degree of

certainty the tides at that other point from the observed

(or predicted) tides at the reference station.

79

The relationships between the tides at any two sta-

tions can be established from the ratio of their ranges

and the difference in the times of occurrence of high

and low waters. In other words, it is assumed that the

unwanted noise has perturbed observations equally so

that when the range ratios and time differences are deter-

mined, the unwanted noise is eliminated.

To produce range ratios and time lags cotidal

models, we require

(i) the mean range RmO at the reference station

and the mean ranges R . at discrete points mJ

(~j' Aj); the range ratios are then given

as'·

( 4.42)

(ii) the mean time differences between the times

of high and low waters at the reference

station and at the discrete points given in

minutes of time.

If the sounding reduction is to be done with range/time

analytical cotidal model, the reduction aligorithms can

be summarized in the following steps:

(i) The tide is observed at the reference

station to cover thP- time interval M

(the soundings are also performed

within the same interval of time). A

least squares approximation of the

observed series at the reference

80

station is done so tl1 at :t t any time t in

the interval, the height of tide can be

predicted.

(ii) At each sounding location i, the depth (D1 ),

the time (t) and the geodetic coordinates

( <Pi, ;>..i) are observed.

(iii) With the observed geodetic coordinates

( <Pi, ;>..i), the range ratio ( r i) and time

difference (correction to time) are pre-

dieted using the analytical models.

(iv) Using the corrected time at the reference

station and the approximating polynomial

from step (i) above, the height of tide

(h0 (t)) at the reference station is

predicted.

(v) The height of tide at the observed loca-

tion i is computed from the relation

h 0 (t) x ri .

(vi) The reduced sounding is

d. =D. - h 1.(t) . 1 1

(4.4::,)

(4.44)

lt is more convenient and simple to work with range/

time cotidal models because (i) unlike the amplitude/phase

models where 2 x NC()N (NCON is the number of constltw,nts

used) analytical models are needed to describe the tides,

only two models are needed to completely describe the

tides, (ii) working with rangejtime cotidal models allows

81

us to use the observed tides at the reference station to

reduce soundings instead of the predicted tides .

V TEST COMPUTATIONS AND THE RESULTS

5.0 Data

To test the proposed analytical scheme, there was

unfortunately no adequate data immediately available. How­

ever, the tidal information for secondary ports on the Bay

of Fundy, published in the Canadian Tides and Current Tables,

1978 by the Canadian Hydrographic Service was minimally ade­

quate for testing the analytical range/time cotidal models.

This tidal information is given with reference to the

Port of Saint John. In Table 5-l, the data as extracted

are tabulated for 35 secondary stations (Figure 5-l).

The predicted tides for the Port of Saint John from

January l-15, 1978, were extracted from the same Canadian

Tides and current Tables, 1978 and treated as observed

Lides in the computations. The zero hour of the day the

observation started is taken as the origin of time and

times are gjven in hours from the origin of time. The

observations are treated such that the period of the

sounding exercise is covered, in other words, it is

assumed that the tides were observed at Saint John through­

out the period of the sounding. In Table 5-2, the tides

as supposedly observed are tabulated and from Table 2-l

the following 7 major constituent frequencies are uscJ.

Symbol

M2

s2

82

Fn-quency ( deg. fhr)

28.984104

30.00000

Index l~o.

0065 0001 0015

0040 0060 0129

0140 0150 0160

0170 0190 0215 0225 0235 0240 0245 0247 0250 0260 0270 0285 0290 0300

one Latitude Long1tude Mean Range t·1ean T1 me Time(ZT) Range Ratio( r) Diff. (min)

-0 I 0

Saint John +4 45 16 -66 04 25.10 1.0 0.0 Outer Wood Isl. +4 44 36 -66 48 16.60 0.6614 -28.5

Welshpoo1 +4 44 53 -66 57 16.90 0.6733 + 5 ,l)

St. Andrews +4 45 04 -67 03 22.60 0.9004 +15.5

Partridge Isl. +4 45 14 -66 03 25.00 0.9960 -10.0

St. Martins +4 45 21 -65 32 30.15 l. 2012 + 9.0

Herring Cove +4 45 34 -64 58 33.25 l. 3247 +19.0

Cape Enrage +4 45 36 -64 47 35.40 1 .4104 +17.0

Grindstone Isl. +4 45 44 -64 37 38.30 1 . 5359 +20.0

Hopewell Cape +4 45 5'1 -64 35 39.90 1.5896 +19.0

Pecks Point +4 45 45 -64 29 38.70 1.5418 +19.0

Joggins ~~harf +4 45 41 -64 28 38.15 1.5199 +18.5

Cape Capstan +4 45 28 -64 51 33.05 l. 3167 + 11.0

VJes t Advocate +4 45 21 -64 49 32.90 1. 3107 - 1.0 Cape D 'or +4 45 18 -64 47 36.55 1 . 4562 +16.5

Port Greville +4 45 40 -64 56 36.70 1.4622 +30.0

Diggent River +4 45 24 -64 27 39.50 1.5737 +33.0

Cape Sharp +4 45 22 -64 23 37.95 1. 5120 +48.5

Five Isl. +4 45 23 -64 08 43.05 1. 7151 +56.0

Burnstooat Head +4 45 18 -63 48 44.30 1 . 7649 +67 .0

Avon Port +4 45 06 -63 13 45.05 1./948 +32.5

Cape Blomidon +4 45 16 -64 21 29.80 1.1873 +46.0

Scots Bay ~4 45 19 -64 26 37.10 1. 4 781 +14.5 - -·----

Table 5-l Bay of Fundy -Tidal Information on Secondary Port.

Remark

Ref. St.

* * * * * * *

* * *

I '

'

I

co w

---- -

1 Index Location Name Zone L:;titude Longi tuje 1'1ean Range ~1ean Time Remark No. Time ( ZT) Range Ratio t r) Di ff. (min) 1

0 I 0 I

0305 Baxter Harbour +4 45 14 -64 31 37.4 1.4900 +12.0 * 0312 I 1 e Haute +4 45 15 -65 00 34.15 1. 3606 0.0 * 0315 Margaretsvil1e +4 45 03 -65 04 31.75 '1.2649 -12.0 * 0320 Parkers Cove +4 44 48 -65 32 26.60 1.0598 -14.0 * 0325 Digby +4 44 38 -65 45 25.25 1. 0060 - 9.0

0330 Deep Cove +4 44 24 -65 50 24.00 0.9562 -15.5 * Ct;

0335 Sand Cove +4 44 30 -66 06 21 .15 0.8426 -18.0 * .:~

0336 East Sandy Narro. +4 44 29 -66 05 19.10 0. 7610 -37.0 * 0337 Tiverton +4 44 23 -66 13 17.45 0.6952 -45.0

0340 West Port +4 44 16 -66 21 18.10 0. 7211 -34.0 * 0345 Lighthouse Cove +4 44 15 -66 24 17.90 0. 7131 -34.0 * 0353 Church Point +4 44 20 -66 07 18.10 0. 7211 +1~.0 * 0355 Meteghan +4 44 12 -66 10 16.90 0.6733 +18.0 * _j

-· -- ------ ··--- --------~------------- ~-~ ~

* Data used in test computations

Table 5-1 (cont'd).

t:M -"""()~ \i'g"\%4'Y ij~~~

J)t ?a [l ,_-, J 6ran.l 1 ~j Muu

]),, )_ ~)'. (; \~ !As~~'£ Island

t./tl 0 ~ ~ l:l OUTU

wooo '·

'f'ARUOUTH

}\ I

-N

~

1 , d 1 f~

• 0

~

PlfUI.alr!fi[Nf TIDE GAUCU: LOCATION

TUIPO.AfiY TIOI GAUG( LOCATION

Figure 5-1

Bay of Fundy-- Tide Gauge Locations

,0 o ao .., and Sections for Co-tidal Modelling ICAl.C 01 IIIL[S

C):) CJl

86

Station: Sajnt John

Coords.: Lat. 45 16'N

Time (Hrs

03.833

10.250

16.250

22.350

28.667

35.00

41.250

47.250

53.500

59.917

66.083

72.417

78.583

84.833

91.083

97.417

103.583

109.915

116.167

122.333

~L-------=~6~6~~o~4~·=w­ong.

.Height(m) Time (Hrs)

7.4 128.500

1.3 134.833

7.3 141.083

'1.2 147.333

7.4 153.417

1.2 159.833

7.3 166.083

1.2 172.167

7.6 178.25

1.2 184.917

7.3 190.917

1.1 197.00

7.7 203.333

1.0 209.667

7.4 215.750

1.1 222.083

7.9 228.167

0.8 234.667

7.5 240.750

0.9 246.915

Time Zone: + 4

Date: Jan. 1-15, 1978

Height(m) Time (Hrs) Height(m)

8.1 253.167 8.4

0.6 259.333 0.2

7.7 265.500 8.0

0.8 271.917 0.6

8.3 278.000 8.1

0.3 284.333 0.4

7.8 290.500 7.8

0.6 296.667 0.8

8.4 302.833 7.8

0.2 309.083 0.7

8.0 315.333 7.6

0.5 321.500 1.0

8.5 \

327. 750 7.4

0.1 334.167 1.0

8.1 340.250 7.4

0.4 346.500 1.3

8.5 353.000 7.1

0.1 359.167 1.3

8.0

0.5

Table 5-2 Tide Observations.

87

Symbol Frequency (deg:fhr)

13.943036

15.041069

14.958931

30.082137

28.439730

5.1 Computations and the Results

The computations have been completed in three steps.

First, least squares approximations were done to determine

the coefficients of the polynomials that will predict the

range ratio (r.) and the time difference (correction to 1

time) at a point Pi ($i' Ai). Second, a least squares '·

polynomial approximation of the observed time series at the

reference station (Table 5-2) was completed to determine

Lhe coefficients of the polynomial that will predict the

height of tide 60 (t) at the reference station at any time

t E M. Fina1ly, using the results of the first two steps.

the observed geodetic coordinates at a point P 1 (~i' Ai)

and the observed time at the location, the height of tide

at the ship was computed for the determination of the

reduced depth.

5 .1.1 Determination of the Coefficients of the Approxim_ating Polynomials

Of the 35 secondary gauge stations spread around the

Bay of Fundy, 21 of them that are located around the main

body ofthe Bay were used. Because of the intervening

88

0 peninsula which bifurcates the Bay at about longitude 64 55',

the tidal wave propagation have been greatly affected along

the two branches. A single analytical cotidal model for

the entire area could not therefore be produced. The Bay

has been divided into three sections numbered I, II and III

in Figure 5-l. We have used the 21 secondary stations to

model section I (those stations marked with * in Table 5-l

under remarks column). It should be noted that the origin

of the local Cartesian coordinate system is approximately

at the centre of the area being modelled (¢0 = 45° 05' OON,

:>.. 0 = 65° 35' OOW). The data at the reference station

(Saint John) was not fixed giving a total of 22 data points

for the approximation.

Using equation 4.11, it was dedueed that the highest

degree of polynomial possible with 22 data points is 3,

giving a total of 16. coefficients and 6 degrees of freedom.

This does not however mean that the polynomial of degree 3

will give a better approximation than polynomials of de~ree

l or 2. In Table 5-3 the degrees of the polynomials and

their associated a posterori variance factors are tabulated.

Two of the functions (Range ratio and Function A) have their

variance factors reach a minimum at degree 2, while the variance

factor of the third function (Function B) varies more

slowly at degree 2. The conclusir,n is that the polynomial

of degree 2 will give the best approximation with th:s

data.

The approximation for time lag required some extra

n

1

2

3

89

Degree of Std. Dev. Range Ratio Function Function Freedom of Obs. "2 A = R __ ~os(v) B = R sin{v) ( df) al (m)

ao a (J2 0 0

18 0. 1 0.94497 14.02610 20.60978

13 0. I 0.84280 12.40109 13.66546

6 0.1 1.14585 17.50086 13.43903

Table 5-3 A Posterori Variance Factor for Various Degrees of the Polynomials.

90

data manipulation. First, thP time differences given in

minutes were converted to angular measure using the

relation

12 hrs

1 hr 30°,

1 min 0.5''.

(the Bay of Fundy tide is mainly semi-diurnal). Attempts

to approximate the time lag converted to angular measure

yielded large variance factors which of course decreased

with increase in the degree of the polynomial. Unfortunately

the highest degree of polynomial with the data available

is 3. The conclusion reached was that the time lag dis-

tribution is not simple enough to be approximated by lower

degrees of the polynomials.

From Chapter II

A cos wkt + B sin wkt, (5.1)

where A 1

R (5.2) 2 cos Clk

B 1 R sin (5.3) = 2 Clk .

ak is time lag (or phase lag), and R is the mean tide range

at the station.

Also

~ = Arctan (B/A) , (5.4)

91

(5.5)

A and B can therefore be evaluated at each station using

equations 5.2 and 5.3 respectively. We can now seek for

the polynomials that can predict A and B at any point P. l

(¢i' Ai). Once A and Bare predicted, the predicted time

lag (phase lag) can be obtained using equation 5.4. The

associated variance (assuming no correlation between A and

B eg. cAB = 0) is given by

2 = raa12 2 + raa12 2 aa l-aAJ oA laBJ on ,

(5.6)

where

B =

A2(1 + (B/A)2) '

Cla I 1 ()0 - 1 + (B/A)2 X A = 1

A(l + (B/A)2)·

(5.7)

(5.8)

2 d 2 d" t· . fA dB t· 1 aA an oB are pre 1c 1on var1ances o an respec 1ve y

from the least squares approximations. For weighting, i'

was assumed that all the stations h~ve been observed inde-

pendently with equal amount of care. The standard error

of the observed range was set at O.lm.

If observed data is used, it is pertinent to note the

following:

( i) The standard error of the observed mean ra11be

should be computed from the observed data

using the relation

(5.9)

92

where R Dis the daily mean range, R is the m m

mean of mean ranges, n is the number of obser-

vat ions.

(ii) A and B should be computed from equation 5.1

in the least squares sense using observed

heights and the dominant constituent frequency

in the semi-diurnal or diurnal band, depending

on the type of tide.

In Table 5-4, the Fourier coefficients and their asso-

ciated standard deviations for the range ratio and time

lag are tabulated. The last four Fourier coefficients

in the range ratio and function A have been eliminated,

and in function B, two of the Fourier coefficients have

been eliminated in the middle. In Table 5-5, the original

coefficients of the polynomials are tabulated. Because

of the discarding of the last four Fourier coefficients

in the range ratio and function A, only five original

coefficients can be recovered. In function B where the

Fourier coefficients discarded are not the last ones, all

the 9 original coefficients were recovered. (Note, each

Fourier coefficient was tested against its standard

deviation)

To compare the analytical cotidal model with other

cotidal charts, the area was divided into a rectangul:1r

grid of 10' latitude and 10' longitude (Figure 5-2), and

the values of range ratios (r) and time lags have been

93

R A i~ G E R A T I 0 T I ~1 E LAG d Coeff.(Cr} ac Coeff.(CA) aCA Coeff.(C3 aCB

r

1.049 0.01957 3.970 0.07508 0 0

0.4623[-5 0.330[-6 0.1783E-4 0.1265[-5 0.4906[-5 0.1237[-5

0 .1940E-10 .6191E-ll 0.7554E-10 0.2375E-10 0.7091E-10 0.2321E-10

0.2126E-5 0.5570E-6 0.8618E-5 0.2138E-5 0.3977E-5 0.2089[-5

-0.2648[-10 0.1350E-10 -O.ll40E-9 0.5178E-10 0 0

0 0 0 0 0.1728E-14 0.1004[-14

0 0 0 0 0.2869[-9 0.1042E-9

0 0 0 0 0.2024E-14 0.1582E-14

0 0 0 0 -0.5218E-19 0.3350[-19

'----

Table 5-4 Fourier Coefficients After Discarding those of them greater than their Standard Deviations.

94

R AN G E R A T I 0 T I M E L A G

Coeff.(Cr) oCr Coeff.(CA) OCA Coeff. (C8 °CB

1. 120 0.03537 4.264 0.1357 -0.5691 0.16978

0.4041£-5 0.5680£-6 0.1550£-4 0.2179£-5 0.8730£-5 0.3614£-5

0. 1364E-1 0 0. 7644E-ll 0.5374E-10 0.2932£-10 0.1484E-9 0.4318E-10

0.1247E-5 0.7150£-6 0.4835E-5 0.2742E-5 0.1695£-4 0.5526£-5

0.2648£-10 0.1350E-10 -0.1140£-9 0.5177£-10 -0.2617£-9 0.1938£-9

0 0 0 0 -0.3887£-14 0.2208£-14

0 0 0 0 0.4269£-9 0.1391£-9

0 0 0 0 0.5786£-15 0. 1834E-14

0 0 0 0 -0.5218E-19 0.3349£-19

L__

Table 5-5 The Original Coefficient of the Polynomials and their Associated Standard Deviations.

95

~ s:::

I';( N Jo.

~~

I G)

an .a G) E IIi. ;::, ;::, z -~ 1.1..

't:! Jo. 0)

~

\ Ckl

Range Ratio

----- Time Ditf.

<!) Qr i q:t-n of· Coord a.

~· Ref. Stn.

'~ c

16. 5 9 ' 1& 5mites lirm pt 2 1

I

Figure 5-3

Bay uf Fundy-- Range/Time Co-Tid:al Curve from the

Analytical Co-Tidal Models

~

0 O'l

f',, L f, ! l. lr:- L :• ,. ,\ (j E ,... A I I w ~ II 1\ C I l r~ t: LA<; ;l A T l r· t: C f-' ll) tJ L 11\ 1 !;)

f·,u L1qllLDC.

4 ~. 1 tc... c 7 (..

~ 4toCCCC':C

44ot:.!..3:::.:c

'• t.,4,(;tuc7C

t.~ol{l:(/(

l .;:.;.c,cc;o

7 44oc.::3~3l

<.,; '• 4 • t. t..c c 1 c ':I '• t , i {.;u c. 7 l

1 j 4:.cCJCCL

l i .;t;.t:ZJ:Z.3(i

l c· r;tj .c...t.ct7C

i J 4!.. db ct;C

l !j ~;~.ccu . ..:c . '.; 4.; • t.:: L!.! i:

l ~...~ 4 ~ • .:; ..; . .) 3 j c

l I t,;,lU.t'l'

I· .. 4~•'-u;c..:c

1 ~ I, 'I • t. :; J 2 :_: 1:

i. \,.. c't ~ I .;. .; _! ~..,; (i

LC.I\CITUDc

-tt.l(;6c7o

-u: .ltet7o -(~ oltt;67C

-tc.lcct:/0

-t.t .ccccuc

-H.CCCCOO

-tc.ccocoo -tto.:OJC)C

-t:~.e:.::::.::~c

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(J • c ij 0 2!) (.) 1 tO ....:]

0 • r~ IJ c .:j J .J 1

0 • -:::, . ) -, 'J J () 1

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c· ::..

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·~ '; , l r.; L (; ·; C

ij::.~cuw.Jl

I; : ' ~ .: ..) ._; ,) c

·~ ~ • ~ :: J ·' J (

,,~.~(1;.;!~.7(

-4 :-:-. • ~ !.. ·;,; L ~ (

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~ . ~ . - -· '• ~ . ..:. ..: ...:.. ...: ...: (.

t.~.lf_(J(t(

"I ~ • :.. '· ..... \.. "'" c ·: ~ • - .; J ·.:. j \..

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4 .1.~ " :: ~ oJ ...... ,~

:., ~ • L (!... .... ""' (

·t ·•. ·~ -~~~ ·- ._:

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- c t • 1 (: •: l 7 c -t::c.2::.;:::;o

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Table 5-6

1 .• 2~1711..: •)o316SL-Ol -~.1 7c43C

l.l';i!J233 u.35·'31D-Jl -1Goi.ll<::2.;<;

1 .284t3b5 u.4210u-Ol 21.126ttS

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1.27C7-,;G 0 •2')cJ'1C-01 -1.217lt:<;.

1.263743 Co JCjiJ~C-i) 1 ·-..:e0i.H.4c4

1 .3357t.(: 0 • :.: ').:: 2 ._;- Q 1 lt..4·:.::2!:1.jQ

1 •. :351~7 c.z~::::u .. -o1 7.2)c4t7

l.;jt~::47 0.3:24/C-01 •.<:.le4·li:

1.3<;13£-'S Q • ,, J 1 ,. i: - J 1 lQ.-,,?(:lCS

l • .::S:7l~C: ::.3110<.--01 l·'·l7c.l::

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c.:jQ.!I.j41 c.3r.)luC-Jl -:c2.MoS,;..21

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.;.747ll7 c • J .; ]. '~ l; - ') 1 - 3 ~ ' 0 r·J ':1 !.; (; j

Predicted Values at Grid Intersections

'j • j?. 2 7 '] J t

·.J • .j[.2~:~) (,l

f 17 (. ::- ,~. ....... (' , •J • .....J ';) ::_} -.. ... t.... ..,I ...

J • j (; ~ 3 ,; .:, 1

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) • :; 1 :: :; ,;

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' J'

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0 • ,, l l (. ) c l

(.!)

00

/

//

, . ..r-·-·-· i i

1....... ,.i .. / ~: ......

U.S, A.

Amplituae of the Main Semi•Diurnal Tide in Feet·· ·2'----

Phase Lag of the l't'hin Semi-Diurnal Tide· -150°'---­

bo0• Approx. 1 hr. in time)

Figure 5-4

The Average Progression of Semi· Diurnal

Tide in the Bay of Fundy (Dohler,1966).

\0 \0

100

predicted at each grid interst::·('tion. The co-range curves and

the eo-time curves were plotted as shown in Figure 5-3.

Figure 5-2 shows the grid numbering, and in Table 5-6, the

predicted values at each grid intersection and their asso-

ciated standard deviations are tabulated. The cotidal curves

from the proposed analytical models compared favourably with

the cotidal curves (Figure 5-4) taken from 'Tides in Canadian

Waters' [Dobler, 1966] showing the progression of semi-diurnal

tides in the Bay of Fundy.

5.1.2 Least Squares Polynomial Approximation of Observed Time Series at the Reference Station

In this case, the heights of the tide defined at dis-

crete times (ti) in the time interval M are given and it is

required to determine the coefficients pf the polynomial

that will best predict the height of tide h(t) at any other

time t s M. A one dimensionai trignometric polynomial (Eqn. 2.30,

Chapter II, Section 2.2) and the 7 constituent frequencies

listed on page 82 have been used. The number of coeffici<:'nts

is given by

U = 2 Neon + 1 (5.10)

where Neon is the number of constituent frequencies being

used. For weighting, it was assumed that each height was

observed independently, with equal amount of care and

precision, and oh(t) = 0.05 m. The weight matrix is

therefore

[. 1

P = Diag .--2 mxm 0 h

1

(5.11)

101

In Table 5-7 , the Fourier. coc~fficients and the

recovered coefficieQts of the approximating polynomial

of the -observed time series, and their associated standard

deviations are tabulated. Two Fourier coefficients were

discarded in the middle of the series thus all the 15 ori-

ginal coefficients were recovered.

5.1.3 Tidal Reduction

For this set of computations, simulated sounding ob-

servations (corresponding in location to the 22 data points

and with all observations made within the time interval M)

were used to illustrate a proposed reduction algorithm.

At each sounding location i, the depth (Di), the time (t)

and the geode~ic coordinates (~i' A1 ) or the local Cartesian

coordinates (xi, yi) are observed.

The arguments of the approximating polynomials for range

ratios and time lags are the local Cartesian coordinates

(x, y) and the argument of the approximating polynomial for

the heights of tide at the reference station is the time (t).

With the polynomial coefficients and their associated standard

deviations stored in the computer, only the arguments (xi, yi)

are needed to predict the range ratio (ri) and the time lag

(tci). The time lag is, in a sense, the correction to be

applied to the observed time at the ship (sounding location

i) to get the equivalent time at thP reference station.

With the equivalent time at the reference station computed,

the height of the tide at the reference station is predicted

- ---· ·------·------FOURIER COEFFS. AFTER TEST COEFFICIENTS OF THE ORIGINAL AGAINST THEIR STD. DEVS. POLYNOMIAL

Coeff.(Fc) Std. Dev. Coeff.(C) Std. Dev. Or a,..

4.279 0.5861E-2 4.279 o.6227E-2

-5.339 0. 9072E-2 -1.951 0.1423

2. 335 0.02048 2.226 0.1081

0.4590 0.9760E-2 -0.1756 0. 2113

-0.0496 0.8784E-2 0.2592 0. 3972

0.0360 0.8436E-2 0.0678 0.0100

-0.0909 0.8295E-2 -0.1031 0.8507E-2

-0.1338 0.8309E-2 -0.2305 0.0319

0.1!:>84 0.83601::-2 0.0335 0.0322

0 0 0.1392 0.0382

0 0 0.0832 0.0229

-0.4193 0.0954 0.4132 o_ 1602

0.9868 0.1508 -0.0302 0.3586

0.4479 Q_Q840 0. 7741 0. 1272

0.4300 0.1259 0.4300 0.1259

Table 5-7 Coefficients of the Polynomial for the Observed Time Series at the Reference Station.

--

103

using the known time as the argument of the predicting poly-

nomial. The height of tide at the ship, which is the

required reduction, is obtained using equation 4.43. The

reduced sounding is computed using equation 4.44. Applying

the law of propagation of errors, the standard deviation

of reduced sounding is given by

1/2

(5.12)

where OD· is the 1

standard deviation of the depth sounded,

Ohi(t) is the standard deviation of the predicted height at adi CJd.

the ship, l and l 1 The heights of the tide an1

= ahi(t)

at the ship, required to reduce the soundings, are tabulated in

Table 5-8 alorig with their estimated standard deviations.

The predicted time lags and range ratios are compared with

the original data set (observed values) as shown in Table

5-9. At this stage it is important to mention that normality

in the distribution of residuals was assumed and chi square

tests on the variance factor performed at 95% confidence

level. The test passed for the range ratio and observed

time series approximations but failed for the time lag. There

are several possible reasons for the failure of this test

and as such a definite conclusion cannot be made without

performing several other statistical tests [vanicek and

Krakiwsky, Chapter 13, 197M]. However, it can be concluded

from our earlier discussions (~ection 5.1) that eithe the

time lag is not simple enough to bP approximated by the

lower order polynomial or the info~mation available is

104

not sufficient to approximate the time lag. In thP pr<>sent

2 circumstance it may be safer i<l assume o0 known and equal

to 1, so that

I2 (5.13)

f 1 UI\L f.lLUlll(~~

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.:

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,,

1.)

1 1

1 ~

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1'

1 7

1 1 ~

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.: 0

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o't 'f • L· ·~ C ~ ( C - (. 1.; • t (: c .: ( !J

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·.; ,ue) .;c.~ol..i ~Q.7J7

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u .. :~v :;?..2!JC J ;~ ... , c = 1. (.. t: (j ::.:. c 2.2 :3 2. ( 1;:. s ~~.l~_·J j ~. 7 :~ 0 ::4 ,;:! r :·

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1 ~ ~ ~ 3J \ q ·; • -~0 0 I'' t. \. t (.

1 u • .~ ,JU l ~;2 • .; c 0 1!.::?. c ":- 1:t

1 s, c ~.:,o 1 s ~: • c; 1 ., 1~-~.l'tl-4

1 3. (; '-:.t ·J 1 ~(~. ,·~ ~ ·J \ 5(; • J(JL

Table 5.3 Ti da 1 Reductions

I L'L AT f<LI- Ph, 1;. AT [ L>

b. ~~-r, 1. 403

7 .Ill 1 • 4 t e

1. 2 1 e 1 "~· ::!3

c.SJ5 1. JJ5

t>. j 1 '• • ; ) 1;

'•·.'3'-;,1) 1 .lt..u

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l.O;'.t <) .d2'l

t.J.BlY 0 .. u Uti

.-!. t 71 0. t 4 'l

J.2~5:~ (),(7J

(~ • 7 b ;~ (). 121

1 ,] 1 s o. c::: ... j

1.2C5 0. 7S2

;:..r;cc;; u • t: ~I

1. 2'-1 't 1 • C2 I

,, • 1 1 ~ 1 • I 7 u

<. c 1 1 1.~4b

7. 3 14 0 I • 4 Jt,

t.J. 1) 2 1. ~~· t

5 .. cc,.: 1 • o.:: r:

T 1 ,n:: AT 5H I~

<;.616

I c.Hd

1 l ol 05

c;.255

e. t:'+6

4.€3'4

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(..!I C.

!:.A2t3

.;.8b':

2.4S.l

2. !)J :)

z .oo 7

0. iit; 1

C .. -='06

~ .. Jt,._;

1 • .324

2 • 4 It!

2.714

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t. eu 1

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ST,JC V

O,E<;7EL: 'JO

c.~2o::~o 'J·J

c,.;;;7~c oo

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C,4~4CLl CC

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~.51~4D •JJ

0 , ':"; l L ~) 1J () •J

J. 4 'J., 'Jl) ) 0

C. J:.i2 1fG UJ

J,J172G ).J

c. s...,v i L,; J ,)

0.4.!B~C JO

c.:· .. l.~:; ~r'

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..:.!iC44C J.J

C .t~4Cil CC

c.~241D :Jc

1-' 0

106

-"r-eo"--" Station Obs. Predicted Di ff Obs. . Predicted. Di ff. Index No Time Lag Time Lag · Range Rat1o Range Rat1o "><

0065(22} 0 -3 +3 1.0 1. 032 -0.032 0001(14) -28 -32 +4 0.661 0.655 +0.006 0015 ( 15) +5 +3 +2 0.673 0.752 -0.079

" 0040( 16) +16 +16 0 0.900 0.827 +0.073 0060{ 17) -10 -7 -3 0.996 1.027 -0.031

0129(18) +9 +8 +1 1 . 201 1.170 +0.031 0140(19) +19 +20 -1 1. 325 1. 346 -0.021 0150(20) +17 +12 +5 l. 410 1. 404 0.006 0225(21) +11 +10 +1 1 . 317 1. 386 -0.069 0235( 1) -1.0 +8 -9 1 . 311 1.403 -0.092

0240(2) +16 +8 +8 1. 456 1.418 +0.038 0305(3) +12 +14 -2 1. 490 1. 538 -0.048 0312(4) 0 +5 -5 1. 361 1 .335 +0.026 0315( 5) -12 +1 -13 1. 265 1. 306 -0.041 0320( 6) -14 -16 +2 1. 060 1.100 -0.040

0330( 7) -16 +1 -17 0.956 0.911 +0.045 0335( 8) -18 -26 +8 0.843 0.828 +0.015 0336( 9) -37 -24 -13 0. 761 0.829 -0.068 0340:1 0) -34 -26 -8 0. 721 0.668 +0.053 034!i 11 ) -34 -31 -3 0. 713 0.643 +0.070

0353( 12) +18 -10 +28 0. 721 0.778 -0.057 0355(13) 18 +5 +13 0.673 0. 721 -0.048

--

Summar~:

Time Lag . o< Diff.< 12~ Range Ratio . 10.0051 < Diff.< I o .11 Time Lag, RMS of the Diff. (observed-predicted) = 9.44 min Range Ratio, RNS of the Diff. {observed-Predicted) = 0.0505 .

Table 5-9 Difference Between Predicted and Observed Values.

* Numbers in brackets corresponds to the serial numbers in Table 5-8.

VI CONCLUSIONS AND TmCOMMENDATIONS

The objective of this work has been to produce analyti-

cal cotidal models, using observed data or existing cotidal

charts, which could be stored conveniently in a computer

so that when observed sounding data are input, the

output would be reduced soundings. The principal advantages

of the proposed analytical scheme are the following.

(i) The analytical models can be obtained and

updated using the observed data in addition

to that of already produced charts. This

allows up-dating the model when more obser-

vations are available.

(ii) This scheme does not require large computer

storage space. For example, instead of

storing many digitized numbers, the digiti-

zed values are used to determine a few co-

efficients of the best approximating poly-

nomials.

(iii) 'These models allow for the rigorous propa-

gation of errors. With associated esti-

mated standard deviations, the reliability

of the final result can be easily obtained.

(iv) A degree of flexibility is offered. It i~

convenient to use data from existing co-

tidal charts, observations or a combination

of the two.

107

Least squares polynomiaJ approximation is applied

to either

(i) recover a function F(x) from a known set of

its values, or

(ii) to replace the known function in further

computations by a more trackable polynomial.

The problem of least squares polynomial approximation as

applied in this work is that defined by (i) above. It

would be interesting to view the problem as in (ii) above

and apply it to the Laplace tidal equations to obtain the

necessary polynomials.

From the test computations using the data on the Bay

of Fundy, the computer effective run time is 28.46 secsand

the storage space is 336,136 Bytes for Least squares poly­

nomial approximation for range ratios and time lags. For

the polynomial approximation for time series at the ref­

erence station the effective run time is 23.03 secsand the

storage space is 299,288 Bytes. For the Tidal Reductiou

therefore we have a total of 42 coefficients and their

associated standard deviations to store in the computer.

In the program to execute this for 22 sounding stations,

the time of execution was 0.76 sec and the storage space

used was 14,480 Bytes. The result also shows that the

water level at a location (~i' Ai: can be predicted with

a standard deviation (obi) of 0.5 m or better.

It is recommended that the prediction of tides at

the reference station with the polynomial should be done

1J8a

strictly within the time inte~val M used in the least

squares approximation of the observed series. When extra­

polation is required, it is advisable to use the amplitude/

phase analytical cotidal models and carry out the predic­

tion using the procedure described in Chapter II,

Section 2a2a

Finally, since the data immediately available was

not adequate to fully test the proposed analytical schemes,

it is suggested that proper data be obtained to facilitate

complete testing.

REFEHENC ,:s

'Admiralty Manual of Hydrographic Sttrveying,' Vol. 2 (1968) Hydrographer of the Navy, Taunton, Somerset.

Atlantic Tidal Power Engineering and Management Committee, (1969), 'Report on the Feasibility of Tidal Power Development in the Bay of Fundy, Appendix 1,' Tides and Tidal Regime.

Balogun, A.A.: (1977), 'Parameterization of Systematic. Errors in Terrestrial Geodetic Networks! M.Sc.E Thesis, University of New Brunswick. Fredericton.

Bye, J.A.T and R.A. Heath.: (1975), 'The New Zealand Semi­diurnal Tides,' Journal of Marine Research, Vol. 33.

'Canadian Tide And Current Tables,' Vol. I, (1978), Canadian Hydrographic Service, Ottawa.

Cartwright, D., W. Munk, and B. Zetler.: (1969), 'Pelagic Tidal Measurements,' National Institute of Oceano­graphy, U.K.

Christodoulidis D.: (1973), 'Determination of Vertical Crustal Movements From Scattered Relevelings,' M.Sc.E Thesis, Univ8rsity of New Brunswick, Fredericton.

DeWolfe, D.L.: (1977), 'Tidal Measurement Program of the Bay of Fundy-Gulf of Maine Tidal RegimP,' Light House #15, The Journal of the Canadian Hydrographer's As so(' ia t ion.

Dobler, G.: (1966), 'Tides in Canadi:;.:, Waters,' Canadian Hydrographic Services, Marine Science Braneh, Department of Mines and Techn1cal Surveys, Ottawa.

Dahler, G.C. and L.F. Ku.: (1969), 'Presentation and Assess­ment of Tides and Water Level records for Geophysical Investigations,' Symposium on Recent Crustal Move­ments, Ottawa.

Dronkers, J.J. and J.C. Schonfeld,: (1954), 'Tidal Computa­tions in Shallow Water.' Research Division, Rijkswaterstaat, the Netherlands.

Dronkers, J.J.: (1960), 'Tidal Computations in Coastal Areas,' Delta Project, A symposium; American Society of Civil Engineers, New York.

109

110

Dronkers, J.J.: (1964). 'Tidal 1 ·omp::tations in Rivers and Coastal Waters,' North-Holl~nd Publishing Company, Amsterdam.

Dronkers, J.J.: (1970), 'The Morphological Changes in the neighbourhood of Rotterdam Europort and the New Approach Channel,' Paper presented at the Centenary Conference on Ports and Harbour Managements, Calcuttr1.

Dronkers, J.J.: (1970), 'Research for the Coastal Area of the Delta Region of the Netherlands,' Coastal Engineering Vol. III, pp. 1783-1801.

Dronkers, J.J.:(l972), 'Tidal Theory and Computations,' Hydraulic Dept. of Delta Works, The Hague, the Netherlands.

Freeman, N.G. and T.S. Murty,: (1976), 'Numerical Modelling of Tides in Hudson Bay,' Journal of Fisheries Research Board of Canada 33: 2345-2361.

Gallagher, S. Brent and W.H. Munk,: (1971), 'Tides in Shallow Water and Spectroscopy,' Dept of Oceano­graphy, University of Hawaii, Honolulu.

Garrett, C. ari'd D. Greenberg,: (1976), 'Predicting Changes in Tidal Regime: The Open Boundary Problem,' Journal of Physical Oceanography, Vol. 7, March 1977.

Garrett, C.J.R. and W.H. Munk.: (1971), 'The age of the tide and the Q of the ocean,' Deep-sea Research, Vol. 18, pp. 493-503, Pergaman Press, Great Britain.

Godin, G: (1972), 'The Analysis of Tides,' University of Tornnto Press, Toronto.

Gordeev, R.G.; B.A. Kagan and E.V. P0lyakov: (1976), 'The Effects of Loading and Self-Attraction on Global Ocean Tides: The Model and the result of a Numerical Experiment,' Journal of Physical Oceanography, Vol. 7, pp. 161-170.

Hendershott, M. and W. Munk: (1970), 'Tides,' Scripps Insti­tution of Oceanography, La Jolla, California.

Hendershott, M.C. (1972), 'The Effects of SQlid Earth Defor­mation on Global Ocean Tides', Journal of GPnphysicnl Research, Vol. 29, pp. 380-402.

Hendershott, M.C. (1972), 'OcPan Tides,' Paper presen'ed at the lst GEOP. Researd1 Conference on Tides, Columbus.

1 : l

Krakiwsky, E. J. and D. E. \\'ells. ( lD71), 'Coordinate Syst(~ms in Geodesy' . Lecture ~:ote:; #16, University o [ New Brunswick, Fredericton

Krakiwsky, E.J: (1975), 'A Synthesis of recent advances in the method of Least Squares,' Lecture notes #42, Dept. of Surveying Engineering, University of New Brunswick, Fredericton.

Lambert, A.: (1974), 'Earth Tide Analysis and Prediction by the Response Method,' Journal of Geophysical Research, Vol. 7, #32, pp. 4952-4960.

Larsen, J.C.: (1977), 'CotiJal Charts for the Pacific Ocean near Hawaii Using F. Plane Solution,' Journal of Physical Oceanography, Vol. 7, pp. 100-109.

Luther S. Douglas and C. Wunsch,: (1974), 'Tidal Charts of the Central Pacific Ocean,' Journal of Physical Oceanography, Vol. 5, 1975.

hlosetti, F. and Manca.: (1972), 'Some Methods of Tidal Analysis,' Inter. Hydro. Review, Vol. 49, Monaco.

Munk, W.H.: (1968), 'Once Again- Tidal Friction,' Journal of Royal Astronomical Society Vol. 9, pp. 352-375, Sunfield and Day Ltd., Sussex ..

Munk, W.H. and D.E. Cartwright.: (1966), 'Tidal Spectro­scopy and Prediction,' Philosophical Transactions of the Royal Society of London, Vol. 259, pp. 533-581, May 1966.

Munk, W. , Snodgrass F. and M. Wimbush. : ( 1969), 'Tides Off~hore: Transition from California Coastal to Deep-Sea Waters,' Geophysical Fluid Dynamics. Vol. l, pp. 161-235.

Munk, W.H. and B.D. Zetler.: (1975), 'The Optimum Wiggliness of Tidal Admitt8nces,' Journal of Marine Research, Supp. 33: pp. 1-13.

Nassar M.N. and P. Vani~ek.: (1975), 'Levelling and Gravity,' Technical Report No. 33, University of New brunswick, Fredericton.

'Our Restless Tides,' (1977). U.S. DPpartment of Cor~ur.t.:l'ee, National Oceanic And Atmo~.-: heric Administration, National Ocean Survey.

Schwarz, H.R.: (1973), 'Numerical Analysis of Symmetric Matrices,' Prentice-Hall, Inc. New Jersey.

112

Thomson, D .D. : ( 1974), ' Mean Sea Lev•,l,' Dept. of Surveying Engineering, University of New Brunswick. Fredericton.

Tinney, B.: (1977), 'Automated Tidal Reduction,' Light House #15, Journal of the Canadian Hydrographers' Association.

Vanicek, P.: (1970), 'Further Development and Properties of the Spectral Analys~s by Least Squares,' Department of Surveying Engineering, University of New Brunswick, Fredericton.

Vanicek, P.: (1973), 'The Earth Tides,' Lecture Note #36, Dept. of Surveying Lngineering, University of New Brunswick, Fredericton.

Vanicek, P.: ( 1974), 'Introduction to Adjustment Calculus,' Lecture Note #35, Dept. of Surveying Engineering, University of New Brunswick, Fredericton.

Vanicek, P. and D.E. Wells,: (1972), 'The Least-Squares Approximation and Related Topics,' Lecture Note #22. Dept. of Surveying Engineering, University of New Brunswick, Fredericton.

Vanicek, P. arict E.J. Krakiwsky.: (in prep.), 'The Concept of Geodesy'.

VonArx, S.W.: (1962), 'An Introduction to Physical Oceano­graphy,' Addison-Wesley Pub. Comp. Reading.

Wells, D.E. and E.J. Krakiwsky.:(l97l), 'The Method of Least Squares,' Lecture Note #18, Dept. of Sur­veying Engineering, University of New Brunswick, Frec1~ricton.

Wemelsfelder, P.J. (1970), 'Mean Se: Level as a fact and as an illusion,' Symposium on Coastal Geodesy, Munich.

White, J.C.E. (Lt. Commander): (1971), 'Hydrographic and Tidal Information for Deep Draught Ships In a Tidal Estuary,' Paper presented at the XIrrth Inter. Cong. of Surveyors, Wiesbaden.

Zetler, B .. D. Cartwright and W. Munk.: (1969), 'Tidal Constants Derived from Response Admittances', Sixth International Symposium on Earth Tides, St rasbourg ..

113

Z~tler, D.B. and R.A. Cumming: (1066), 'A Harmonic Method for predicting Shallow-Water Tides, 1 Journal of Marine Research, 25(1). pp. 103-113.

Zetler, B. et al: (1975), 'Mod~ Tides,' Journal of Physical Oceanography, Vol. 5, 430-441.

A P P E N D I C E S

I OUTLINE OF THE LEAST SQUARES APPROXIMATION THEORY

Least squares polynomial approximation is applied to

either

(i) recover a function f(x) from a known set of

its values, or

(ii) replace the known function F(x) in further

computations by a more trackable polynomial.

The problem of least squares polynomial approximation as used

in this report is that defined by (i) above. A brief out-

line of the least squares approximation theory due mainly

to Vanicek and Wells, [1972] is here given.

Given:

(i) a function F defined on a finite set M

., 1 X X 111--ll 2 M discrete

M _ [a, b], M compact

(ii) a base'= ~ 1 • ~ 2 ... ~u . a set of u

linearly independent prescribed functions

from the functional space Gm,

(iii) a weight function W, defined and non-nega-

tive on M,

then the problem of least squares approximation is to deter-

mine the vector of coefficient (C 1 . c2 ... Cu) of a genera­

lized polynomial Pn which minimizes the weighted distuuce

P(F, Pn) defined as

115

116

P(F, Pn) - (X~M W(x}(F(x) Pn(x)) 2) 1/2 .

M discrete (I. 1)

P(F, Pn) - (J W(x)(F(x) - Pn(x)) 2) .l/2

M .M compact (I. 2)

The approximating polynomial is given by

n Pn = I C.l)J.

1 1 . (I. 3)

i=l The scalar product of two functions G, H E: Gm is defined as

I W(x) .G(x) .H(x), M discrete

---- XE:M <G, H> = ..__

f W(x) .G(x) .H(x). M compact

M (I. 4)

If the product of two functions G, H E: Gm is zero, then the

functions are orthogonal. If the base functions are ortho-

gonal,

<l)J. ,,, .> 1 'I' J

i = j

i :f j

If i j, it means

or

2 <w. w. > = 111/J ·II o. · , 1 1 1 lJ

where 0 ij is known as Kronecker delta

--1 i = j 0 .. =

1J . ...___

0 i :f j

(I .5)

and is defined as

(I. 6)

Returning to the problem of le:•st squares approximation we

are seeking :for the coefficients c 1 , _c2 ... Cu of the poly­

nomial Pn that would make the distance I IPn - Fl I the

minimum. This means minimizing the Eucleidean distance

I W(x)(F(x) - Pn(x)) 2 Xt:M

with respect to c1 , c2 ... Cu The condition is written as

Min c1 , c2 , ... cu s p 2 (P, F)

== Min cl, c2, ... c € E ) W(x)(Pn(x) - F(x)) 2

u l..·

XEM

Min u 2 cl, c2, ... c € E I W(x) L (C.l/J.(x) - F( x)) . u . 1 1 1 XE:M 1==

(I. 7)

When the partial derivatives of the above w.r.t individual

C 's are equated to zero, the minimum distance is obtained.

Minimizing we have

a I Xt:M

a c. 1

== 2 I

u [W(x)( I C.ljl .(x) - F(x)) 2]

j==l J J

u

XEM W(x) I (C.l/1 .(x) - F(x))

. l J J J=

o I C .lj! . ( x) . J J

aci

2 I W(x) I C.l/J .(x) - F(x) ljli(x) XEM j==l J J

u 2 I w ( X ) I c .l/J . ( X )l/1 . ( X )

X j J J 1 2 I w ( X ) F ( X )l/J i ( X )

X

0 . (I. 8)

From the definition of the scalar product, the above 0an be

written as

118

u L <~ .• ~.>C.= <F, ~i>.

j=l 1 J J CI. 9)

Equationi.9gives the system of normal equations which can

be solved to obtain the coefficients c1 , c2 , ... Cu. Putting

I. 9 in m·atrix form we have

Letting

and

U = [<F, ~i >],

the solution of normal equation is given by

-l . C=\N U.

( 1.10)

( I.ll)

( 1.12)

( 1.13)

N is the Gram's matrix and Gram's determinant det(N) ~ 0

because we are dealing with linearly independent base

functions '¥. Equation 1.13 therefore has a unique solution.

If we are dealing with orthogonal system of base

functions ~*, then

- * * I *·1 2 N* = DiagL <~i ~i>] = Diag( I ~i I ) .

The solution of the normal equation becomes trivial and

is given by

* 1· *I. 2 C* = <F, ~i>/ l~i I , i = 1, 2, ... u. (1.14)

Each Fourier coefficient C* can be solved independently.

The system of base functions f often encountered are

not usually orthogonal. The system can however be orthogona-

lised using Schmidt's orthogonalization process. The

procss works as follows:

i) choose

ii) define

* 1)!1

119

X t: M (I. 15)

* * tjJ 2 ::: tjJ 2 + 6 2 l ~I 1 ' X £ M, f3 E E 2 1 (.!.16) '

* Multiplying the above equation I .16 by WtjJ 1 and summing up

all the equations for all the X yields

* * * * * <tjJ2' tjJl> == <tjJ2 tjJl> + 6 2 {tPl, tPl> '

(I. l 7)

* * To make the system orthogonal,< tjJ 2 , tjJ 1 > must be zero. The

unknown coefficient 62 1 can be determined from '

(I. l~)

iii) Define next

* ,,, * D * D B t/1~ == l)J3 + 6 3,2 '~'2 + fJ3,1 1Pl, X EM, f-13,2' 3,1 t:

(I. 19)

* * Multiplying by W1)! 1 and w~ 2 yield respectively

* * * * * <l)J 3 ' l)J 2 > == < t/13 ' 1/J 2 > + B 3 , 2< t/1 2 ' tP 2 > + f3 3 , 1 < t/1 i ' tP ~ > ·

By reason of orthogonality,

* * <1)! 1' t/12 > = 0 .

We therefore have that

that is

* * * < tJ; 3 tJ; 2 > + B 3, 2<tJ; 2 · tJ; 2 ' = 0 '

* "' * B 3 , 1 = < tjJ 3 · tj; 1> I <tJ; 1 ' t/J 1>

* * * s 3 . 2 =< tJ; 3 tJ; 2 >I<~ 2 ' ~~ 2>

(I. 20)

(I. 21)

The process can be g·eneralized for any coefficient 8 .. thus ,> J 1

* * * B j i = < tjJ j 1/J i >/ <1/J i tj; j> (I. 22)

Expressing the original system in terms of the orthogonal

system we have

* tj; 1 = tj;l

* * * * tjJu =- Bu,l wl- £u,2 tj;2 · ·· - Bu,u-1 t/Ju-1 + tJ;u

Putting it in matrix form we have

* l/Jl 1 0 0 0 tj;l

tj;2 -62 1 1 0 0 tj;~ '

= * ( I.23) tjJ3 -s3,1' 83 2' 1 0 tjJn

' ,:)

* tjJ -(3 u 1' -B s ... 1 tVu u . u,2 U, 3

B .. is defined Jl

by equation I.22.

Letting

1 0

-(32 1 1 '

B uxu -(33.1 - (3 3 2

'

-Bu 1 '

-su,2

equation 1.23 is written as

* ljJ == BljJ •

1~~

0 0

0 0

1 0

-su 3 ... , 1

(1.24)

B is the transformation matrix that transforms non ortho-

gonal system to orthogonal system. It is an u x u tri-

angular matrix and the determinant det(B) r 0.

If we have that

using equation I.24, we can transform the Fourier coefficients

into the coefficients of the origjnal bas.; functj_ons, thus

* M"

(BII' ) 1 C

C == (BT)-lC* . (I. 25)

II DRIE.F DEI:>CR!PT!ON OF TliE COMPUTER PROGRAMS USED

The computer programs used in the computations are in

three parts, namely

(i) Least squares polynomial approximation for

cotidal curves,

(ii) Least squares polynomial approximation of

observed time series at the reference station,

(iii) Tidal reductions.

II.l Least Square Polynomial Approximation for the Cotidal Curves

Figure A-1 is the flow chart describing the program.

INPUTS

lst card, FOR.MAT(5X, 7I4)

ID - The dimension of the polynomial

N - The degree of the polynomial

M - Number of data points for the approximation

NPP- Number of grid points for prediction. If

there is no prediction NPP = 0

INDEX - Code for the type of function to be approxi-

mated. If index= 1, the polynomial approxi-

mation for range ratio (amplitude) is per-

formed. If index = 2, the polynomial

approximation for time lag (phase lag) is

performed.

Figure Il-l Polynomial Approximation of Cotidal Curves - Flow Chart

READ DATA J I

Compute Req.No. ot Co- Yes c;;e warning ~ efts. Is df. negative?

Ho CALL CHARTE

(converts ~. A to x, y)

CALL VANDE Matri:=J (Forms Vandermonde • s

--Poly. Approx. For Range

or Phase?

I RANGE

Define \tJeights

I

Solution by orthogonalization?

NO

J

PHASE

'·

SPLIT: FA = ~R cos v Define FB = iR sin v - weights

I

,-- CALL ORTHO

- CALL APPROX. -~ ~ L.S. Approx with O.B.F)

I (does L.S.Appr~x with r·L~

Prediction at Gn d --~ Y~_s ____ l-~ Points? j

~NO -~

c· PRINT RESULlS -]

I CALL CA;!H:_jE CALL VANDE CALL pr·ED

~-~---~----------------- --------------

~

124

ID - Code for orthogonal or non orthogonal

solution, 1 - for orthogonal solution

2 - for non orthogonal solution.

ITEST - Code for testing Fourier coefficients

0 - for no test

1 - against its Standard Deviation

2 - against 2 times its Standard Deviation

3 - against 3 times its Standard Deviation.

2nd Card: FORMAT(5X, 15, 5X, Fl0.6, 5X, Fl0.6

5X, Fl0.6, 5X, Fl0.6)

This card contains the identification

number, latitude ~ and longitude A of the

\ reference station, the range ratio ( ampli­

tude) and time lag (phase lag) at the

reference station.

n cards: Format as in the second card. Each card

contains, station identification number,

latitude (¢), longitude (A) of the data

points, the range ratio (amplitude diff.)

and the time lag (pbase lag diff.) at

each data.

NPP cards: FORMAT(5X, 15. 5X, Fl0.6, 5X, Fl0.6)

If there are nn prediction at the grid

points. these ~ard~ will be omitted.

Each card contains the grid point numJer,

the geodetic coordinates of the grid

12o

points ( ¢, ,\).

SUBROUTINES:

SUBROUTINE CARTE; computes the local cartesian co-

ordinates (x., y.) given the geodetic coordinates l l

of the points ($i' >-.), the geodetic coordinates 1

of the origin of the local system ($0 , ;,0 ) and

the dimensions of the ellipsoid.

SUBROUTINE VANDE - computes the prediction matrix

given the geodetic coordinates (¢, A) of the pre-

diction points, the number of prediction points,

the dimension of the polynomial and the number

of coefficients.

SUBROUTINE APPROX. - does the Least Square approxi-

mation of the function given the number of coeffi-

cients, the number of data point, the Vandermonde's

matrix, the weight matrix and the functional values.

SUBROUTINE ORTHO- orthogonalizes the Vandermonde's

rna trix using Gram Schmidt methc·d, computes the

Fourier Coefficients of the orthogonalized matrix,

derives the coefficients of the Vandermonde's

matrix, computes the variances of the Fourier Co-

efficients and the variance-covariance matrix of

the original coefficients.

SUBROUTINE PRED. - predicts t!le function values

at the grid points and computes the variance-co-

variance matrix of the prediction.

~*····~······~~····~·~~-~···1~************************•***~*****************************************' $~UU lKEN~A/G

1 ;~

3 4

5

u 7 u

c ~·····~····~~·~·~·1··········~····································· C * P~~(~~~E lC ((~5T~UCT A~ALYTlCAL COTIOAL ~HARTS USING ANALYSED * L • TiCAL CC~STA~l L~ RANGE kATlGS A~D TIME DlFF. REf,TG A STAhDAR* C + C ST~llC~ Ch C1GlT1ZLC VAL~LS F~UM EXISTl~G COllDAL CHART * c • * L + T~l5 ~~ES T~E LAT. AhO LCNG. c~ Cl5CRETE POINTS.THE AMPL.AND * C • PHA~E LAGS l~ ~ANGE RATlG~ A~D Tl~E DlFF~. ~T THE PUlNTS AS * L + 1 ~E I~FUlS TU CCMPUTE THE CGEfflCI~~TS CF THE BEST PREDICTING * L + PLL~~C~lALS Pl ANY ClrER PCI~T. * c • * L + GLIDE lU 5U~E NCTATICNS U~tC IN lrE PhOG~AM * L + INPLT: * (_

~~ l. (.

(..

L c (.

(.

L c (. (..

(..

<.. l. L l c L c c c L c c. L (. (.

c

+ FHI- LATITUUC Gf STNol * + ALL~ - LChGITUDE CF ST~. I + FHoFG - A~PLITUOL AND PHASE LAG.·LR RA~GE RATIO ANC TIME + OIFF.l~ ~~~~TES CF TIME. ~ lU - Dl~ENSION UF lHE AFPROX. * ~ - DEGFEE Cf THE PCLY~UMIAL * ~- NU~EE~ CF OESE~~ATlCNS * 10 - ~CCE FGR ORT~CGONAL CR NCN CRlHOGONAL SOLUTION. + 1 - FOR URTHOGC~AL SOL. ~ 2 - PCR NC~ U~ThOGUNAL SCLo * llEST : COUE FOR TESTING FOURIER COEFFS. * 0 - NG TEST * 1 - AGAINST ITS STD DEV. * 2 - AGAINST 2 TIMES ITS STD. DcVe + 3 - AGAINST 3 TIMES llS STD. D~V. * ~PP! - ~0. OF GRIC FGINTS FGR P~EClCTION.lF THEHE lS * ~0 PREDICTION AT GRID POI~TS NPP=O

* Ol.TP~l:

* * * * * * * * * * * * * * * * * • L - hLo LF <..~EFFICICNTS RECUIREO. t~HCN

EXCEEC ThE ~C. OF CBS TH~ PROGRAM CoCA,CE - ~ECTG~S GF CCEFFlCIE~lS.

TrE NU. UF COEFF* • • •

IS AUORTEDJ

* * • * i SEE SLLRClTINE GRTHC FUR MUR~ EXPLANATIG~S OF OUTPUT NUTATIUNS * • * ~ * ······~·~~·~~·~~~·~··********~·········••***~**********~*********** * MAIN P~OGRAME

lM~Lltll REJL*8(~-ro0-Z) ~l~E~~~l~ Phl(~OJ.ALON(50JoFH(SO),FG(50).A(50o50loPC50o~0) * • C u \i /l r.; ( S C • 5 C ) , V ~ '' (50 , 50 l o C ( 50 ) o t:N ( 50 • 5 C ) , X ( 50) , Y ( 50 ) , V ( 50 ) , * ~{~Ol.Nl~(5aJ,NG~l0(5Q),XPC50),YP(50),PMT(50o50loPMC0(50o50)

" 11'<. t ~tl, >vAf.P c=o, CP'I;t-~1(1' ~LPT (50) ol\l:f'-iG(50),PM(50o50)oPFH(!30)iPF~(50) CH,t:I\!:IC~ ALFHA( ~0 ,5c.: ow(50) ,FC(!30) oSl.~FC(50) ,SGN(50 ),STCP(!50) •

* D(5J),~IGtvf.FC!:O)oSIGI~AH·.50) lJ I t.< E 1\ !: 1 C N f • A ( 5 0 ) , r- U ( 5 •J ) , J= t'\ 2 ( 50 ) , F I:J 2 ( 5 C ) , 'W A ( 50 J • W t=· l !3 0 ) , C A ( 5 0 l • * Ca<~Ol,CLA(5C,50),COUC~0,50),PFA(50),pFB(50)•VARPA(50,50),

+ ~A f; iJ li { 5 C , 5 () ) , t: ( 2 , 2 ) , A f<! 50 , 50 ) , F R (50 ) l~CA=lCI~ICE=ICC=~O

11HC=20<: 2C:5 • OCO Fl=.3el41~9C::!!:lJO

1-' ~.J m

9 lv

(.

c c.:

11 12 14(;

(. (. (.

c.: lJ 14 1 5 lo 17 lts 19 20

~ (.

c ~~ 22 20 c 23 10 0

c c (.

24 25 H::

{.

c.: c

27 28 101

2'i..l 10 2 30 31 :!2 3.3 3

c.: 34 .35 1.31

36 37 13 2 3e 39 130 40 41 12i 42 4.3 44 45 11 1 46 4

c ~

1'1.=!: SIGMJI=l.CDC

f<L Jl C I 1\ P f\ 0 G F !1 P. t:: S P E. C IF I CAT I 0 I\

~ E ll C ( !:: , 14 b l J C, 1\ • ~, NP P, 11\ CCX • I 0 , l TEST Fl I'M A 1 ( !: > • 7 l 4)

CO~PLlE ~u. CF COEFFICIEhlS tNC CE(HlE OF PGLY Al\0 CEGR~E UF CF f~CECOMe IF Of IS ZERO CR NEGATIVE GIVE WARNING AND STOP

L=(l\+1),.*10 IDF=M-L

IF!lLf-eLI:.C)lt-:Et~; DC FRli\T, 1 F~CGRAME SPECIFICATICN INADEOUATE' ~TCF lL~E oa (1\C IF

CO 1\ 1 II\ l.. E

hEAD 11\ CATA Cl\ ThE STAI\DARC OR ~EF. STATION

~EJD(~t2C~)I\~MC,Ft-:IO,ALC!I;Q,FHO,FGU FC~MAl(~Xwl5o5X,Fl0.6o5XeF10.6,5XeF10.6t5X,FlO.c) FC~Mtl(5X.I~t5Xof(0.6o5X,Fl0.6)

READ IN DATA F~CM OlHE~ STATIONS

CC 1 l=loM f;EJl0(~,.2C0)1\l.iM( I ),PHI( I) tALON( I) ,FH( I},FG( 1) CCt-Tl"UE

PRINT .ALL Tt·E JNPUT DA"TA

H'lNllJ1 f(f.~JIT (/.1,4Xt 'NO' ,ox, 'LAT lTUOE 1 ,lOXt 1 LONGITUDE' tlOX,'RANuE FiA

>tTlO'.lOX, 1 1IME LAG') . Fl~MAT(EXel4t7X,FlQ.6,SX,FlC.6,S~~F10.€t9X,Fl0.6) Fhii\l102tN~~C,PHlU,ALO~C.F~C,FGC CC .J 1 = 1 ,M P .. II\ 110 2 , N U f.l ( I ) , F H 1 ( I ) • ALt..: II; ( 1 ) , F 1- ( 1 ) , F G ( 1 )

CCI\ IIN~E

Ff'l", 12 1 FC~MA1(11t5Xt 1 CEGotF PLLYo 1 oEX, 1 NOoOF OE~e 1 t5X, 1 NUoL~ CUEFF•'•

• 5X, 1 CEG.CF f~ECI.:~'} F~IN11~2,No~tLtlDF ~C~~Al(//,8Xti2,12X.IJ,l5Xti~.20Xtl3) FF<!I'.llJC FL~MPl{l/t5Xt 1 CARTESlhN CGURD, OF TH~ GIVEN STATI0N5') Fj.-;!,...1127 fG~MAl(l/e~x.•srN. NL.' .ex,•x-ccc~o.•,tsx.•v-cacr.o.• 1 CALL ~A~TE(f.I,Pt-I,ALQN,PI-lOtALONO,X,Y)

DU 4 ! = 1 • ~ F~ 11\1111 ,NU,. (I) ,X( l ),"((I) f- 0 ., fJ ,q ( I • 6 X • 14 I 4 X • f 1 5 • (J • 7 X • F 1 5 • 6 )

tlJI\Tll\l..E

<.L~PLII.: 1'(11 fl N ATfd )<

t-1 1:\:) --.)

4 1 4d

4<;; 50 51 52 53 ~4 5~ ~{:

57 5C:J 5"' 60 {; 1 (;2 (;J (;4

(;~

t::()

6 -, {Jb {:;Cj

7C 71 72 73 74 7 ~1 76 7 I 7b

7 •;

E!O b1 u ~.

u<e

8~

e '~ t:= 80 t 7

bb 8'i ()O 91 Y2 93

c

5

1 4 1 :: 7

12E

L c. c.

4 /~

,_, 7

103 L (._

<.. <..

c.

H

fj

c

c L.l !.: 1-= 1 • 1--1 CC :.> ..;=1oL

A( lo.JJ::C,lJO COI'Tli\UE llJF=I'tl DO 7 I R Cf-= 1 , I" ICCP:C CU 1:: I=lolCF 1 A= I- 1 CC 14 J=lolCP JA=J-1 ICCF= ICCf.+l P( lRCf'ollDF)=X<lRUP)**IA*Y( lRDPJlt*JA

((f\llNl..E CCNlli'LE

Ct.;I\Tli\UE F~ll\1121:: FL.I~MAT(I/o~~.·~EI'DE~CND ~AThlX 1 ) FF<INT,' ' CALL IVLllDU.lCAof'oLJ

CcTER~INE TrE V~L~ES AT THt DISCRETE POINTS

lf(Nff-oECoC} GC TG S7 CC 44 1=1,1\FF f; E AD ( ~ , 1 C 0 J 1\ GH 10 ( 1 ) , ALA T ( I ) , AL C l'o G ( l )

CCNIINLE lALL CP~TECI\PP,ALAToALLI\G,FHIOoALCNOtXP,YP)

CALL ~AI\CC(I\PPoLolCPoALAToALCNGoPHIG,~LCNO,XP,YP,PM) CCI\111\LC

CO 15 ISIG=l,li\OEX IF(lSIG.EG.lHrEI\ CC

FRli\T, 1 t+ POLYNO~IAL AFPROXI~ATICN FOR "RANGE RATIO' H-<11\TlO::

FGFMA1(4Xo '**************••***•*******************' )

lF A~FLlTLOE U~ fA~CE RATIC IS TC APPHOXlMATEU PROCEED, CT~E~WISE GC TC ST~TEMENT ~C 46

lF(lC.EG.lHC lC 46 FC~~tliCI\ Cf WEIGHT MAT~IX CG E (=;ltM CG E ..:=1tM F(lo.Jl-=C.OO CCI\llt>UE CO 'I J.:1,M J= [ F(J.J)=lCOoCDO Cli\111\UI::

c Ft:I•FLf.;t-' lH~ LEAST SGL:AHt.S t.PPfWX BY CALLING THE SUBRGUTit-.1::. APPROX c

10~

lOt

CALL ~PF~U~(Lo~tAoFoFHoE~.C~V,AC,U,CCVARoAFVF) 1=1-\It-.110~ FG~MA1(1CXo 1 ~ECTCR GF CCEFFICIENTS' CALL t-CL1lJ(C,lC.AoLol)

Pfdl\l1C6 Fl.} f.~,. 1 ( 1 c X. 'I~[~ l Cl; A L. '!;X t I ... E c T Cl''

f-' t-V 00

94 95 <.16 91

S!:l <,<;;

100 101 102 1 03 104 1 05 106 1 07

108

10':J 110 1 11 112 113 114 11 ~ 116 11 7 110 1 1 ':,1 120 121 122 123 124 125

126 127 128

1 2'-J 130 1 31 132 13.3 1 34 1.35 1.36 137 1 38 13:1 1 A(\

107

10e

11 0

4t

57 c c c c

47

4E:

4S

51 c

(ALL t-CLlO(\i.lCA,~.l}

PF<lN11C7oAPVF FUF~A1(10Xo 1 JI FCSTE~lO~l VARIA~CE FACTOR =',FlO.El Pfdi\TlCE

fC~M-1(10Xt 1 \/~~lANCE COVARIANCE MATRIX OF THE COEFF1ClENTS 1 )

CALL ~GL1D(C(VJR,1CA,L,L1 PRll\lllC FCFMt1(//t5~t 1 FCL~NC~lAL APP~UX CF ThE GIVEN FU~CTILNS') CALL WOLlO(ACtlDAoMoll IF(1C.EG.2)G( 10 51 CCI\111\l.E CO 57 l=lof>l 'll(l)=lOC.OCC CCI\Tli\I.JE

PERFORM LEAST SCI.JJIRES APPRUX I.JSII\G CRTHCGCNAL EASE FLI\CTICNS lY CALLII\G l~E SLEROI.JTINE CRTHU

"' CALL lhT~O(~,L.SlGMAoAoiDAoSlG~FCoVFCoNPColT~ST,V,CGVARoFH, ~,(,ALP~A,FC,SI.JMFCoSONoSTDPoiW) Fh 11\11:::: CC 47 l=loL PRU..T134,FC(l) eSli~FC(l) CC.I\111\I.JE:: I=HlN113!: Cu 4E l=ltL

H111\T126,C(l) CCI\1 1M.!:.: P~lNlt'~C. (f (OEFFoCF C~lGN.POL~.AFlER TEST =• ,N~C PfdN11J7 CALL f>CllO(CC~If'.ICA,LoLl ~f'll\11.3t QC 4<; 1=1oM Pfdl\113c,V(I l CCI\T lM.E Pf'll\lo 1 A Fl.STEGRI VARIANCE FACTCR= 'oVFC CCI\1 11\LE

C IF Pf'E:ClCllCI\ IS f.;EL.UlR£.:0 ((ILL ThE SulJROuTlNE PHI::D (.

c c l.. c

lF(I\F~oEGoC) GC TO ~e CALL FI'EC(NPP,L .. ~M.C,COVAR,FMT,PFHoPMCOoVAR) tLH CG

1~ AFP~C) IS FOR TIME LAG (PHASE LAG)Pf'GCEED

P~1Nlo 1 it FCLYNCMlAL PPRGXl~ATlCN FOR TIME LAG 1

iJfdNll03 (K::iJ o5*Fl/l€Co f:U:c!:olC/3o2cCE E r•:: C • 1 2 5 >I P I I 18 C • E I~ f.= C • 1 C DC 6 C 1 :: 1 , M t:H::f C (I) >ICK R=l'UHH(IJ/i2o fP ( 1 l::fi HCCS (PI-) F!J ( 1 )::I·HCS if\ (PH) lAo 1:!. f I I ,- 1 • / 1-' ,; r; ll • :;:

!-' 1:'-J (.;)

1 4 1

14 2

143 1 4 '• 1 'I !;i 146 l 4 7 14t.l 14'J 15i) 1 51 152 1~3 154 1~5 15u 1~7 158 1 sg 1 ( \,) 1 (;; 1 162 UJJ 1 t 4 1 t 5 lt.o l (c I lcB 169 170 1 7 1 1 7 c. 173 1 I 4 175

1 76 17 7 1/IJ

1 7 '} 1 tj:J

1 u 1 1 t) 2 1BJ li.l4 1 ct 186 lti7 1etl 189 19U 1 91 1 ~2

~t-3 (I).: 1 o/EF.f.; :H2 c C FC~ t~ThCtC~~L Et~E FU~CTIC~~ STAEME~T 52 IS EXCUTED c

c

GC CU,lli\LE

lF(lCoEColJGC 1U 52 CIJ 17 1=1 ,M CG 17 .. =1 ,tJ F(l,J)=C.CO

17 CO~lli\~E CC. le l=l,f'J

.; = 1 P( loJ)=\\1<(.1)

lc tCI\TII\LE Pf'IN114C CALL APP~OX(LotioAoPoFA,(~,CA,VoAC,~,CCA,APVF) FE;I~llO!; CALL t.'CLTD(CP,lDAoL.l) FRINllOI: CALL tiC l.lO ( 'v tl CA or-\ .1) FF<ll\llC 7 tAP \If DG 1!: 1=1,Pt DU lt J=l,J< p.( loJ>=o.oco

H CC~lll'-i{.E cc 1<; 1-=l,tJ -~=1 F ( 1 • J ).:: II E ( 1 )

lS LC~llNl.E f-:blNllJ<; CALL APFROX(L,N,A,P,FH,E~,<;e,V,ACoUoCOB,APVF)

Pkll\110f. C~ll 1-CLTD((E,ICAoL.l) F h it, 1 l 0 t tALL ~C\..1() ('v .ICAtl-lol) FR II\ T lC 7 ,AF\tf lf(lCoEGo2lGC TO 56

52 Cll\1lNLE C fl~D PCLY~OMlAL APFGOX. FOR FUI\CTIONS FA.FE c

1 '• c

*

53

!", 4

Fr.;If\1140 fUhM~l(// 1 5X, 1 FCLY~CMlAL AFP~OXl~ATlCN FOR fUNCTILN FA') CALL LRTbU(~,L,SIGMA,A,lOA,SlGMFC,VFC,NPC,ITEST,v,CQA,FA.~A.CAo ALFhA ,FC oSl.JtvfC ,SCN oSlDP t lw J Ff,(Nll..:.::

C(. 5.:: 1=1 ,L r:H IN 1134tFC ( l) ,Sl..MFC( l) (.L~l 11\lif. Pr;ll\11~!:

CC ~4 l=l,L ~Hll\T13t.c.t. ( l> CCf'.TIH .. l F~l~l.'l\(. lf CCE~~.OF C~lGN.POLY.AFTER TEST =•,~PC t-:1111'.11::7 ( .A. L L I• U L 1 0 ( C C A , 1 C A , l. • L ) FM{I\11Jf: CO ~!; l=l,N PFll\TlJt:,V(l)

f-' w 0

193 1<:.14

195 19u

197

19U 1S.9 200 201 202 203 204 205 206 207 20fl 209 210 211 212 213

4:14

215

216

217

~=

c

13':. c

cl

t.;2

63

( 5(:

c c c (

c. c c

,.

CUI'-T lf\lJ[: Flllf\1,'/l. PCSlELF<I VAI<l/11\CC fACTC.R=',VFC

r:~ltdlJS f;:f-<M/11(//,~.X,'I=CLYI\LMIAL A~:-"f'GXl~ATICI\ FOR FUNCTlCN F13')

CALL LRH'O(,.. ,L,SlGMA,AolDI\tSfGMFC,VFC,NPC, lTESToVoCOO,FuoWOoCl:lo ALPHA,FCoSL~FCoSCh-,STDPolW) I=Rll\113:! CU cl 1=1,L Ff' 1NT134 ,rc ( l) ,SUMFC( I) CLI'lll\l-l t=r;INliJ= · CC t2 l=loL PfdNTl.;t,CE( I} CCI'THUE 1=~11\T,'I\C CF PGL'V.AFTE:R TESi= 1 oi'<I=C Ff;l!'.11J7 CA~L ~GllD(CCEolCA,L,L) Pf\1Nf13€ CC 63 I=ltl~ P li l N 1 1 3 ~ , V ( l J COd lt-.lJE P~li\T,'A PCSTEC~I VARI~I\CE FACTC~='oVFC

C.LI\111\l.E

IF Tt-t:F-E lS /'.U FJ;EClCT ICN STATEMENT NO 'iS IS EXCUTED

lF(I\FF.EC.C) GC TO se PRED1C11CI' CF FL~CTICNS FA,FE AT GRIC POINlS

CALL ~REC(I\PP,L,F~oCA,CCAoF~T,PFA,P~CCoVAkPA)

<ALL FREC(/'.FF,L,FN,CEoCCEoFMToFFE,PMCG,VARPBJ

CCMFUlE l~E P~ECIClEC liME LAG ANC ASSOCIATED VAHlANCES 218 DO t.4 1=1,/'.FF 219 FFUU=CHJN(Pfe(I)IPF,6(1))/CI< 2 21) E: { 1 , 1 ) ::: ( 1 • I ( 1 • -+ ( PF E ( 1 ) I P F-A ( 1 ) ) * * 2 J J *(- P F 8 ( l ) / PF A ( l ) * * 2 ) 2 2 1 1: ( 1 , :2 l = ( 1 • I ( 1 • -+ ( P f B ( l ) I P FA ( I ) ) * * 2 ) ) * ( 1 • / PF A { I ) )

C CC,..PLTl ~JRlJNCES 2 2 2 VA f; P ( 1 ) = t: ( 1 t 1 ) *>I< 2 *VA I~P A ( l t 1 J + 8 ( 1 t 2 ) ** 2 * V AR P 8 ( 1 , 1 ) 2 2 3 S I 0~ A f ( l ) = 0 S CRT ( VA RP ( 1 ) ) / C K 224 64 CGI\ll~LE 225 t::"C lF 226 ~€ CC~lii\LC 227 15 CC~Tl~UE 228 Phli'-T•'*** P~ECICTlON MATRIX ****' ~29 CftLL ~CllO(F~oiCA,~PP,L) 2~0 Fh1!'.112t 2:!1 Fr.It-.1127 232 1~6 fCF~t1(/loSXt 1 CARTESI4~ CUOROS. CF GHID FUlNTS'l 233 ~C 5C l=loNPF 234 Ff:;tN1111,NGI<lC(l)oXP(I),'tP(l) £35 5C Cl~l!Nl.E 2 .3 c .-: :~ r " r 1 2 5 2 .:i 7 1 2 t F C f. i• t T ( 1 1 ' , / I • 5 X , 1 P I~ E:.l.) I C 1 E 0 R A N G E R A 1 l 0 S A 1\ lJ T I M E L A G S A T T H E G

HHD Fll"lS') 2.3u Ff. 1~'<114 1

r-' v:: r-'

2.39 240 £'41 242 ~43

244

245 246 247 £:48 24<;; 2:::o 2 51 <:.52 2~3 2!:;)4 2e~ 2 tc 257 c. t (j .259 260 261 c: 6;:: 2 t:J

2o4

26~ 2o6 2 f:; .,

L

* 4!:

b(

b7

uf.: 1 ..•

.) --1J4 13t lJ(; 13 7 13c 1 4 l

* 1 '•" ,. 1 4:

CU 45 [.::1 ,1\FP ALftl(l)=ALAT(l)*lBO./Pl Alli\C( I )~ALCI'G( I )*1130,/P I 5IC~~~(l)=OSCRl(VAR(l,l)) F ,:. I N 1 1 4 2 , N C: I~ 1 U { I ) , A L ft T ( I ) , ALi J N G ( 1 ) , P F H ( 1 J , S 1 G M A k ( I ) , P F G ( I ) , !:l<.:~Jit-=(1}

CCI\111\li£:

CU tf: l=l,L J=I \\I~ IT E ( 7 , 1 '* 5 ) < ( I ) , (. C V AR ( l , J )

CCNllNLE CC 67 l=l,L .J:: 1 V.RlTE(7t14~lCA(I)oCUA( I,J)

((.t\ 1 [1\t,.f_ CG u: l=l,L J= I llh ITt: ( 7,14!:) ( (; (I), CLD ( l, J) CCI>~ I lN Lt::

f- C J;f>i A 1 (II, 5 X,' FL l..h l t:;R C CEF F I C lENT 1 , 10 X, 'VARIANCES 1 )

FUf.,.Al(/,8.li,E11.4,10X,Ello4) fLf.NJl(//teX, 1 VECTOR UF ORIGINAL CCEFFICIEI\15 1 )

FCFf>I~1(1,5X,[11.4J FC~M~l(//o5Xw 1 VARIANCE-COVARIANCE ~ATRIX OF THE CUEfF. 1 )

FC~~A1(11o5Xo 1 VECTCR UF ~ESICUALS 1 ) F c f; M ~ 1 < 1 1 • 4 >~ , 1 r- c • , ex , • L" r 1 r u o E • , s x , • LC "G 1 T u o E • , 1 ox , 1 nANG E RAT 1 L 1 t4X, 1 Sl<;MA 1 o7X, 'l!IYE LAC' o5Xt 'SIGMA') FLf.~Al(/,3Xol4,5~.Fl0.6t6X,Fl0.6t6XoFl0.6,6X.Ell.4ofX,FlOo6t tXoE11.4) FOf..~~l(!:X,O:Ello4) STCP Ef\C

2b8 SU~~lliTlt>L CtHl~(M,ALATtAL~N,PHICtALGNO,X,Y)

~:b9 27C 2 71 <::. 72 27:3 2 74 215 276 277 <:.Hl 'i. 7";; 2UO 2 f.J 1 2b2 2H3 2 tJl~ 2B~ 2 l.l tJ 2U .{

c C 1~15 S~B~ClTJ~E CCMPLlES THE CARTESIAI\ COCf;DS. x,y, fRGM GEO~RAPHICAL L ClCRC~ •• Lf.llTLLE Af\D LCNGITUQC,. c c

4

I~FLJ(ll REPL*8(P-r,O-Z} ClfVEf\~ICI\ /L~T(50) ,ALOtd50)i,X(50).Y(50J r.~=e.::7u<;co.~co r• B = t; .; ~ e ~ t: 3 • E c o P1=:!.1415<:>2<!;DC EC= ( f.oA**2-f<E **2) lr\ A**2

FHIF=FrlC*Fl/100. tLC~R=~LCI\O*Pl/180.

X M = { ( 1 •- H. ) >to FA J I ( D S 0 f~ T ( 1 •- E C * ( 0 S 1 N {PH 1 R l * * 2 J I** ::3 ) X 1'. :. f:; A I 0 S C R T ( 1 • - E ( * ( D S 1 t-. ( P H 1 R ) * * 2 ) ) ~=CSC.f.TPM*.llf\J

DC 4 l=ltN ALAT ( l )::ALIT (I )>I<Pl/180 • ALL~(l)=ILG~(ll*Pl/lUO.

X(l>=f.ll (ALAT( 1)-PhlR) 't( l)::f<*CCUS(PH[f<)*(ALGI\(1)-ALCNR)

LCf~Tli\UE FEll..rl\ EI\C

I-' ':.u t·"

(.

2tld c.. (.

c..c 2 b'i 290

fYl 292

. 29.J 2~4 40 2~5 21,1(; 2S7 291.i 2f:l':i 3vO J01 .302 30.3 4:! .304 I~ 2 305 41 30(: 307

<.:

308 \... (. (.

c 30Y .310

311 312 313 .314 .31~ 31o 317

31U c.. L c L L \...

31<) 320

3 ;2 1 3C.2 423 .] 2 (f

-: ••t::

Tt-l~

fL~~ULTI~E Vft~CL(~P~oltlCP,ALAT.~LC~G,PHIU,ALC~U.XPeYPoPM) SUEf.;lLT ltd:: Ci.:iJ.Fl. TES 1hl Pt<E;;:; lCTlCf\ MATRIX PI>',

IMfllC.ll ~EflL*B(A-~.G-Z) Cl~Ef\5IC~ P~(5Q,5Q),ALiT(5Q),PLC~G(50),XP(50),YP(50)

tU 40 l= 1 ,f';PP CU 40 oo~=1 eL FM( leJ).:Q.CDO CCI\ll~l.E CC '1 1=1 oi\PP rccr=o LU 42 t<-=l,ICP I<A-=1<- 1 [( 4..:i J::l, lCF .;A=.;- 1 lCCf-= ICt:P-+1 ~M(lolCt~)=XP(l)**KA*YF(l)*~JA

C l t- T ll'd .. C C{f\ll~I..E <..(~Tlf\I.JE

HEllJI-<1\ EI\C

~U~~CLTJI';E P~EO(NPF,LoP~tCoCO~A~oPMTePFe~MCC,~AH) 1hl5 5L~RCLTli\E F~ECICT~ TtiE FUNCTICN VftLU~S AT THE G~ID POINTS Af\C CC~F~ltS TI-E F~ECICTION vARIAI\CE CCVARIAI\CE MATRIX.

IMFLICIT Fc~L*E<~-HoO-Z) Cl~E~SIC!'; F~(50,5Q),C(5Q),PF(50),PMCU(50o5C),COVAR(5Q,50)

~ ,v~~(~C.~t),fNl(~C,:C) lRL~=ICft=lCE=IDC=50 CALL ~~LL~(PFelOC,PM,lDAtCtlDOt~PP,L,l) CALL T~~SC(FMT,IOE,PM,lOA,~PP,L) CALL M~LLC(FMCOolOC,PM,IDAoCCVA~,ID~ti\PP,L,L) C-LL ~M~LC(VARtiOC,PMCO.IDAoPMToiD8,NPP,L,NPP)

F[lLR~ Ef\C

~LE~CLTl~E ~FP~CX(L,MoAo~oFtENoCtVoAC,lJ,COVAN,APVF)

····~··~~··~····•************************************************ * T~IS S~~~UUTINE CCES TI-E LEAST SQUARES APPRUXIMATION * ~ CF T~E GlVCI\ fL~CTIC~S A~O RETURNS THE VECTCR OF COEF * * t"lCH.NlS J~GETHLI~ V.I1H THE VArl. CGVAE t-'ATRIX * ··~·~··~~·····•••***************~···~··•**~***********************

IMI-LIC.IT f<E~l>tc(/o-1-,L·-Zl c l ~·F.: " s I c. 1\ f. ( ~ 0 • c:: 0 ) • p ( ~ (.; • 5 (.) ) t EN ( 50 ' 50 ) I A c (50 ) • c ( 5 J ) • f ( 50 ) ' * lLv~~(3Ce5C)oA1~(50,5Q),AT(50,SO),~TP(le50).VF(l),U(50),

* ['.\l(~OltlWd(!:JJ,'vl{1,.3Q),V(50,1J If.<[~:: ICA= lCI.!=lC<.;=~O

CALL lt.~!:lJ(Ill,lCE.A.lDAtt>'oL) CALL ~MLLlJ(ftTPtlUC,Al,lCA,F,ID8oL 1 ~ 1 N) CALL ~MLLO(EI\,lDC.ATP.IOA,A.IDBoL•~•L) r A I t i ' ' I I l \ I • r f' 1'". - A T C) I i""" A . C I I~ U _ I . U . -t \

r-'

.J2(; :i 2 l 3 2Cl .:J2S. .3 :3 1,) 3 31 .332 333 .334 6

!:)

LO ~ l=ltl lO ~ J::l,L l N ( I , J ) = f: I'; ( I , J ) * l • C 0- 2 C CC.I\llf\LE

CALL 1-'li\\D(l:l\,lkC.II,L,DETAtlV.ltl'tl2) ClJ c I=l,L DC t. J::l,L E: N ( I, J ) :: E 1'-. ( I, J ) +1 • 0 0- 2 0 COI\111'-.l.E

335 CALL ,.ML;LO(C.lCC,EI\,IUA,UoiDE,Lololl C CC~PLTE ~[SCLAL5

3.Ju CALL t-Ml.LD{.AC,ICC,.A,l!JA,C,IDC,~,Ltl) 337 (.ALL t-Sl.;EU(\I.ICCoACol!JA,F,ICC,Mol)

L ~C,.PLTf A FCSl~RlC~l VAHIANCE ~ACTCR 33U CALL lf;l\~lJ(\lol•\t,IDAot-1.1) ~3<.J CALL ,.IJLLD('-ll=.l,VT,1,PtiD8,l,M,M) 340 CALL NMLLD(\F,1ovlP,l,\I,IDBoloiJtl) 341 IDF=t--L J42 .AP\F=\IF(l)/ICF

L C(t-PUTE VARIANCE CGV.AklANC~ MATRIX CF CCEFFICIENT 343 CC 10 l:loL 344 CO lC J=loL 34!::1 CC\IJIF([,J)=.tFVF*Et..(I,J) 34t lC CGI\111\UE 347 16 ~El~~l\ 34H Et-.C

34 <J

350 .j~1 3 5 ;_ J <~ ~

3 = '• J"-"' 350 357 35!3 35<; JlO Jul J62 3(J3 3t;4 J(: ~

(.

5 Ul::i f.. L \,; T If\ E C H C. L C ( .A , 1 h C A , N A , C ETA , * ) c c H[ LS( Cf Tl-15 ::iUU<CUl INL IS CPITlLt-.AL L MAlhlX lN\tEh~llf\ ~SING ChCLESKI lJECCMPGSITlUN c L l~l=ll A~~LM[~l~ C. A= .Af,f<AY Clf\lAlNI"G 1=~51;'1\/C DEFINITE S"t'IO~ET~l<.. INPUT MATHIX C lRC.A = ~0~ Cl~E~Sill\ CF AkRAY CONTAINING INP~T MATRIX c NA = ~IZC C~ l~F~l IJATRIX C L~lF~T ~~~U~E"l~ C !JElA = DElE~~~"~NT CF lh~UT MATRIX C A: CCNTAlN: ~~~ERSE CF INPUT ~AThlX (11\P\,;T DESTROYED) L * = Ef<kG~ f<llUf~f\ (T.tKEN IF NA .LT. 1 OR lF DETA .LT. SING) c

DlUEL~ ~FECISllf\ ~.CETAoSUM.SORToOSGRl,A8SoDAUS,SlNG OlMEI\~ILf\ l(l~(~oNAJ SGRT(~Uf\IJ=C~Qkl (Slfll) ~~S(CET.AJ = DAE~(CElA) t:ATA ~lt-<.dlC-!:C/

C L~CLESKl CECUMFlSITlt~ Of INPUT MATRIX INTO TRIANGULAR MATRIX •. :(f\A ell. 1) GG lC 18 GE:TP = ~(1tl) A(l.ll = SGf.T(t(l,l)) IF(NA oEGo 1) GC 10 6 CC 1 1 = 2 ,1\A

A(lvl) = A(!tl) I A(l.l) DC 5 .J = 2ti\A

SL."' = c. j 1 ::: J -DC ~ I< = l,Jl

;; SLM:: ~LM + A(J,l<l ** 2

uC0C7260 00007270

\)0007280 00007290 OCOC7300 uooo1:;1o 00007320 00007330 00007340 oooo73oO 00007~50 00007370 00007380 00007390 J0007400 00007410 00007420 00007430 OOOC7440 00007450 00007460 OC007470 00007480 OCCC7490 0 oc -, 1500 00007510 00007520 00007530 00CC7540 00007550

,.... w

.3(;6 367 3U:l 3t~ 370 371 372 373 .H4 375

37il

377 .3 7U 37<; 3eo 381 382 3 8.3 384 385 38ti 3b 7 3t•tl 389

.3Yil J<,l 392 3~3 .3Y4 395 3<;t.. 3~7 J~d .3<:;9 400 401 402 403 404 405 40(: 407 408

409

3 4 c::

A(J,J}-= ~(.Of:T(A(J,.J) ·- SLM) lJ E 1 fJ -= 0 E 1 A * ( A ( .; , J ) - 5 \., 1•1 ) lf(J oll..lo NJ'I) GC TL; 5 .J2 ::: ,J + 1 lJ(. 4 1 = J2ti'fl

SLM = C. DC ~ K = 1 , ..J 1

Sl~ ~ ~L~ + ~(I oK) * A(.J,~} A(loJ) = (t(l,J)- SU~) / A(J,Jl

CCI\li"LE

6 lf(AO!(CETA) oLlo SlNG) GC TC lf C 11\~EHSICN OF LC~E~ TRIANGULAR MATRIX

CU ? 1 : 1 t I\ A A(l,l) == lo .I A(l,l)

7 CCNlll\uE lF(I\A .EGo l) C:C TC lC

1\ 1 ;:; 1\A - 1 ou s J ::: 1 • "1

Jc = J + 1 DC c; I = .;2,".4

Sl.M ::: Co 11::: I - 1 DC: € K = J , 11

£l SLtJ. =SUM+ A(t,KJ * A(K,J) S 11 ([ , J .t .: - J! ( I, I) * SUN

C CCNSTRUCTIQI\ OF 11\~ERSE OF INPLT ,...ATRlX lG DC 15 J:: t.NA

IF(J .t:a. 1) GC 1C 12 J1 = j - 1 DC 11 1 ·= loJl

11 A<t .. n = J'I< • .IJ 12 DC 14 l = J,,.._,

.': lM ~ C • CC 1:! I<= loNA

1.:! SL,...: Sli,_ i A(K,~) * ll(t<.,J) l<'l A(l,J) = Sl..fl lt <.CI\Tll\l.E

HCILRI\ 1~ ~~lll(f,l7) DEl~ 17 Fl~MAl(lOX. • 5lt-GLL-~ MATR[X 11\ CbGLOo OET =1 oE20.5)

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f..iETl.j-;1\ 1 E::"D

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00007€60 00007670 OC007€BO 00007690

00007700 00007710 00007720 00007730 00007740 00007750 00007760 00007770 00007780 00007790 00007800 00007810 00007820 000078.30 00007840 oooo7eso 00007860 00007870 00007880 OOOC7890 OOOQ7c;Oo OOOC7910 0000'7920 00007930 00007940 00007950 OOOC7ii60 000C7970 OOCC7<;tJO OOOC7c;<.;o

S~U~CLrl"l C~T~C("ofloSJGMf,PHI,~ROoSlGMAFeVFC,NPColi\D~X,V,SL~D,Fo~OOOC7370 &, ,1\Lf- i.At <.,!:Uiw'C.SC2r'~TDP,lw) 000073~0

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11, 5TCP - VEC10k AGAl~ST ~HIC~ ThE AOSOL~TE VALUES GF ~CURIE~ 00007730 tuEFflClt~1S AHE TESTED J0007740

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:+ PHltec,ee> L TlST FO~ f\[(tTIVE EEG~EfS CF F~EEDOM

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ALP H A ( J • K J ::: - S C 1/ ~ ( 2 ( K ) ALPhA(I<.J)=~LFt-I(~,K)

:; lCNlli\UE L DElE~~l~E TtE fCU~lER CCLFFICIE~TS FOR THE GRT~UGONALIZED MATRIX

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7 SC3=SCHF(ll*'f.(U*P ((K)=~C .. USC.C(K)

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14 L:(l l=Dt l)+ALPHJI(I,J)*C(.JJ 1~ LlNlii\LiE

L COIIP~TE TI-E ~AhlAI\CE CF THE FCURIE~ COEFFlClEI\TS AND ThE ~ MAlRlX CF TtE CtEfflCIEI\TS

[.;( :~ 1=1.1'-' Ct 1~ J=1o/o

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OOOC8110 OOOC/3120 OOCC8130 00000140 occcet5C 00008160 oooce11o OOOC8180 OOOC81SIO 00008200 00008210 OCGC8220

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139

II. 2 Least Squares Polynomial Appr;1xima tion o1:: ObsP!._"~Pd Time Series

Figure A-2 is the flow chart describing the program.

The program uses any number of constituent frequencies - ICON

and the required number of coefficients is computed from

INPUTS

lst card: FORMAT Free, contains the following:

M - number of observations

ICON - number of constituent frequencies

ITEST - code for testing Fourier Coefficients

0 - for no test

1 - test against its Standard deviation

2 - test against 2 times its Standard deviation

3 - test against 3 times its standard deviation

ICON cards: FORMAT(lOX, F15.6)

Each card contains one constituent frequency.

ICON cards: FORMAT(5X, Fl0.6, 5X, Fl0.6)

Each card contains the nodal(modulation}factor

and the astronomical arguments required if

harmonic constants are to be computed. If

harmonic constants are not being computed,

these cards should be omitted.

M cards: FORMAT(5X, Fl0.3, 5X, Fl0.3),

each card contains the observed height and

the time of observation.

140

SUBROUTINES:

The Subroutines used are APPROX and ORTHO as in the

prevtous case.

141 ~igureii-2 Polynomial Approximation of Observed Time Series - Flow Chart

L~TA~~--=]-~ COMPUTE REQ. NO. OF cOEFFS. YES iS DF NEGATl VE? .

I F0~1 VANDER1•10NDE' s f!ATRl X

CuEFlNE WEIGHT~ J 1

IS SOLUTIOi~ BY ORTHOGONALIZATlON?

[CALL APPR~

l-

YES

PRINT HARNING

CALL l ORTI10 ~--1

YES C0~1PUTE li 1---------)- HAR~1Ui~IC

CONSTAt!~~

AKE HARt·10N I C I CONSTANTS HEEDED?

\No _ _____j

~- ·------~---'----

PRINT RESULTS ~

...----·--- ------------·----

~~·~·····~·~········1·~~·~········~·•***************************~·~·····························

1 2

j

4

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Cl~E~~{C~ ~H~SE(2Q),HK(20),VK{20J,VKR(20)~XKAPA(20),FK(20) *· , S ll· r-. ~ ( 2 C J , E ( 2 C ) , P b 1 ( J 0 )

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LU 20 1=1 oi(CI\ f..[tc(~oll.2h( !)

I 0 r. r. 1\ 1 ( 11) X , F l 5 • 6 l (. Lad 1 1\ L [ LC ::2 1::1, 1CCI\ , f..[ JIC ( ~, 12 1) F I< ( 1), 'vt< ( 1

FL f·tJ Jll (!:)., r- 1 C • (;, SX, F- 1 u • c) \ K j, ( 1 J = \ K ( I ) * P l I 1 8 C • u.r--111\LE

~EIIU l~ TF-E CESEH'vEU liME AND HEIG~T. 1\CTE THE ORIGIN FUR TIME (AI\ [I; 1AKU4 II!: lU1l~U hUUf< llf fi1EF lkST CAY UF UUSERVAlll]I\--11-'E TltJ[ iS II\ HCUf;S.

cc 21 1~1 ,t.J ~EAC(~oll~Jth(l),l(ll

FC~Mtl(!:>oflCo4t5X,~lQ.~) <..LI\llt\LC

Ef.F!Jf;:.:Q.C5

FRlNl LLT THE lNPLl DATA

FhlNlt'*~* CCI\STITUENT F~ECUENCJES +*****' t;C 22 l=lolCCI\ Pld"T112.w<t l C.L"lli\I.;E Pl'll'\1114 FLi~t-'Jil(//,5.>.,•rt::IGHT G!.!Slf<VEC 1 ,5Xo 1 TI~E OF Ct3So 1 )

CC 2.:: I=l,f' ~~IN1124,Fr(l),T(l) t:C J.;fol /J 1 (!;;)., F 1 C • 4 ,1 0 X oF 1 C •:; j CCI\111\UE

rc~v£~1 F~EO~EI\CIES TL RACIAN NEASLRE CC 2<.. K=l.l((.l'\ ~(~)~~(K)*Fl,lcCe C.CI\llf\UE

t'lJf:({IIPlH.I'\ (.f v.ONC£1-!0ND MAHHX

co <:4 1=1·N CU .C~ .J=lol P( l,J)::C.CCO CGl\111\l.E DC ;:5 1=1 ,tJ, .J=l f(J,J)=l.CDO K.:: C Cu £7 .,:2,Lt2 I<.:: I< • 1 1- ( 1 , J }·.: t:C ( 5 (II ( K I '!< T ( I ) J

CCI\T lt\LE I<= c CU <:H .,:,!,Lo2 l<=l<il I( loJJ:CSli\OdKH<l(Il J

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FH 11\T, '*** VA"O[~CNC MATHIX **~'

CALL ~CLll(A.lCA,~,L)

lF(llJ.EG.l)CC TC 35

tOf.ti/-T ICN Cf V.E l<:t-T MAT~ l.X

co ;;o I=l.tv cc ::o ..i=l.l> ~(l,Jl=O.CCO CCI'llt\LE cc ;;} 1=1,/Y J= I

F ( I , ..; l = 1 • / [ I· F 0 r; * * 2 CC/\111\LE

FEf.~CR~ T~E LEAST SO~ARES APPRCX. EY CALLING THE APFI<CX

CALL ~PF~UX(L,/YeAoFtFH,ENtCoVoAC,U,CCVAR,APVFJ H'lt-Tll6 ru~~A1(//,1C>,'VECTCR Cf- CCEFFlCIEI\lS') CALL I>ICUTC IC, iCA,L, 1 J Ff;I"Tlll F-Cf.f. AT (//,10)' 'RES llJLAL I' lOX.' \I[C TLR.) CALL /YCulC (VdCA,Mol ). ~ R l f\ T , 't~ u /Y E E F 0 F 0 E G I< E E S L F F ~ E E: 0 G ,., -= ' , 1 0 t­FRI"TllU,.OFVF

SUUFWUT INE

FOH1 Al(//.lCX, 1 A PCSTEf<lGf--<1 VAfdAI\CE: FACTCii =',flOov) FFit-1119 fG~~Al(//,lOXt 1 VA~IAf\CE CLVARl~NCE MATRIX GF TrE COEFFo 1 )

CALL ,_.(L;l[(CG\IIf<.ICA,L,L~ lF ( ICoE:Co2)GLJ TCJ 40 <.LI\111\l..E CU JG 1=1 .~

~l( l )::lo/CF-F(f;>H2 tLI\Tlf\oLE

(ALL lRlt-C(,..,L,~PW,.AoiCAtSIG~tVFC,hPC,ITEST,v,COVAR,FH, .T,(,ALPt-AofC,5U~FC,SC2,5TDFol~l Fldl\112~ CC :;7 l=l,L PH INT 12t ,FC ( 1),5LMFC( 1) CCf\Tli\Ut. FHih~o 1 FlURIER CGEFFo -A PGSTEC~l ~.f.= ',SIGN Fhii\To 1 ~C LF CF-IGiNAL CCEFF. AFTER TEST= • ,NPC F r; I 1\ T 1 2 7

ClJ 2U l=l,L Fldf\TlcE.C(IJ CCI\1 11\LE

f=f-11\112<; C ilL L ~ G l 11)( C C V J lq 1 C A , L , L l Ff.INTlJO

lJO ::s (::1 •"' FRI"TlC:E,'dl) <..C"lli\LE

F~lhl,' .A ~CSTEC~I V.F. CHlGlNAL FCLYNO~lAL =' ,VFC

,_. ~

1 04 105

lOt 1C7 tee 109 1 1 0 1 1 1 1 12 1 1 J 114 115 1 1 (J

117 11 d ll':J 120 121 1.22 123 124 125 126 127 12tJ 12':J 1 3u 131

1~2 13.3 134 1"<"' --13(: 137 l:Jll 1.:1Y 140 141 142 14.3 144 145 146 147 1 4tl 14~ 150 1!:1 152 15.3 154 1 c. c:

v-

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PRE.:DlCllC.I\ CF 1-t::lGttTS I..Sli'IG THE PlJLYI\U"'l~L FLR SOME TIMi..: 11\TE:~VALS

111'lTE(E,1..32) 1'tf~1Tl(t,lJ2)

CC 4~ l=lo':C TP=l Flll(l):::l.O t.:C 4E t<.::1.tCt"' J(=2*!<: .J!;=2>tl<t-1 t-:hl(.J():CCCS(I\(1< l*lP) Flll{J~)=C::>ll\ (V. (K l*TP) CLI\ll"l.JE Sl.t<=C ~ l.. IV 'v A= ) CL 47 IF=lol Fh:((JP)*PI-l(IF) PV=Prl( lFl*.>t~li<CC.vAR(IP.lFl ~ufo=!:L.IV H·H ~l..fW'v/l=SL~V/l-+FV CCI\T 11\UE ~TO=C~Cf.l(Sl.t-VJI) ~<ol:(l1E(6,131)1Po5l..M,STD CCf\111\UE CC 4E: 1=1,L 'A F< IT 1: ( 7 , 4 2 )( (l } • CU V AFd l .I ) CCI\111\l.E FC ~IV .111 (!: >, C: l le 1, 5X, E 11 • 4, Fuf:MAT ( '1' .l!:X,.'t--CU~LY FI-<LCICT 101\S IN ThE: TIME 11\iiERVAL' l rORMAl(l.lo~.>c,•·tlPIL IN HCl..HS 1 o5Xo 1 Ffi. t-EIGHTS 1 ,5Xo 1 STD. EfiJ.i. OF

• r r: • I· L I ll- T :;.: • l f-u J... M /ll ( I , B ~ , f (; • 2 t <;X of 6 •::! o 1 2 X t F 1 0 • 5 ) Fl..i ~ M II 1 ( 1 / t 5 >< • 1 f U l.l< H. R C 0 1: F F 5 o 1 · t1 0 X • 1 \1 11 f' l AN C E S 1 l fL~~Jil(/,5X,Ell.4olOX,E1lo4, fC~MAl(//tt~.•vEClC~ CF C~lGlN~L CCEFFICl~~lS') t-Cf.MJl(.lo5X,El1.4) . FL~fWJll(//,~)1,•\/Af.OlANCE CCV.Af<lAI\CE ft',ATRl.>< CF CRlGli'<AL CUlf'f-•') FC~MJ1l(.l.lo5Xt 1 VEC1GR OF ~E:SlOUAL5') IF(hrC.ECeC)~Q TC 41

CG :!3 1.:2,Lt2 J=I-+1 IA-=1/<: FH/I~~(lJ)=CilTA~2(C(J),C(lJ) l(l/11=1•/(FI<.( lA)-4<DC.CS(PHASE(lA))) t K ( l A ) . .:: ( ( I ) H3 ( l A ) \Kf.(lAJ:2.+Fl-V~R(lA) ~Kilf.A(lill=VKR(IA)-P~ASE(lA)

1.: P ( 1 , 1 ) = ( 1 .I ( 1 • + ( C { .J l/ C ( I ) ) .; ~ 2 ) ) >t ( - 1 * C ( J ) /C. ( [ ) * * t: ) t.!l= ( 1, 2) = ( 1 .; ( 1 • + (C. ( .J) /C ( l) J * :t2) l * 1 .IC (I) ::: 1 OJ. H ( I. /J. J::: t: P ( 1 , 1 ) * * 2 *CCV A R ( 1, 1 ) -+ BP ( 1 , 2 ) * * 2 *CU V Af< ( .J • J) t:A(1 dl=ltlfl<( lAJ*DCCS(FI-IASE(IA)) E A I 1 t ~: l :: (( 1 H D S 1 f\ ( PH I~ 5 E:( I A ) ) / ( F K ( I A ) *DC 0 S ( PHASE ( l A ) ) * * 2 ) ~ l l 1'1 /l t ( l t } = E J. ( 1 , 1 J + * 2 *C. t: VA I' ( l , 1 ) + E A ( 1 , 2 ) * * 2 + 5 I G M AF ( I A J

t..LI\111\LE I-I; l I' T 1 1: <:

r c ~~ M. .ll < 1 / , = )I • • c c "s 1 1 l u E" T • , 5 x , • A~ F L 1 1 u o E • • 5 x , • s 1 G I'A A AMP L • • , 1 ox , * 1 PI-I\!:L LtG 1 .~X, 1 S1Gft'A Pt-ASE') lH.: 24 1<=1 t [(C~

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CCI\T 11\Uf: C.U·T lf\.UE

~TCF ENC

Suf~CLTli\L ~FP~GX!L.~,A,F,FolN,CoVoACoUoCOVARoAPVFl

~~~*~~~~~*~~~~~4~*4***~~····~************************************ * lHlS 5LU~OL1[N[ CCES Tt-E LEAST SCUARES APF~UXIMATlGN * >+ U- Tt·C G lVU; FLI\CT lUI\S AI\LJ RETUR/\S THE VECTOH Uf COEf * * f- 1 C I l N 1 S T L C E 1 t- f: f< 11\ 1 l H T t- E V ~ 1-< • (. C VA E MAT f~ (X * ~*~·~··~~********~************************************************

li\-\FL l<. IT REPL*ci.0-1-,L-Z) C I l'l 1\ ~ I C 1\ ~ ( t iJ , (, 0) , P ( 6 0, t. 0 ) , EN ( t C , (; 0) , A C ( 6 0) , (.. ( 6 0 ) t F ( 6 0) , * CwvP~(6CotC),AlP(c0,60),Af{6CotO),~TP(l,60J,VF(lJ,U(60), * Iwl(tC) .IW.C:(t:O),"l(lo6C),V(6Col) lFiC/1= lC/J= lCI::=ICC-=60 CALL 1~/\EU(PltlCEoAtiOA,~tL) C 1\ L L t< i>1 I.. l 0 ( ~ l P , l C C • A T , I C A , P , 1 0 8 , L , M , tv ) tALL fvML.LO(EI\tlLCo/ITF.ICA,AoiDEol•"'•L) CALL t-'Ml.LlJ (L. t ll.C ,A1P .IDA ,F t IDBtLoMol) CALL tvii\"()(EI\,IhO.A,L,OtTA,ll'lltl'1.2) CALL t-11-IL:LO(CtlOC,E..I\tlDAtUolDEtLol.l)

~ CL~~LTE ~ESCLALS CALL />IM\..LO (AC,lOC,~.IOAoC,ICC,~,L,l) (tiLL t<'SUEIJ (V, ICC oACo lOA.F, lDCtMo 1)

C CCI'FLTE P ~lSTERlC~l v.A~I/Jf\.CE FACTC~ (~LL J"~f,~UI\ItloVelUAtf'iol) CALL r>MLLD(IJlPtleVTtlePoiDE!tloM,~) CALL ~~LLLJ(vF, loVTPoloVelD8oltfVol) ICf=fw-L .APvF=VF ( 1) /I CF

c Cl~P~Tl VARl.ONCE CCVARIAI\CE MATRIX CF CO~FFIClENT

t.

(_

CC lC 1-=l,L CC lC J=loL CU'oftf. (I oJ) =~FVf *EN ( J ,J)

1-: CCI'.TC1\l..E I ( f; L 1 l. r; 1\

CI\C

S LU ~f.. l f 1 f'. [ CRT 1- C ( f'. , t<, 5 l G M A ~ F H I , f.' R C , S I G MAF- o V F C , NPC , 1 f\. uE X, V, !J UM () t f , W 0 G 0 C 7 3 7 0 ooC,tL.f.HAo (,!:UfvC,SC?,SlOF, [V.J 000073F30

L Tt-lS !:UU<lLllNt Cfdi-G(Cf\AlllES THE t.'ATfdX PHI USING T~E GRAM-SCHMIDT <.)0007390 C Mfli-CC, <:u.q=LlE~. TI-C: fUURlE:k CCEFFlClEt\TS lJF THE OHHOGGNALllt:.U MAThlXOH.l07400 L UL~I~lS TI-l L~~FF-IClEf\TS C~ ~t-loCCMFUlES TI-E VARIANCES CF THE FUUHlER OUOC741G C CC[ff ICllf'.l~ ANC 11-E vARIANCL-CCV/I~l~NCE MATRIX OF THE CCEfFICIENTS 00007420 L li\FLlS : vCOC7430 L 1. ~~il(tt11ILI\AL- (CULL: P.E f\..1\CTICN SlJ8FRCCf<AM lNSTEAC)- ANN BY M OCCC7440 c C.UI\T,-11/\lf'.C: TH E.:ASE fUNCTlCi-iS EVALliATED FGR EACH Ct:l~ERVATIOI'. 00007450 t. ;;. N- H-E I'<Lt<UE:~ cr U..:Sf:~VftTILI\S JCOC74b0

I-'

'"" Gl

192

lc,i:J

1 <;14 195 1',{:

1 <;. 7 1 y d lS~ 2(..0 201

·202 203 2 0'4 ~05 206 207 2013 <2 0 Si '1 0 ' 1 1 212 21..:! 1:14 215

2lt 21/ .• , 1 .. ..~

c c <.:. c c c

(.

c L L c c

..: . 4 • ,-:J •

c. 7 • t: •

<; •

~ - l~l ~~~DE~ UF EASE fU~CTIC~S (EQ~AL C~ G~~ATEH THAN ~ - ft V~CTCR CF LE~GTh ~ CC~TAINl~G l~L COMPGTED ~EIGHT F - FU~CTlCNAL VALLES

2) JOOC7470 FU~CTIONSOCCC7480

00007490 OCOC7~00 000075\0 00007!:20

5lG~A- T~E A F~lC~l VARIA~CE fAtTC~ ~RC - l~E ~AXl~UM RU~ Dl~E~~lC~ GF FHl MRC - l~E ~AXI~UM CULU~N DIME~SICN Of PHI

INCE)t ~(~Mil~ C~TlCNAL lE~r FC~ STATISTICAL SIGNl~ICANCE OG007530 U~ FLL~1ER LUEFFIClENTSa••• 00007540 lF Q,51A115TICAL TEST fOR fCuRltR CC(FFlClENTS A~ANCUNEO OCOC7~50 IF l,TES1S AGAINST UNl TIME ITS STANUA~O OEVIATICN 00007560 IF 2,TES1S AGAINST TWICE ITS ST.DEVIATIUN OCOC7570 lf J,TES15 AGAINST TH~EE Tl~ES 115 ST.OEVlATlL~ OOOC7EUO

l~PLlCIT R~tL4E(--b,O-ZI 00007750 C 10. l~ - ~~liE CCCE CF T~E CC~PUTE~ 00007590 C OLTFLTS : 00007600 L 1. ALHi.C- AI\ MHC EV tJ MATI~[X COI\lAII'<ING Tt·E ALPHA'S U::it;D IN COMPLT100007f10 C T~E CHl~OCC~ALllEO MAl~iX AND IN CC~PUT[hG THE CUtFFIClEhTS OF PHOOOC7f20 c 2. (- l~E ~ FOU~IE~ COEFFICiENTS OF THE L~l~OGUNALIZED MATRIX OCOC7E30 c ~. C- l~E M CUEFFICIENIS OF THE IN~ul ~ftl~IX Phl 00007640 C 4. SU~C- 1~E VARIAI\CES Of THE FOURIER COEFFICIENTS 00007650 C ~. SUMu- THE V.C~IANCE-COVARIANCE ~AT~IX CF THE COEFFIClENrS OOOC7t60 C to SC2- l~E 50Lt~E5 CF Th~ hUR~S OF THE CRlHOGCNALlZEG MATRIX 00007670 C 7. SlGMAF - T~E FCU~lER PCLYhOMIAL A POSTECRI VARIANCE FACTOR OCOC7680 C t• V- l~E N ~ESlC~ALS 00007€90 c L c (.

<;. \iFC - H-E lHlG 11\AL f-GLYNC~lAL A POSTEORI VARIANCE FACTOR 00007700 lOa hPC - ~LMEEH CF l~E CUEFFICIE~TS CF THE ORIGINAL ~CLYNOMlAL 00007710

AFTE~ TtE STATISTICAL TEST IS PERFORMED 00007720 11. STUP - ~CClU~ ~G.C[~ST ~H[Ch T~E AESLLvTE VALUCS OF FUURltR 00007730

c CUEFFIClE~lS ARE TESTED 00007740 Dl~E~Sl(~ AlfHt(eOo60),.(60),F(6Q),C(60),Q(cO),SuMC(60lt

* S U fJ C ( t. 0 , t 0 ) , !: C ~ ( C. 0 ) , V ( t C > , S T DP ( 6 C ) t F 1-i 1 ( 6 0 • 6 0 ) C TESl FUh ~E~ftTl\E CEG~EES OF FREEDOM

It' (1'\aLloM) GC 10 lCO 1<=1 ~LPhA(~ ,l").:l.t;C

C UEJE~~~~E T~E ALPHA'S FLR COMPLTATlON OF CRThCGC~ALllEO MATRIX 10 CC 3 .;:t<.,M

lF(J.I\E.I<) CO lC f ALPhA {I< , I< ) : 1 • C 0 GO 10 ~

t S(l=C .CC SC2(K):::C.DC sc3=o.oc cc ;:: t~l.~ P=P hI ( I ,I<.) lf(KaEG.lJ C:U TC 4 1<1=1<-1 CO :, Jl=l,Kl

L F=Piftl~~~(Jl,K)~F~l(l,Jl) 4 SCl=SC.l+II(IJ1<PI·d{loJJ*.,.:

S C J., :.; C 3 t f ( 1 J *Ill ( I H P £ S C 2 ( t< ) = ~ C 2 ( I< ) + ~ ( I ) * F * * ~:

~LPI- A( .. oi<J=-SC l/SC2(K) ALP H ft ( K , ...1 ) = _.l F t- J. ( .; • I< )

, CCN'T 11\\...E C UllE~~~~L l~l flU~l(~ CCEFflC(E~TS FO~ lHE C~ThUGG~ALlZED MATRIX

<- (K J=~C.?/S(;:( I<) K =K + 1 IFtlll_r-i_.;;l (:[J l( .24

OOCC7800 00007810 00007820 OOOC78JU 0000784 0 00007850 OCOC7860 OCOC787C 00007880 00007890 OOOC7SOO 00007910 OCOC7<.i20 00007940 000 C7~5 0 00007960 00007970 OCCC7<;8Q 00007990 000080•.)0 1)0008010 00008020 a:ccecJc OOOCB04U OCOC8050 OOOOtlOuO OOOC8070

_o o o c e Otic

f-1 ~ ~J

C:. 1 '.1

<' 2 0 .: 2 1 2 ~-- 2 L ;: .J i L 1~

~25 Z2o 221 c c:e

2.2'7 z..;o c. .: 1 <;..;2 ·t::...--2..14 ") -~ r::; ,_ __ 23(.; 231 cJe 2::.'} 2ql)

2 '• l 242 £:: '• J 244 24S ~- 4 t.: 247 24B

24'-1 2SO 251 2t2 253 25 4 z:.:t~ 256 2!;7 2.5t 2~9 2 {,t)

2tl 2b?. 2(;3 '6 ,. 2t5 2Ct 2<J7 26 f: 2t<; 270 2 '1 .-.. -,. .·_;

IHI<.LT.;:) CU lC 10 C DElE~~~~~ r~E ALP~''S L~lC (N CGMPUTI~G lhE CGEFflCIENTS UF PHI

.JI<.=K- 1

000Cd090 JOOCelOO OOOC8110 CCCC8120 OOOC8130 OOOC81'~0 OCOC8150

s ~l=l< J~·=JK-1 " ,J::: 1-- - J K - l CC c Ll'<~l,JJ

JL=JL-1 t f. l P 1- A ( J I< , K l= A l F r A ( J K , I< ) +A L F h A ( J Y , J L ) *A Lf-' f Jl ( K , J L )

lF(Jt<.f\Eol) Gl 10 S I f' ( I< • l T • t• l C 0 1 C l 0

C UElEh~l~E Tfl L"Sl FCL~l[~ COEFF!C!E~T 34 SC2CKl:=o.uc

~CJ=G.CC c c 7 1 = 1 • ~~ ~::::PI-l(l,kl

K 1= K- I CC 1 .J:l,Kl F::P+Al~t-.JI{.;,KHFhl.( I,J) ~ C2 ( K ) ::: ~ l 2 ( I< ) + ~ ( 1 ) * P * * 2

i ~C3=.SC3-H(l)*l\(l)*F C(K):::~C::tSC.C(I<)

C UtlE~~l~£ Tf't CCEFFICIE~TS UF PHI lCEI<T=l lCOl~1:::C

lOCC CCNTll\u[ cc 13 1=1.~ [([):(({)

IF(l.E:O.tv.l GU lC 13 l I= I-+ 1 cc 14 J=lltt'

14 CCI }:C( ll+ALPhP(l oJl*C(J) 12 CC.Nlll'lJE

<.. Clt~F~ll T~E VA~IA~CE CF T~E FGU~IER COEFFICIENTS AND THE C M~l~lY Lf J~l lllffJClEN~S

UlJ l!;; I-=1•" CG 1~ J=I,r.

1 ~ ;; lJ~·1 C ( I , J ) .: C • D 0 ~C4=C .DC cc 22 1=1,1\ FI'=C.CO cc cl .;=1·"

c 1 F r-= F 1'0 t C ( J ) * F 11 1 ( 1 , J l . \1(1 l=f( lJ-FI\ v2=v<Il~*2

~2 ~t4=SC4-+\2*~< ll SlGMAf=5C4/l~-,.~1CCl~Tl*SlG~A v f <.. = ·; 1 G 1-1 /l F IF( lt;f:Kl.l:(,o2) vFC-=~C4/{N-I\I=C)*!:lGMP 1 F ( 1 i'~ C E X • ~ G • 0 l I' P C .: N Cl cU l=l,N SUM({ l)=~I(NAf-ISC<:( 1) lF( IDEKToEColl GC TC 28 lF( C ( 1) .,u~ .CDC) Sl~C( I )::O!JC

2E CCNTli\L[ cc <:3 I=I,/v' DC 2..3 ,=1,1 DC 22 ·,::J,I

~~ SUM0(0,K)=SLMD(-,KltALPHACJol )*PLPHA(K,ll*SUMC{l)

000C8160 OCCCel70 OOOC8180 OOOC8190 DCCCc200 00008210 CCCC8220 OOOC8230 OOOC8250 JCOC8260 0000!:3270 OCGC!:J2tjQ 00006290 occcs:::oo 00008310 IJOOC8320 OOOC8330 OOOC8340 OCCC8350 oooca:;6o OOGC8J70 OOOC8380 00008390 00CCe40C OCOCS410 OCCC€420

VAniANCE-COVAOOOC8430 Q000t:l440 OCOC8450 000013460 OCOC€470 OOOOtJ4oO OO<JC8490 OOCC8510 00008520 OCGG8530 00008540 JC008550 JCOC8560 00008570 J0008580 OOOC8590 OCCCE60C 00008610

00)C8630 00JC8640 OOOC8650 oooost;uo ococet:·tc 00008680 00008690

'--' ,_. ('f.)

27J 274 ~7!: 276 277 27tJ

~7<:;

2tJO 201 2b2 2t13 2t14 2e~ e:.ee: 2b 7 2e8 2t1Y 2<;0 2<;1 .1::.'12 2s;3 294 2<.i5 2iii(: 2.~1 298 2~9 300

cc ~4 1=1·" 1 r= I+ 1 lf(li.Glo/11) Gl 1G ::C CL G4 J=lTw/11 .

2 4 !; 1.-M C ( .; , l ) = ~ L M L; ( 1 , J ) 30 CCNllt-L.E

L UPlll~AL C~ElK FU~ STATISTICALLY SlGNlflCA~T FOURIER COEfFICIENTS 1 F ( l r-. C E )< • E C • 0 ) C U l C 4 C

32

31

..3~

40 lOC lC~

$1,\J

If( lCEI<1oE:Co2l (C TL 40 PlNCE)=CtLC~T(l~CEX) cc ::1 l=ltf.l ~ TD 1-l ( I ) =f. II\ C E .l< * C S G t' T ( S U 1-1 C ( 1 ) ) lF(C.AL;S(((ll).LleSTDP{1)) GC TC 32 <;C TG .::1 ((lJ=COC lCGL.I\1= lCOI.t--1 _.1 ~l.MC( I) =COC CCNTli\UE t-~C-=0 DC .33 1=1,,_. Jf(((l).I\E:oCDOl NPC=l CCN T 1 1\LE 1CEKT=2 GC lG lCCO f.CETI..I-11\ ~f;l TE (l~ ol 00: J F-ORMJ.IT(IO'•'*E:r<f:CH* 1\EGATI'YE CEGREES OF FREECCM 1 )

f'ETUI'I\ E t-D

*** ~CNSTITLEI\T F~ECL~NCIES ~·~~+* 2Ee'lt!4104 3C.CCCCCC 1J.s4::o::L 15oC4l~tS 14.S5C!S31 3CoC8Ll37 .C€.43S7:!C

~t.: [ljHl t.H.: ~ E I' \IE 0 7.4JCC 1.30CC '· ..30 cc 1.2()(;(.

lo40CC le20CC 7.:JOCC 1 • 20 OG 7.(;JCC 1 • 20 c; 0 '· 30 c c lelUCC I. 70 C C l.GUCC 7.40CO 1-1tiCC

Tl~t: cF ces. 2 .s:: 3

10e2!:C lfo2EC ~t:.J~O a .t::c7 ~5.00C 41e2!:C 47.25C ;~.sec :'i.S17 66.083 H o417 7e.5c:! 84oe::3 sl.cc:: <;;"lo417

OOCC8700 0000b710 OCOC8720 ·JOO 08730

.OCOC8740 00008760 OOCC8750 JOOOS770

JOOC87d0 OCOC8790 00008600 acoceeto 000Ct3820 OCOC8830 JCOC8840 00008850 OOOC8860 OOOC8870 \JOOC€880 OOOC8890 00008900 OOOC8910 00008920 oocc9g.3o 00008940 00008950 OOOC€960 00008<;70 ococec.oao

1-' ,!>. '.

150

11.3 Tidal Reductions

Figure A-3 is the flow chart describing the tidal

reduction computations.

The program uses as input the following:

the results of computations 1 and 2, that is;

the coefficients of the approximating poly­

nomials Cr, CA and CB and their associated

standard deviations.

- the observed data at each sounding namely:

the depth sounded (D), time of sounding (t)

and the geodetic coordinates(~. A) or the

local Cartesian coordinates (x, y). The

observed data at each sounding are punched

in one card and read into the computer one

card at a time.

The SUBROUTINE. PREDICT used in this program is different

from the SUBROUTINE PRED. The subroutine predict uses

prediction vector and predicts for one point at a time

while the Subroutine Pred uses prediction matrix and

predicts for all the points at once.

lGl

Figure II-3 Tidal Reduct10ns - F"low Chart

READ DATA

READ OBS. DATA (one set at a time)

CALL CARTE

(converts <P, A to x, y)

COf.1PUTE PREDICl)ON

VECTOR ( PHl}

CALL PREDICT (Predicts range ratios, time lags and heights) r

.•

C0~1PUTE HEIGHT OF TIDE AT SHIP

HWCE CDr·1PUTE REDUCED DEPTH

PRINT RESULTS

NO

*~~••***************~******************************************************************************FAtll ~JUB lKEN~AIG.ll~E=8tPAGE5=10

1 .., c_

3 4 ~ (;

7 8 y

iO 1 1 1&.: 1 3 }q

1~ lb 1 7 1tl l'J 20 21 22 23

c ~i@@~~~ii~iii@ia~ii~~~~~i~~~i~iiiiii~~~~~iii~ii~ii~i~~~~@W~&i~@~£@@ L L c L L (. (

c c c c L L c (.

(. (.

(.

c c L <.: L L c ( (.

c <... c (.

c

(,

7

8

<,; 1 0

f:f1CGf..A"' TC llwPLif.IEI\T T IUAL RlUUCTIGt-.5 USli\G ANALYTICAL CCllUAL CHART IN f1EAL Tlt>'E.

I'< 0 T A T I C 1\ S 1\=CECFC~ Uf FCLY. FUR RAI\GE ~ATIO L=I>Uf.I.UF CUEff-S, IN THE PULY. LT=I\U~.Cf CU~FFS.IN THl PGLY.OF THE TIME SERIES AT THE R~F. STATIC~\ 1\CLN= NUf.l tF CCNST(TUENT FREG USEO I~ REAL TIME APPRUX AT REF. STN. lTYPE= l~E TYPE UF INFU~MATICN

1- FOf1 LAT, LGNG.GIVEN ~- FOR X,Y CCC~~S.GIVEN

ISFLll =CODE INDICATINC WET~ER T~E TIME LAG IS SPLIT INTO fU,.,CliCI\ A AI\D 13

1 f'UR 1\U SPLIT C:- FO~ ~PL[T

CT= FG[Yo CUEFFSoFUR HEAL TIME AT lH~ REF STNo C~=PCLY. CCEfFS.FG~ HANGE RATIO VACToVAC~= ASSOCIAlED VARIA"'CES FOR Cl,CR RESPECTIVELY CA.CE=CCEFfSoUF PULY. FOR TIME LAG

~AC~oVACB=~SSCCIATED VARIANCES FOR CA C~ RESPECTIVELY CECLAL,XLA~O SECLAU = LATITUDE OF REF, STN. IN DlGREEoMIN,

A"'O ~ECCNDS DEGLCC,XLUMO,SELLOU = LOI\GITUDE CF R£F. STN.IN CE~RECS t>'IN.

/lNC SECOI\DS ToXMll\ = OESERVEC TIME II\ HOURS AND MlNSo CEPT~L = ObSERVED DEPTH IN METRES D~GLA,XLAMl"oSECLA = OBSERVED LATITUDE lN DEGREES,MINS,AND SECS. DE~LLtXLCMll\tSECLU = UBSERVED LU~GITUDE IN DEGREESof.IINS.AND SECS.

i~i~~~~~aii~ii•~~@~~i~~~~~~~~~i~~~&~i~~@~~@~~~@~~~i~~@~~~~ii@~iE~

IMPLICIT REAL*8(A-H,O-Z) C I f<i E "5 lll\ C T ( J 0 ) , C R ( 3 0 l • PH I T ( 3 0 ) t PH I R ( 3 0) , CA ( 3 0 ) , C l3 ( 3 0) • * VACT(JO},VAC~(3Q),VACA(3Q),VAC8{30)oH(300),STEC300),PHI(300)o

~ ALlNG(JCC),~(lOI,TL(30C) f.:l=3.1~1!:92t!: CK=0·5*Pl/18Co f'E~D(!:o4C)L,LT,I\,t..;CON,ITYPE,ISPL1T DO 1 K = 1 , NCC 1\ J:;L:.J~C (5, so >w< I<) ~(KJ=~(IC)*Fl/180• <..CI\111\UE DC 6 I=l,LT f.< E. Jl D ( ~, 4 5) C T ( 1 ) , vA C T ( 1 ) C(.,._T 11\UE DC 7 1 = 1 ,L J=; E. ..0 0 ( ~, 4 5 ) (f<( I ) , VA CR ( I ) CGI\llHJE CU 8 l=l,L ~[; /J D ( 5 , 4 5} C I> ( l ) , VA C A ( 1 ) CCI\Tli\UE lF(lSFLlToEGol)GO TO 10 CC <) l=loL .. t: ~ 0 ( !: , 4 5) C c ( 1 ) , VA C 8 ( l ) CCI\Tl"\JE CCI\Tif\.Uf.:

1-' :J"I '.

24 25 26 27 28 29 .30 Jl .32 33 .34 J5 36 J7 JO J<J 40 41

42 4.3 44 45 46 47 41'j 49 50 51 52 5-:! 51+ 55 56 5"7 58 59 6u 61 62 6J 64 ~5 66 67

68 (19 70 71 72 73 74 75 76 77

c (.

c

RE~D(~,44)DECLAO,XLA~U,SECLAQ,OEGLCO,XLOMU,SECLQu CALL CEG~AD(CEGLAU,XLA~C,SECLACoPHIO) CALL L~G~AC(CCGLCO,XLO~O.SEtLUO,ALON) \\~ITE(6o!:7) 1'of'ITE(Go!:61 I<UL,...l=O

2 CO" t 11\ U E

~ 4

1 1

l 2

lF(ll~P(.LC.l)TrEN DO REID(~,5~JT,>MINtDEPTHC,OEGLAtXLAMIN,SECLA,DEGLO,XLOMIN,SECLG lf"(T.LToOoO)C::O TIJ 39 KGLI\l=KCLNT+1 CALL CEGRAO(CEGLA,XLAMIN,SECLA.XLAT) CALL ~EG~AO(CEGLG,XLGMli\,SECLO,XLGNG) CALL CARlE(XLAltXLC~G,PHIQ,ALC"oX,~J ELSE DC ~EAD(~,(C)T,XMlN,OEPTHC,XeY

IF(T.LT.C .. O)GU TO 39 END IF

CCMFUlE T~E VECTC~ PHI FOR PREDICTION

IOF=I\+1 l= c Ou41<=1tlDP KA=K-1 00 ~ .;: 1 ,[ DP JA=J-1 1=1+1 FHlR(l)=X*~I<A*'**JA CGI\Tlf\UE CC"Tli\U£: CALL FW~ICTCLoPHIR,CR,VACR,RtSTOR) CALL FHDlCT(L,PHIRoCA,VACAtAtSTDAJ lF(l~PLIT.EOol)GO TO 11 (ALL FHDICT(L,PHIR,Ce,VACO,B,STDE) lC=DATAI\(8/A)/CK ~ 1 = ( 1 e/ ( 1 • + ( E/ A ) Jt * 2 ) ) * ( -fY A:+ :+ 2 ) B 2.: ( 1 e/ ( l• + ( E I A J * * 2 ) ) * ( 1 ./A) VA1C-=B1~:+2:t.STDA+B2**2*STD8 IF(l~PLI1.EC.2JGC 10 12 TC=A VAH-=~TCA CCNIINl.E

TL (KCl.N T ): TC TCr=lC/tC. T 0 t-1= 1 + X M 1 N I c C • TAl"= TU1-TCh

c . C Ct~P~TE rHl FC .. THE PREDICTION OF HEIGHT AT THE REF. STAtiON c

3

FHIT(l)=l.O I<= c 00 3 l=2.Llt2 lA=l+l 1<=1<+1 PHlT(l)=CCGS(~(KJ*TAR) PHJT(IA>=OSl~(~(K)*TAR) CC.:"Tli\LiC CALL FFH.:lCT(LT,PHITtCTeVACTtHTGtSTDH) t•T II)L=IITC.4<1'

1-' ·-·t

w

7<3 19 h0 Ul b2 8:3 U4 Ub 86 IJ7 t:IJ 8Y 90 '.11 92 93 <)4 95 '>6 ~~ <}f:j

~9 100 1 0 1 10?. 1 0 J 1 () ll

1 0 ~

1 06 1 ') 7

(

1 011 (

c c L (.

lOY 1 1 0 l 1 l 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7 1 18 l 1 <; 120

1 2 1 (.. (. (.

(..

L

3~

1 J 4~ 4 1

40 :>0 4!; 55 r:, 0 4 ")

':-7 r. r)

(.I

~TDE\=O~GRT(~**2*STDH~HTC**2*STD~) UEFTr=DtFTHC-HTICE STCEf-<=u!SGHl ( o0l**2+STDEV**2l 1- (I< m;l\ r )=HT ICE

STE(KCL,..,T)::::STCEV PH 1 ( I<CUI\1) =Xli\T ALCI\G{KCLNTl=XLCI\G XLAT=XLAl*lECo/PI XLCI\G=XLlNC*l80o/PI 1\R IT E ( 6 , t: 1 ) I< C U 1\ T , X LA T , XL CN G, C EP T HU, TO H • TAR, tHO, P, HT l DC, S T DE. V Gi..l TC 2 CC t- T 11'-UE "'0BS=KCLt-T ~~1TE(6,42} DO 1:: KlLNT=loi\CES 'ftf; lTE ((; o41 IKCUI\T.TL(Kl)lJI\T) <..Ct-Tll\l.E f-Uf;MAl( 'I' ,ex, 'PRLOICTCD Tlf./E LAGS') F G Fo 1'1 AT ( I , 3 X , I 3 , 5 X , F 6 • 2 l FQf;MAl(e>.C:l:~) fC:FoMJ!l (SX, flO.tJ FOf;IYAl( e.)C,;::E11o4) FUFMATCex,sFc;.;::) FORMAT(5X,3F6.2,?F!?.G) FOPMAT(5X,6F6o?) F oF""' AT < • 1 • , / /, c; r , • T T ,.., "1 • • "x , • R r:: n uc T r r"' c; • 1 rnr.•1AT( //,~X,'NU'_.' ,4X, 'l.AflTlJf)l"'t ,()X, 'LONGITIJf)F"' ,4X, 'ORS, nFrTH'

'~olX,'TIMr: AT SHIP 1 ,?Y,'TT••;: <\T PF:F 1 ,?Xo'TJ!'lF AT PI=F 1 ,?X,''PP, R<\TlO *',;:>X,'TTDF. AT SHtn•,?X,'ST 1)f'V'l .

F f1 R MAr ( /, ~X , ! 3 , ::>X , F 1 0 , r) , < v , F 1 ? , 6 , 5 X , F 7, ~ , 7 X , F 7, 3, r, X , r 7 , ~ , 7 X , F 7 '*, '3, 4 X oF 7 • ·~ , (,X , >- 7, 3 , ? X , C 1 ' o '+ )

STO!'J r'\fn

';ll[lROlJT JNF: CAPTE(fll A.T,I\I_O'~,PHTO,ALONC 9 X,Y} . THIS SUDROLTINE COMPUTES THE CARTESIAN CCORDS. FROM THE LAT.

AI\C LCI\GlTUCE ·~~**********••******************************************************

lNFLlCil H~IL*S{t-h,C-ZJ RA=C: ..!76206o4CO s:;e:t:356!:E3odCO Fl:;!,l41~92f5DO EC=(HA**~-~E+*2)/RA**2 X~=((l.•EC)+~~)/(OSQRT(l.-~C+(OSlN(Phl0)++2))**;:) XN=RAIDSGRT(l.-EC*(DSI~(FHIGJ**2l) I'=CSCFoT (.XM>+Xtd .X=R+(ALJll-PHIOJ Y=~+CCOS(PHlGJ+(ALON-ALONlJ) I'Ell..FI\ ENC

SUERCLTl~E F~CJCT(L,PHI,C,VAR,PV.STDEV)

THIS Sl.ERlUTINE PHEDILTS VALUES OF THE FU~CTIOI\ USli\G THE CGEF~. CF ThE FCLYI\CMI~L A~C T~E EASE FUNCTIGNS. **********~······***************************************************

f-L CJl ~~

1Z2 12J 11:!4 125 126 127 128 129 130 131 10: 2 1."33 134 13ti

1 :n

1J7 1 30 139 140

1 0

(.

1MFL1Cll REAL*€(A-H,C-Z) CU.t:I\!:ICf\ Fhi(30),C(3')),VAR(;:O) ~~~=c.o · ~lit-'VJI=C .c CC lC l=ltl Flii\=Frl(l)H(l) VA=PI-1 (I >**2 ~VIlf.< (I) ~IJ~-=~LtHFUf\ SU~VA=Sl.fJVA+vA. <.Cf\Tlt-.~E PV·= Sl.M ~T[I::\=SLtJVA hl.:ll.f;l\ E"C

Sl.bFClllf\E CEGRJID(A,B,C,Cl.T) L L LCf\V~hTS CE,, ~INo SEC TG hAUlAI\S c

$GU

ltJPLlClT RcllL*€(A-t-,U-Z) Ul.T=<A+B/bO.ODO+C/3600.0UOJ*DARSlN(1.000)/90.0DO F~Tu~l\ El\0

f-'o c..n \,;1

156

III CANADIAN DEFINITIONS OF CHART AND SOI.!NDIW; DATUMS

This appendix has been added to supplement the information given in

Sections 1.0 and 1.1 of this report. The information given here has been

taken directly from the Hydrographic Tidal Manual 1970 [Energy, Mines, and

Resources Canada]. The descriptions and definitions presented concern tidal

waters; for similar information regarding non-tidal waters, the reader is

referred to the above mentioned reference.

Chart datum is the datum plane adopted for a published chart. It

is a low water datum which by international agreement is so low that the water

level will seldom fall below it. It is the level above which tidal pre­

dictions and water level records are based. The datum is only used within a

gauge location and differs from place to place depending on the range of tide

or water levgJ_.

For tidal waters, the Canadian Hydrographic ,Service has adopted the

level of Lower Low Water Large Tides (see Figure III-1) as its reference for

chart datum, and Higher High Water Large Tides as a reference for elevations.

A sounding datum is the reference surface to which soundings are

reduced during the course of a hydrographic survey. It is the datum used when

compiling a "field sheet" for a survey. It may or may not be the same as

chart datum.

When selecting a datum, the following must be considered:

(i) the datum should be sufficiently low so that under normal weati1er

conditions there will always be at least the charted depth of water,

(ii) the datum should not be so low that it gives an unduly pessimistic

impression of the least depth of water likely to be found,

(iii) the datum should be in close agreement with those of neighbouring

surveys.

157

The following are the definitio~s of various reference surfaces

(datum planes) and water level variations in tidal waters used by the

Canadian Hydrographic Service.

Graphical representations of several of these are given in

Figure III-1.

(i) Higher High Water Large Tides (H.H.W.L.T.) is the highest predictable

tide from the available tidal constituents, with the astronomical

(nodal)factor fk close to unity.

(ii) Higher High Water Mean Tides (H.H.W.M.T.) is the mean of the predicted

heights of the higher high waters of each day.

(iii) Lower Low Water Mean Tides (L.L.W.M.T.) is the mean of the predicted

heights of the lower low waters of each day.,

(iv) Lower Low Water Large Tides (L.L.W.L.T.) or Lowest Normal Tides (L.N.T.)

is the lowest predictable tide from the available tidal constituents,

with the astronomical (nodal) factor fk close to unity.

(v) Mean Water Level (M.W.L.) is the mean of hourly water levels for a

period of observations.

(vi) Mean Tide Level (M.T.L.) is the mean of all high and low water heights

over a period of observation.

(vii) Charted Elevation is the vertical distances between an object and the

reference surface of Higher High Water Large Tides.

(viii) Charted Depth is the vertical distance from the chart datum to the sea

floor.

158

z 0 0 w -

BENCH t- ~ ----- --- -------~ - ~ MARK <( > w

ELEVATION J: _.

H. H.WL.T. u w DATUM

H. H~W.M.T. LL. w 0 C) C)

..J z z w <( <(

WATER LEVEL AT TIME T > 0:: ~ w ..J ~w w

!ct 0 ::lO 0 1-- M.W.l. G. S.C. VERTICAL w ~-j: ------- .... - :r: w DATUM .... w

CJ) c c :E - w z 1-- -..J I--C) <t w 1--co o::O:: w :>-~

u. ~ <(~ ~ 0 - L.L.W.M.T. 1-

CJ) 0 t- J: w

J: 1--C) a. - w

CHART & SOUNDING) UJ c L. L.W.L.T J:

DATUM

REFERENCE SURFACES AND WATER LEVEL VARIATIONS