Net effect of wind on recreational tidal streams in Florida

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Net effect of wind on recreational tidal streams in Florida
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Moreau, D. H.
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Abstract:
The object of this study was to determine the effect of wind velocities on tide heights in a small Florida estuary. In particular, the purpose was to perform a statistical analysis of wind and tide data collected during a period from July 1, 1963 to June 30, 1964, at Cedar Key, Florida. Studies of this project were divided into two phases. First, an analysis was performed to extract from the. record an estimator of the astronomical tide, that periodical tidal variation due to movement of celestial bodies. An expression was found that accounted for 87 percent of the total variability of tidal heights. Second, studies were made to relate deviations of observed heights from the estimator developed in Phase I with wind velocities. Those deviations correlated poorly with observed wind velocities, possibly due to several factors: (1) wind observations were instantaneous readings, (2) location of the tide gage, and (3) too much weight given to small deviations.

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WATER RESOURCES

research center

Publication No. 7
The Net Effect of Wind on Recreational
Tidal Streams in Florida
By
D.H. Moreau


University of Florida
Gainesville


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UNIVERSITY OF FLORIDA


-demo- A&

















TIlEi NET EL` 'ECT 01F' WINI) ON
RECREATIONAL TIDAL STREAMS
IN FLORIDA

by

D. H. Moreau
Assistant Professor
University of Florida


PUBLICATION NO. 7


of the

FLORIDA WATER RESOURCES RESEARCH CENTER


RESEARCH PROJECT TECHNICAL COMPLETION REPORT

OWRR Project Number A-006-FLA






Annual Allotment Agreement Number
14-01-0001-903 (1967)


Report Submitted: October 13, 1967





The work upon which this report is based was supported in part
by funds provided by the United States Department of the Interior,
Office of Water Resources Research, as authorized under the Water
Resources Research Act of 1964.

















Abstract


TI E NET EI 'ECT OF WINI) ON RECREAT IONAL TII)AL STREAMS IN FLO RIDA


The object of this study was to determine the effect of wind
velocities on tide heights in a small Florida estuary. In particular,
the purpose was to perform a statistical analysis of wind and tide data
collected during a period from July 1, 1963 to June 30, 1964, at Cedar
Key, Florida.

Studies of this project were divided into two phases. First,
an analysis was performed to extract from the record an estimator of
the astronomical tide, that periodical tidal variation due to movement
of celestial bodies. An expression was found that accounted for 87 per
cent of the total variability of tidal heights. Second, studies were
made to relate deviations of observed heights from the estimator devel-
oped in Phase I with wind velocities. Those deviations correlated poorly
with observed wind velocities, possibly due to several factors: (1) wind
observations were instantaneous readings, (2) location of the tide gage,
and (3) too much weight given to small deviations.







Moreau, D. H.
THE NET EFFECT OF WIND ON RECREATIONAL TIDAL STREAMS IN FLORIDA
Completion Report to Office of Water Resources Research, Department of
the Interior, October 1967, Washington, D.C. 20240
KEYWORDS: wind tides/ estuary/ harmonic analysis/ astronomical tides.

















J II Ii 'ilJ UC'T ION


The objectives of this study was to determine for a small
Floridian estuary the significance of wind induced deviations of tidal
heights from tidal heights due solely to astronomical variations. In
particular, the study had as its purpose an analysis of wind and tide
observations collected by others over a time interval from July, 1963,
to June, 1964 in conjunction with pollution studies in the Waccasassa
River estuary That data consisted of hourly observations of wind
speed and direction (N, NE, E, SE, S, SW, W, NW) and of magnitudes and
times of occurrence of high and low tides as measured at Cedar Key,
Florida during the time interval.

Studies of the data were made in two phases. First, it was
necessary to extract from the data that portion of tidal height vari-
ability due to the astronomical tide, that tide which results from the
periodic motion of celestial bodies. After a suitable expression was
found to estimate the astronomical tide, deviations of observed tides
from estimated tides were found by subtraction. Those deviations were
then correlated with observed wind velocities, resolved into components
that were perpendicular and parallel to the coast line at the mouth of
the Waccasassa estuary. Because of the difficulty of visually deter-
mining the proper orientation of perpendicular and parallel axes at the
mouth, correlations between deviations and wind measurements were exam-
ined for various angles of rotation of axes relative to North-South and
East-West axes. Studies were also conducted to evaluate the predictive
value of proceeding wind velocities on deviations of actual tides from
astronomical tides.


ASTRONOMICAL TIDES


Studies of the U.S. Coast and Geodetic Survey (USC&GS) of the
Environmental Science Services Administration list 26 periodic constituents
of astronomical tides. Each constituent has its own frequency, amplitude,
and phase angle; the latter two characteristics vary with latitude, longi-
tude, and geomorphology of their location. From lengthy records of tidal
observations and a knowledge of gravitational effects upon tides, the
USC&GS routinely predicts tide heights at numerous stations on U.S. coast
lines. The relative importance of those 26 periodic constituents vary
from one location to another. Tides at Cedar Key, Florida are predicted
from calculations for a gaging station at the entrance to St. Marks River,
Florida. Predictions are based upon an expression of the form













h(t) = a +
0


26 2H W.S
E a cos -60 ['t + (V +u) + --
+1 i o i 1


-piL -Ki]


h(t) = a + Z a cos (wit + 3.)
0 1 1
i=1

where


t
Wi
(V +U) i

S
L
Pi


1i


= time in hours from midnight January 1
= speed of constituent i in degrees per hour
= annual correction factors at Greenwich time

= longitude of local time meridian in degrees
= longitude of prediction stations in degrees
= number of full cycles per day of the i h constituent


211
360 1


211
-360 [(Vo+U)i


oiS
+--- -p.L K]
15 P i


Values of u., K., (V +u)i for 1963 and 1964 and pi are available from
the 2USC&GS For the St. Marks River Station and are applicable to Cedar
Key The quantity S has a value of 750 for Eastern Standard time while
the longitude, L, is 83.10 at Cedar Key.

An estimator of astronomical tides at Cedar Key for the purpose
of this study does not require all 26 terms of the above expression. In
this study, a standard statistical measure of variability, the variance,
was used to determine the relative importance of various terms. A well-
known statistical technique, spectral density analysis, was applied to the
data to determine those constituents which contributed most significantly
to the variability of tidal heights3. A plot of the spectral density
function is shown in the figure on the following page. Peaks on the graph
occur at frequencies whose corresponding constituents contribute most
significantly to the total variance of the record. It may be noted that
constituents with frequencies of 0.94, 1.02, 1.92 and 2.00 cycles per
day appear to be the most important. Constituents of smaller frequencies
are not shown.

Coefficients a al, a2, ---, a9 in the expression


q
h(t ) .i -~
1 1


0L~5 (a~+[


Sor est iinhat ini the a1Istr eti-eontiC al tides were tound usinl the method of least







































0 0.5 /0 /5 20
cyc/es ,per c/ay
Segee.- per AO. 3
-Degrees lper Ao&-


SAa6-C 7-,e'4L-


DE.V5/ T /


0r- 77/E.S


A~y -,
~Z//?e /964


N
k
(3


-41/e










squares Those coefficients are listed in Table 1 with the contribution
of each term and the cumulative contribution of all terms also listed. It
may be noted that the estimating equation accounts for 87 per cent of the
total variance of the data, resulting in a multiple correlation coefficient
of 0.93 out of a possible 1.00. Constituent M2, the principal lunar semi-
diurnal constituent, is the most significant term, accounting for 71 per
cent of the total variance. Lunisolar (Kl) and lunar diurnal components
are other important components, each contributing approximately six per
cent of the total variance. Other constituents contribute increasingly
smaller portions to the precision of the estimator.

Differences between observed tides and the above estimator were
computed for 400 observations. A statistical summary of these differences
indicated an average deviation of -0.027 feet, an average absolute dif-
ference of 0.571 feet, and a standard deviation of differences of 0.705 feeL.
Since the tides vary from less than two feet to over seven feet, those
statistics indicate that the astronomical tide estimator is a good one.



WIND INDUCED DEVIATION


Deviations of observed tide heights from astronomical tides
were found by subtraction, i.e.

9
d. = h. -[a + E a. cos (Wt +i.)].
3 i=l 1 i J 1


These deviations were assumed to be attributed to wind forces acting
upon the wave surface the physical mechanics of which are understood
under some conditions The object of this phase of study was to explain
by wind effects a significant proportion of the 13 per cent of total
variability in the tide record unaccounted for by the astronomical tide.

Wind velocities were resolved into perpendicular components,
ui, and parallel components vi, where the subscript i denotes the ith
observation. Three analyses were made as described below. First a linear
model of the form


di = a + alui + a2vi


relating deviations with North-South components of wind, vi, and East-West
components, u.. Results are shown in Table 2-a, where it may be noted lhaL
the East-West component accounted for only 12 percent of the total variabil ity
of the deviations. The parallel component contributed practically nothing to
the precision of estimation. This latter fact may be accounted for by tlc
fact that the estuary mouth is well protected.










TABLE 1


Constituents
USC&GS
No. Designation


1

2

3

4

5

Ul 6


7

8

9


1963


28.9841

15.0411

13.9430

30.000

0.04107

0.08214

28.4397

14.9589

30.0821


(V +u)
+U


226.0

1.2

229.0

0.0

279.9

199.9

257.9

350.1

182.8


1964


326.7

0.9

329.8

0.0

279.7

199.4

269.8

350.3

181.6


Prop. of Total
Regression Variance
p Coefficients Accounted for


36.6

312.6

305.9

62.9

149.4

112.9

25.4

317.8

64.7


1.3281

0.5261

0.5072

-0.3579

0.2006

0.1721

0.1612

0.1444

0.1373


.7099

.0588

.0532

.0209

.0081

.0067

.0046

.0041

.0038


C=uuiative
Proportion of
Total Variance


.7099

.7686

.8218

.8426

.8508

.8574

.8621

.8662

.8699


1963 1964
























TABLE 2-a.


Regression
Variable Coefficient


Proportion of
Total Variance
Accounted For


Cumulative
Proportion of
Total Variance


Perpendicular
Component of
Wind, u


Parallel
Component, V


-0.2246


0.03078


.1202


.0005


.1202


.1207











Second, because of the difficulty in determining the appropriate
orientaLion of axes for wind components, the model above was applied for
;i)ngl'e of rotation of 15, 30, 45, 60, and 75 degrees with the North-South
oriintL;tion. Results are shown in Table 2-b, where it may be noted that
roLaLlon of axes did not improve the precision of the estimator. However,
rotating the axes did change the relative importance of components u and v.

Third, because available wind observations were instantaneous
recordings, and because wind effects may result from winds in earlier time
periods, models of the form


d. = ao + a-u. + a2ui + a 3vi. + a4v. 1


and


di = a + a1 (u + u.i-) + a2(vi+vi-)


were studied, where u. i and v. 1 are wind components observed one hour
prior to the deviation, di. The first of these two models measures the
effect of proceeding winds directly, while the second model averages the
effects of proceeding and current winds. Results of these studies are
shown in Table 2-c and 2-d.

From Tables 2-c and 2-d it may be noted that including effects
of proceeding winds directly improved the estimator only slightly as
indicated by the proportion of total variance explained by the model.
Averaging effects of current and proceeding winds does improve the esti-
mator, but that model explains only 25 per cent of the total variance of
deviations.

Several reasons may be suggested for the low correlations between
deviations and winds. First, instantaneous wind observations were reported
in the data analyzed in this work. Directions were designated only in 450
increments. Average wind conditions over some proceeding interval, say
15 minutes, might better account for surface forces in the estuary. Second,
wind velocities were measured at a single station and may not be represent-
ative of wind conditions over the entire estuary. Third, the method of
estimating coefficients in the various models may place undue weight on
small deviations that may be due to other factors that affect tidal heights
and their observations.












TABLE 2-b.


Proportion of Cumulative
Regression Total Variance Proportion of
Variable Coefficient Accounted For Total Variance


Angle of Rotation = 15 degrees

u -0.2090 0.1159 0.1159

V 0.08782 0.0048 0.1207



Angle of Rotation = 30 degrees

u -0.1792 0.1059 0.1059

V 0.1389 0.0148 0.1207



Angle of Rotation = 45 degrees

u 0.1806 0.1052 0.1052

V -0.1371 0.0155 0.1207



Angle of Rotation = 60 degrees

u 0,2099 0.1159 0.1159

V -0.0857 0.0048 0.1207



Angle of Rotation = 75 degrees

u 0.2249 0.1202 0.1202

V -0.0285 0.0005 0.0005
























TABLE 2-c.


Regression
Coefficient


-0.2174

-0.0554

0.0301

0.0092


Proportion of
Total Variance
Accounted For


0.1195

0.0271

0.0005

0.0001


Cumulative
Proportion of
Total Variance


0.1195

0.1466

0.1471

0.1472


Variable


u.I
i-l

Vi

Vi-1I




























TABLE 2-d.


Regression
Coefficient


-0.1396


0.0228


Proportion of
Total Variance
Accounted For


0.2395


0.0067


Cumulative
Proportion of
Total Variance


0.2395


0.2462


Variable


ui+ui-l


Vi+Vi-1













LITERATURE CITED


1. Saville, Tlhorndike (Principal I nvesitgator). "A Study of Estuarine
Pollution Problems on a Small Unpolluted Estuary and a Small
Polluted Estuary in Florida," Bulletin Series No. 125, Engr.
Progress at Univ. of Fla., 1966.

2. Personal communication: Mr. D. C. Simpson, Chief Tide Prediction
Section, Environ. Science Services Admin., Rockville, Maryland.

3. Blackman, R. B. and J. W. Tukey, The Measurement of Power Spectra,
Dover Publications, Incorporated, 1958.

4. Kendall, M. G. and A. Stuart, The Advanced Theory of Statistics,
Vol. 3: Design and Analysis and Time Series, Hafner Publishing
Co., New York, 1966.

5. Dean, R. G., "Tides and Harmonic Analysis," Chap. 4 in Estuary and
Coastline Hydrodynamics, Arthur T. Ippen (ed.), McGraw-Hill
Book Co., Inc., New York, 1966.




Full Text

PAGE 1

WATER IiRESOURCES researc center Publication No. 7 The Net Effect of Wind 01/ Recreatiollal Tidal Streams il1 Florida By D H. Moreau UlliJJersity of Florida Gaillesville UNIVERSITY OF FLORIDA

PAGE 2

TilE NET EFFECT OF WIND ON RECREATIONAL TIDAL STREAHS IN FLORIDA by D. H. Horeau Assistant Professor University of Florida PUBLICATION NO. 7 of the FLORIDA WATER RESOURCES RESEARCH CENTER RESEARCH PROJ1!:CT TECHNICALCOHPLETION REPORT OWRR Project Number A-006-FLA Annual Allotment Agreement Number 14-01-0001-903 (1967) Report Submitted: October 13, 1967 The work upon which this report is based ,was supported in part by funds provided by the United States Department of the Interior, Office of Water Resources Research, as authorized under the Water Resources Research Act of 1964.

PAGE 4

Abstract TilE NI':'I' I':FFEC'I' OF WIND ON RECREA'I'10NAL TIDAL STREAMS I.N FLORIDA The object of this study was to determine the effect of wind velocities on tide heights in a small Florida estuary. In particular, the pu,pose was to perform a statistical analysis of wind and tide data collected during a period from July 1, 1963 to June 30, 1964, at Cedar Key, Florida. Studies of this project were divided into two phases. First, an analysis was performed to extract from the. record an estimator of the astronomical tide, that periodical tidal variation due to movement of celestial bodies. An expression was found that accounted for 87 per cent of the total variability of tidal heights. Second, studies were made to relate deviations of observed heights from the estimator developed in Phase I with wind velocities. Those deviations correlated poorly with observed wind velocities, possibly due to several factors: (1) wind observations were instantaneous readings, (2) location of the tide gage, and (3) too much weight given to small deviations. Moreau, D. H. THE NET EFFECT OF WIND ON RECREATIONAL TIDAL STREAMS IN FLORIDA Completion Report to Office of Water Resources Research, Department of the Interior, October 1967, Washington, D.C. 20240 KEYWORDS: wind tides/ estuary/ harmonic analysis/ astronomical tides.

PAGE 6

INTRODUCTION The objectives of this study was to determine for a small estuary the significance of wind induced deviations of tidal heights from tidal heights due solely to astronomical variations. In particular, the study had as its purpose an analysis of wind and tide observations collected by others over a time interval from July, 1963, to June, 19641 in conjunction with pollution studies in the Waccasassa River estuary. That data consisted of hourly observations of wind speed and direction (N, NE, E, SE, S, SW, W, NW) and of magnitudes and times of occurrence of high and low tides as measured at Cedar Key, Florida during the time interval. Studies of the data were made in two phases. First, it was necessary to extract from the data that portion of tidal height variabilIty due to the astronomical tide, that tide which results from the periodic motion of celestial bodies. After a suitable expression was found to estimate the astronomical tide, deviations of observed tides from estimated tides were found by subtraction. Those deviations were then correlated with observed wind velocities, resolved into components that were perpendicular and parallel to the coast line at the mouth of the Waccasassa estuary. Because of the difficulty of visually determining the proper orientation of perpendicular and parallel axes at the mouth, correlations between deviations and wind measurements were examined for various angles of rotation of axes relative to North-South and East-West axes. Studies were also conducted to evaluate the predictive value of preceeding wind velocities on deviations of actual tides from astronomical tides. ASTRONOMICAL TIDES Studies of the U.S. Coast and Geodetic Survey (USC&GS) of the Environmental Science Services Administration list 26 periodic constituents of astronomical tides. Each constituent has its own frequency, amplitude, and phase angle; the latter two characteristics vary with latitude, longitude, and geomorphology of their location. From lengthy records of tidal observations and a knowledge of gravitational effects upon tides, the USC&GS routinely predicts tide heights at numerous stations on U.S. coast lines. The relative importance of those 26 periodic constituents vary from one location to another. Tides at Cedar Key, Florida are predicted from calculations for a gaging station at the entrance to St. Marks River, Florida. Predictions are based upon an expression of the form 1

PAGE 7

h(t) a 0 or h(t) a 0 where t wi (Vo+U)i and S L Pi w 1 + + 26 2IT w.S L: a cos 360 [w:t + (V +u) +Tt--poL -K.] i o i 1 1 i+l 1 26 I a. cos (wjt + S.) i 1 i=l time in hours from midnight January I speed of constituent i in degrees per hour annual correction factors at Greenwich time longitude of local time meridian in degrees longitude of prediction stations in number of full cycles per day of the i h constituent 2IT 360 W. 1 2IT w'S --360 [(V +u). +_1_ -p.L -K.] o 1 15 1 1 Values of w., K., (Vo+u)i for 1963 and 1964 and Pi are available from the USC&GS for [he St. Marks River Station and are applicable to Cedar Key2 The quantity S has a value of 75 for Eastern Standard time wh iJ e the longitude, L, is 83.1 at Cedar Key. An estimator of astronomical tides at Cedar Key for the purpose of this study does not require all 26 terms of the above expression. In this study, a standard statistical measure of variability, the variance, was used to determine the relative importance of various terms. A wellknown statistical technique, spectral density analysis, was applied to the data to determine those constituents which contributed most significantly to the variability of tidal heights3 A plot of the spectral density function is shown in the figure on the following page. Peaks on the graph occur at frequencies whose corresponding constituents contribute most significantly to the total variance of the record. It may be noted that constituents with frequencies of 0.94, 1.02, 1.92 and 2.00 cycles per day appear to be the most important. Constituents of smaller frequencies are not shown. a 9 in the expression q 11 (t ) ;1 -t \' ) 2

PAGE 8

9 ----------------B 1-----------------1 6 1-------------" 5 -------------------1--------------------------4' -------------1---1--1----I'. r----31------c--_ -------I 2 ---------------------------1-----------------+---{Il---------I / -------1-------I-------------1 J o 0.5 /.0 /.5 20 o eye les per dcry 7,5 /5 R2.$ 30 ..z>egree S ;Per ,/;OV;--SPECTRAL ...lJENS/TY' OF 71')S Cedar K .,t:::"/or/da Jut /96,8 -Jane /36. 3

PAGE 9

squares4 Those coefficients are listed in Table 1 with the contribution of each term and the cumulative contribution of all terms also listed. Jt may be noted that the estimating equation accounts for 87 per cent of the total variance of the data, resulting in a multiple correlation coefficient of 0.93 out of a possible 1.00. Constituent M2, the principal lunar semidiurnal constituent, is the most significant term, accounting for 71 per cent of the total variance. Lunisolar (Kl) and lunar diurnal components are other important components, each contributing approximately six per cent of the total variance. Other constituents contribute increasingly smaller portions to the pEecision of the estimator. Differences between observed tides and the above estimator were computed for 400 observations. A statistical summary of these differences indicated an average deviation of -0.027 feet, an average absolute difference of 0.571 feet, and a standard deviation of differences of 0.705 feet. Since the tides vary from less than two feet to over seven feet, those statistics indicate that the astronomical tide estimator is a good one. WIND INDUCED DEVIATION Deviations of observed tide heights from astronomical tides were found by subtraction, i.e. a. 1 These deviations were assumed to be attributed to wind forces acting upon the wave surface the physical mechanics of which are understood under some conditions5 The object of this phase of study was to explain by wind effects a significant proportion of the 13 per cent of total variability in the tide record unaccounted for by the astronomical tide. Wind velocities were resolved irttoperpendicular components, u., and parallel components vi' where thesubscript i denotes the ith Three analyses were made as described below. First a linear model of the form relating deviations with North-South components of wind, vi' and East-West components, u i Results are shown iri Table 2-a, where it may be noted that the East-West component accounted for only 12 percent of the total vari;il)j 1 i Ly of the deviations. The parallel component contributed practically nothing to the precision of estimation. This latter fact may be accounted for by tlw fact that the estuary mouth is well protected. 4

PAGE 10

TABLE 1 Constituents Prop. of Total Cu:::c:lat i\'c USC&GS (V +u) Regression Variance i,on of No. Designation 'A 1963 o 1964 P Coefficients Accounted for Total \'ariance 1 M2 28.9841 226.0 326.7 36.6 2 1. 3281 .7099 .7099 2 K1 15.0411 1.2 0.9 312.6 1 0.5261 .0588 .7686 3 01 13.9430 229.0 329.8 305.9 1 0.5072 .0532 .8218 4 S2 30.000 0.0 0.0 62.9 2 -0.3579 .0209 .8426 5 SA 0.04107 279.9 279.7 149.4 0 0.2006 .0081 .8508 In 6 SSA 0.08214 199.9 199.4 112.9 0 0.1721 .0067 .8574 7 N2 28.4397 257.9 269.8 25.4 2 0.1612 .0046 .8621 8 PI 14.9589 350.1 350.3 317.8 1 0.1444 .0041 .8662 9 K2 30.0821 182.8 181.6 64.7 2 0.1373 .0038 .8699

PAGE 11

Variable Perpendicular Component of Wind, u Parallel Component, V TABLE 2-a. Regression Cdefficient -0.2246 0.03078 6 Proportion of Total Variance Accounted For .1202 .0005 Cumulative Proportion of Total Variance .1202 .1207

PAGE 12

Second, because of the difficulty in determining the appropriate ori('ntatlon of axes for wind components, the model above was applied for of rotation of 15, 30, 45, 60, and 75 degrees with the North-South (Jric'lllation. Results are shown in Table 2-b, where it may be noted that roLation of axE'S did not improve the precision of the estimator. However, rotating tile axes did change the relative importance of components u and v. Third, because available wind observations were instantaneous recordings, and because wind effects may result from winds in earlier time periods, models of the form d. a o + alu. + a2u + a3v + a4v. 1 1 1. 1-1 1 1and were studied, where u'_l and vj_1 are wind components observed one hour prior to the deviatiofr, d i The first of these two models measures the effect of preceeding winds directly, while the second model averages the effects of preceeding and current winds. Results of these studies are shown in Table 2-c and 2-d. From Tables 2-c and 2-d it may be noted that including effects of preceeding winds directly improved the estimator only slightly as indicated by the proportion of total variance explained by the model. Averaging effects of current and preceeding winds does improve the estimator, but that model explains only 25 per cent of the total variance of deviations. Several reasons may be suggested for the low correlations between deviations and winds. First, instantaneous wind observations were reported in the data analyzed in this work. Directions were designated only in 45 increments. Average wind conditions over some preceeding interval, say 15 minutes, might better account for surface forces in the estuary. Second, wind velocities were measured at a single station and may not be representative of wind conditions over the entire estuary. Third, the method of estimating coefficients in the various models .may place undue weight on small deviations that may be due to other factors that affect tidal heights and their observations. 7

PAGE 13

Variable Angle of Rotation u V Angle of Rotation u V TABLE 2-b. Regression Coefficient 15 degrees -0.2090 0.08782 30 degrees -0.1792 0.1389 Angle of Rotation = 45 degrees u 0.1806 V -0.1371 Angle of Rotation 60 degrees u 0.2099 V -0.0857 Angle of Rotation 75 degrees u 0.2249 V -0.0285 8 Proportion of Total Variance Accounted For 0.1159 0.0048 0.1059 0.0148 0.1052 0.0155 0.1159 0.0048 0.1202 0.0005 Cumulative Proportion of Total Variance 0.1159 0.1207 0.1059 0.1207 0.1052 0.1207 0.1159 0.1207 0.1202 0.0005

PAGE 14

TABLE 2-c. Proportion of Cumulative Regression Total Variance Proportion of Variable Coefficient Accounted For Total Variance ui -0.2174 0.1195 0.1195 ui _1 -0.0554 0.0271 0.1466 V 1 0.0301 0.0005 0.1471 Vi _1 0.0092 0.0001 0.1472 9

PAGE 15

Variable u.+u .. 1 1 1-TABLE 2-d. Regression Coefficient -0.1396 0.0228 10 Proportion of Total Variance Accounted For 0.2395 0.0067 Cumulative Proportion of Total Variance 0.2395 0.2462

PAGE 16

LITERATURE CITED I. S