Application of statistics to the presentation of wave and ship-motion data

813

NAVY DEPARTMENT
THE DAVID W. TAYLOR MODEL BASIN

WASHINGTON 7, D.C.

APPLICATION OF STATISHMGS TO THE PRESENTATION
OF WAVE AND SHIP-MOTION DATA

February 1955 Report 813

APPLICATION OF STATISTICS TO THE PRESENTATION GF WAVE AND
SHIP-MOTION DATA

by

Alice W. Mathewson

February 1955 Report 813
NS 731-037

ili

TABLE OF CONTENTS

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PRESENTATION OF DATA .......-.-eeeeeeee
OBSERVED WAVE HEIGHTS................
Weather Bureau Data .............ccscceceees
Hydrographic Office Data ................
MEASURED WAVE DATA ........-.: eee

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OBSERVED WAVE HEIGHTS AND MEASURED SHIP MOTIONSG..........cccceceeseessesess

ANALYSIS OF DATA ..0...-esseeesesessseeeseteeeees
DISTRIBUTION PATTERNS ..............-

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FITTING MATHEMATICAL CURVES TO FREQUENCY DISTRIBUTION ................
MEANS AND STANDARD DEVIATIONS ..........cccsccccsssccecscececerecesarececstececssecesseasensreeeessers

CONFIDENCE BANDB.......esesesesseeeseeess

CORRELATION BETWEEN WAVE HEIGHTS AND PITCH ANGLES.......... cece

DURATION OF SAMPLE .........--.:::seee
(COINCIDE IUISIOINIS ‘ccaceccccospacecnoosodoonsdeca5sq0500000"
ACKNOWLEDGMENTS .......--essceeeseeeseeeesceees

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APPENDIX 1 - PEARSON-TYPE DISTRIBUTION CURVES ..........escceceeceeeeeeeeteteneeereee

APPENDIX 2 - DURATION OF SAMPLE

15

Me

NOTATION

Constant defined by an equation in Appendix 1 and used in evaluating Pearson’s
Type I Curve

Constant defined by an equation in Appendix 1 and used in evaluating Pearson’s
Type I Curve

Number of distributions required to satisfy conditions assumed in Appendix 2
Deviation of the mean of a sample from the mean of a population

Frequency of occurrence

Wave height measured from crest to trough

Root mean square value of wave heights

Fraction between 0 and 1

Mean value of the first pn of n wave heights when arranged in descending order
of magnitude

Constant taken from a student’s ‘‘¢’’ table

Constant defined by an equation in Appendix 1 and used in evaluating Pearson’s
Type I Curve

Constant defined by an equation in Appendix 1 and used in evaluating Pearson’s
Type I Curve

Total number of elements which make up the distribution

Number of independent observations required to satisfy conditions assumed in
Appendix 2.

Probability

Constant defined by an equation in Appendix 1 and used in evaluating Pearson’s
Type I Curve

Arithmetic mean of the population

(7 = 1, 2, 3, 4) ‘‘zth’’ arithmetic mean of a sample

Value of the abscissa in Pearson’s Type I Curve

Value of the ordinate in Pearson’s Type I Curve

Ordinate of the Pearson type curve at the mode

Deviation of any value of f from the mean value of frequency distribution
Criterion for the Pearson type curves and defined by an equation in Appendix 1

Criterion for the Pearson type curves and defined by an equation in Appendix 1
Gamma funetion

Variable of integration in the evaluation of AP?
h

Hy

Criterion for the Pearson type curves and defined by an equation in Appendix 1

(¢ = 1, 2, 3, 4) ‘‘cth’? moment about the mean, yp, =
Instantaneous wave elevation

Standard deviation of the population

Standard deviation of a sample

Sf 2

N

Class interval

Confidence bands

Distribution
Frequency

Normal distribution

Population

Probability

Probability density

Probability level

Random

Sample
Standard deviation

Standard error

Statistic

“*¢” distribution

Variability
Mode

Significant wave
height

vi

DEFINITIONS

A grouping of possible values of a variable

The interval within which the ‘‘true’’ distribution will fall with a
certain probability

An arrangement of numerical data according to size or magnitude
The number of times a value occurs or is observed

A bell-shaped curve, symmetrical about the mean and defined by the
mean and standard deviation

The entire data from which a sample was drawn if all of it were
available

The likelihood of occurrence

A quantity which, if integrated over the independent variation, is
equal to 1; see probability

A number which indicates the degree of confidence that can be placed
on a given result, i.e., probability level 0.90 means that 90 times out
of 100, a given hypothesis will hold

The method of drawing a sample when each item in the population has
an equal chance of selection

A finite portion of the population

A special form of the average deviation from the mean, a measure of
dispersion, o = Y(2 f 22)/N

The standard deviation of a distribution of means

The estimate of a number describing the numerical property of a popu-
lation

The distribution of student’s ¢, defined by ¢ = (X, = X) VN/o where X;
is the mean of a random sample of size N from a normal population
with a mean X and a is the estimate of the standard deviation of the
normal population as estimated from the sample.

The variation of the data; the lack of tendency to concentrate

The most frequent or most common value; its value will correspond to
the value of the maximum point of a frequency distribution.

Generally defined as the mean value of the one-third highest waves.
Reference 12 and correspondence with the Hydrographic Office indicate
that th- wave heights estimated by observers approximated the
‘““‘significant’’ wave heights.

ABSTRACT

Available observations of wave heights have been assembled and evaluated
in terms of statistical methods in connection with the study of the service strains
and motions experienced by ships at sea. Curves have been fitted to the distri-
bution patterns, and confidence bands, averages, and standard deviations have
been computed. Distribution patterns for wave heights observed in different parts
of the world are all of the same type with a peak displaced toward the lower wave

heights. Pitching motions measured on a ship at sea also follow this same pattern.

INTRODUCTICN

The David Taylor Model Basin is making a study of the motions and strains in ships at
sea for the purpose of evaluating and improving methods for the design of the ship girder and
its structural components. It is probable that the frequency-distribution patterns of strains
and motions of ships at sea will be similar to those of wave heights. It is also expected
that the year-to-year variability in the distribution patterns of wave heights will be of the same
order of magnitude as the year-to-year variability in the distribution patterns of ship motions
and dynamic hull-girder stresses inasmuch as the latter are, to a large degree, functions of
the wave heights and wave lengths. To verify these expectations, observed wave heights
have been obtained from the Weather Bureau and the U.S. Hydrographic Office. These data
and data measured by the Model Basin on the USCGC CASCO have been studied to determine
(1) the type of distribution pattern, (2) the variation in this pattern over a period of time, and
(3) the mathematical function which will best fit these data. The results of the third phase of

the study are presented in this report.

PRESENTATION OF DATA

Figures 1, 2, and 3 are frequency distributions of wave heights, that is, depth from
crest to trough, obtained from various sources. These distributions are presented in the form
of bar-type graphs or histograms. The ordinates of these histograms give the percent of total
observations or measured values that fell between given limits of wave height as indicated by

the abscissa.

OBSERVED WAVE HEIGHTS

Weather Bureau Data

Figure 1 shows yearly and combined wave-height data which were furnished by the U.S.

Weather Bureau at the request of the Taylor Model Basin.! These data represent wave-height

DRererences are listed on page 13.

og = 20.52 ft? of = 26.01 ft?

Qo) — er If Seat Tey asses a
Hep 1949 1950 ]
20 ic _ Total Observations = 2832 Total Observations = 2741 |
| 21 Observations > 30,3 ft 17 Observations > 30.3 ft
5 i — kane X= 7.50 ft
| |

$ 10

i | r

£ |

oO 5 [| |

3

)

~ ©

ao

Qa

c Tos

Sia 1951 1952

3 90 Total Observations = 2797 Total Observations = 2820
a 29 Observations > 30.3 ft 6 Observations > 30.3 ft
3 15 X =8.45 ft X =7.40 ft

=

2 of = 34.81 ft? z| of = 20.52 ft?

= 10

8

fa)

a5

fo) 5 10 15 20 29 30 35 (0) 5 10 IB oO) 25 SOMaSo
Significant Wave Height, ft

Figure la - Frequency Distributions of Yearly Samples

distributions determined from observations made by weather ships at ocean station ‘*Charlie’’
(52° N, 37° W) in the North Atlantic from 1 January 1949 to 31 May 1958. The observations
were made every three hours by trained weather observers in accordance with instructions
prescribed by the United States Weather Bureau.” The observations are reported as the
average of the significant* wave heights. Only one quantitative measurement was recorded

each time the sea was observed.
Hydrographic Office Data

Figure 2 shows combined frequency distributions of wave heights for periods of 2, 7,
and 40 years tabulated by the U.S. Hydrographic Office? at the request of the Model Basin.
These observations, also at station ‘‘Charlie,’’? were made by German merchant ships from

1901 to 1939. The data are not as reliable as the data presented in Figure 1; because routes

*Generally defined as the mean value of the one-third highest waves; see definitions page vi.

Percent of Total Observations per Class Interval

1949
Sampling Period = 1 year

Total Observations = 2832
21 Observations > 30.3 ft

X = 17.36 ft
o2 = 20.52 ft?

+

1949 - 1950
Sampling Period = 2 years
T r Total Observations = 55731
38 Observations > 30.3 ft

Ie,

Ks TS it
| SSB ie

1949 - 1951
Sampling Period = 3 years

Total Observations = 8370
67 Observations > 30.3 ft

Maye i
og = 27.26 ft2

—

1949 - 1953
Sampling Period = 4% years
Total Observations = 12,365]
73 Observations > 30.3 ft

Mz Tod ih val

ge = Bd IE

15 QQ YQ Es)

O 5 10 15 20 25 30 86035

Significant Wave Height, ft

Figure 1b - Combined Frequency Distributions

Figure 1 - Frequency Distributions of Samples of Significant Wave Heights Observed at
Ocean Station “‘Charlie’’ by U.S. Weather Observers

X is the mean and, Ox is the variance. The observations greater than 30,3 ft were included in the totals given

but are not shown on the histograms.

MEASURED WAVE DATA

were often avoided at times of high seas. Fewer extreme values were recorded.

Figure 3a is the frequency-distribution pattern of measured wave heights produced by

the wavemaker at the Taylor Model Basin. These were simulated to represent a characteristic

confused sea. Only 43 measurements were made.

Figure 3b shows a frequency distribution of wave heights measured at sea by means of

a pressure recorder. These data were tabulated on a form that shows the relation between

wave heights, lengths, and periods.* Since measurements were made for a period of only 30

minutes, it may be assumed that they represent the characteristics of the sea at that time and

1939 - 1943 1934 - 1943

Total Observations = 5358
i——j Sampling Period = 7 years

Total Observations = 528
Sampling Period = 2 years

eo ih X = 3.99 ft
o,= 11.77 ft? o2 = 20.06 ft2

i

lo) 5 10 15 20 «25 30 35 40
Significant Wave Height, ft

1901 - 1943

Total Observations = 18,627

Sampling Period = 40 years Figure 2 - Combined Frequency Distributions

eVGA Ge of Significant Wave Heights Observed at
Ocean Station ‘‘Charlie’’ by Merchant Ships

Percent of Total Observations per Class Interval

a= 28.45 ft?

These Data were obtained from the U.S. Hydrographic

|

Office. X is the mean and a is the variance.

0 5 10 15 2025 30 35 40
Significant Wave Height, ft
at that particular geographic location. It should be noted that the waves were of very small

height (less than 140cm = 4.6 ft).

OBSERVED WAVE HEIGHTS AND MEASURED SHIP MOTIONS

As pointed out in the introduction of this report, wave-height distributions could
reasonably be expected to have the same type of pattern as ship motions and stresses. This
similarity was evidenced by the weather ship USCGC CASCO.° Figure 4 shows the frequency
distributions of wave-height observations and pitch-angle measurements made during this
test. Both were made at 3-hr intervals over a period of one month at station ‘‘Charlie.’’ No

waves less than 1 ft nor pitch angles less than 1 deg were recorded.

357 :
9 Total Observations = 43 '] Total Observations — 688
230) ReUOS | Rees
PA of= 1.99 o2 = 883
i)
S) 28) |
&
Beoh 1
@ 157 ian
ye)
fo)
S 10+ +
e
oe ;
S
1S)
ane)

| 2 3 4 5 6 tr 8 O 20 40 60 £80 100 120 140
Wave Height, in. Wave Height, cm

Figure 3a - Frequency Distribution of Measured Wave Figure 3b - Frequency Distribution of Wave Heights at
Heights Produced by TMB Wavemaker to Simulate a Sea as Measured by a Pressure Recorder

Couiusee! Bea All wave heights for a 30-min period are included.

Figure 3 - Frequency Distributions of Measured Wave Heights

ELA

Total Observations = 237
Total Time Duration = 1 month
of Sampling _

X = 6.29 ft

a” = 7.28 ft?

Total Observations = 3090
Total Time Duration = 1 month of Sampling

X = 2.45 ft
a= 188 te

b
fe)

oO
oO

ey
fo)

~
oa

20

Percent of Total Observations per Class Interval

(al
(0) 2 4 6 8 10 12 14 16 18 (e) 2 4 6 a” © 14 16 18 20
Significant Wave Height, ft Pitch Angle, deg (Peak to Peak)

Figure 4 - Frequency of Wave Heights Observed and Pitch Angles Kecorded on the
USCGC CASCO at 3-hr Intervals over a One-Month Period at Ocean Station ‘‘Charlie’’

2

X is the mean and 0; is the variance

ANALYSIS OF DATA
DISTRIBUTION PATTERNS

Examination of Figures 1 through 4 shows that all have a similar type of frequency
distribution, that is, distributions peak towards the lower wave heights. Similar distributions
presented in Figure 11 of Reference 6 also showed such patterns. The data presented there®
were compiled from charts of observations made by Japanese merchant ships in the North
Pacific during the 15-yr period from 1924 through 1938. These charts’ are available at the
U.S. Hydrographic Office at Suitland, Maryland. Areas of observations were broken down into
2-deg squares, that is, 2 deg latitude by 2 deg longitude. A study of these charts, which
present the data in the form of histograms, leads to the conclusion that, in general, these
histograms are also peaked in the direction of the lower wave heights.

On the basis of a study of the experimental data thus far available to the author, there
is a strong indication that the frequency distributions of wave heights may be approximated
by a straight line when plotted on logarithmic probability paper. Figure 5 shows some of the
patterns obtained from the data of Figures 1, 2, and 3 and Reference 6. This approximation
of the wave-height distributions by a straight line means that they approach a logarithmically
normal distribution, that is, if the frequency is plotted as a function of the logarithm of wave

height, the distribution will be normal.

FITTING MATHEMATICAL CURVES TO FREQUENCY DISTRIBUTION

In addition to the log-normal curve two other types of curves have been fitted to the
Weather Bureau data (Figure 1) inorder to find a suitable mathematical function which might
be used to represent the observations. The Weather Bureau data were chosen because they
appeared to have been obtained by the most reliable and consistent sampling procedure. The
fitted curves are shown in Figures 6 and 7. The first is a Pearson Type I Curve whose shape
is based on the values of the moments yp, of the given frequency distribution and whose origin
is taken at the mode computed from the measured distribution. The curve is defined by the
equation given in Figure 6 and discussed in Appendix 1. The second fitted curve, Figure 7,
is of the form known as the ‘‘random walk’? distribution. It has been shown® that if the sea
elevation é may be represented by a narrow spectrum, the probability that at any fixed location

the wave height / lies between / and A + dA is approximately
—p2

2
P (h)dh =28 e * dh (W
h

where h2 is the mean of h2. If the sea has a narrow spectrum, the elevations & of the wave

surface have a normal distribution, see Figure 8.

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Bios \L_| Yo = 1357, ordinate of mode in number of
ie | Las observations per ft
= \
3 ax = distance from mode, ft
© TIN I
a 0.04|-+ +t +
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0.02 | i |
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I
0 Bie 25 30

10 15 20
Significant Wave Height, ft

Figure 6 - Pearson’s Main Type I Curve® Fitted to Probability Density Distribution of
Significant Wave Heights Observed at Ocean Station ‘‘Charlie’’ by U.S. Weather
Observers from January 1949 to June 1953
ol
[aa
|

oe FEEEEH |

|_|**Random Walk’’ Distribution} } | | | |

aD 2 r
ee 72 = 60.0 BEB Ie)

Probability Density, P(h)
fo)
ro)
a

2)
{e)
B

0.02

a 1 Ls
(0) 5 10 15 20 25 30

h
Significant Wave Height,ft
Figure 7 - ‘‘Random Walk’? Distribution Fitted to Probability Density Distribution of
Significant Wave Height Observed at Ocean Station ‘‘Charlie’”’ by U.S. Weather
Observers from January 1949 to June 1953

|

Figure 8 - Wave Record Showing Elevation ¢ and Wave Height A

It is not necessarily true that a sea for which the wave heights follow the probability
density function [1] will have a normal distribution of &(¢), where €(¢) is the instantaneous

wave elevation.
Reference 8 gives the ratio

co

BO i, aay eri 1 [2]
hp hp hp

h®) denotes the mean value of the first pN of the N wave heights when arranged in descend-
ing order of magnitude, where p is a fraction between 0 and 1. Thus the average of the
‘significant waves’? is A“1/3), It should be noted that, experimentally, it is difficult to find A
from measurements of inasmuch as the average depends to a considerable degree upon the
lower limit to which the wave heights are measured; see Reference 4 for a discussion of this
effect.

The distribution plotted in Figure 7 is fitted with the mathematical curve given in
Equation [1]. The value of 42 which gives the best fit is h? = 60.0." As stated on page 10 the
random walk theory holds only if the sea has a narrow spectrum. It may well be that the
spectrum of the sea for the wave height distribution shown in Figure 7 will not remain narrow

due to the fact that the sampling extended over a period of years.

MEANS AND STANDARD DEVIATIONS

Means and standard deviations were computed for the distributions shown in Figures 1
and 2 and are given in Table 1. It will be observed that the average of the data obtained from
merchant-ship observations is much lower than the average of the Weather Bureau data. From
the standard error of the mean (the standard deviation of the distribution of the means of

samples) of the four yearly Weather Bureau samples, it may be stated that there are 99.7

*The best fit was determined by a Chi square test.

10

TABLE 1

Means and Standard Deviations for Frequency Distributions of Wave Heights

Standard
Standard | Average of cea

Number of
Sample

Observations

—

Computed from Weather Bureau Data (Figure 1a)

1949 2811 7.36 4.53
1950 2724 7.50 5.10 7.69 0.45

1951 2768 8.45 5.90
1952 2814 7.40 | 4.53
Computed from Weather Bureau Data (Figure 1b) J

1 yr 2811 7.36 4.53 |

2 yrs 59390 7.51 4.81

3 yrs 8303 7.82 5.22
mae 12,272 7.76 5.07 |

4830
18,627

chances out of 100 that the average mean computed, 7.69 ft, will be no further away from the
true mean than 1.35 ft.? (3c = 1.35 ft) for the period 1949 to 1952.

CONFIDENCE BANDS

Figure 9 shows confidence bands fitted to the probability density distribution of the
Weather Bureau data. These confidence bands, computed according to Kolmogorov’s statis-
tic,!° show the interval within which the “‘true’’ distribution will fall at a probability level
of 99 percent, that is, in 99 cases out of 100 random sampled distributions, the distribution
will fall within these bounds. The requirement for the use of Kolmogorov’s statistic is that
the sampled wave heights be random and that the distribution of wave heights be continuous.

A plot of the data on probability paper is shown in Figure 9a. The encircled points
were computed from the observed wave heights and the solid line represents a logarithmically
normal distribution. In this figure the confidence bands were fitted to the observed points.
The curve fitted to the probability density distribution shown in Figure 9b was obtained by
taking the average probability density of the class intervals at their centers and fairing a

curve through these points to make the area under the curve equal to the area under the

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12

histogram. The confidence bands were computed for this curve, utilizing Kolmogorov’s statis-

tic

CORRELATICN BETWEEN WAVE HEIGHTS AND PITCH ANGLES

The scatter diagram* of Figure 10 shows the pitch angles measured on the CASCO
plotted as a function of wave heights. Except for scattered observations, this diagram indi-
cates a correlation between pitch angle and wave height which may be approximated by a
straight line. The figure also indicates that the most probable combination is that correspon-
ding to about 3-deg pitch angle and 5-ft significant wave height.

Oo

(Peak to Peak) (For a given observation)

Average of One-Third Highest Pitch Angle, deg

Significant Wave Height, ft

Figure 10 - Scatter Diagram of Pitch and Wave Height Data of Figure 4

A straight line was faired by using pitch angle measurements when ship was headed into the waves; these are

indicated by ©; the numbers give the number of observations.

DURATION OF SAMPLE

An estimation of the period throughout which samples must be taken in order to permit
a statistically valid prediction is often necessary. The details of the computation involved

for two such methods are given in Appendix 2.

*A scatter diagram?

is a method of showing the relationship between two associated variables. In this form the
independent variable is placed along the abscissa while the dependent variable is placed along the ordinate. It
is obvious that if the relationship between the two variables were perfect, every given value on the abscissa
would indicate a value of the ordinate. If there is a direct simple relationship between the variables plotted, the

points will tend to fall on some curve, possibly a straight line.

13

CONCLUSIONS

1. Frequency distributions of wave heights are not normal but tend to peak toward the
lower wave heights. They do, however, have a pattern which is approximated by a logarith-
mically normal distribution.

2. The frequency distribution of pitch.angle for the USCGC CASCO has the same general
form as that shown to be applicable for the wave heights. It is reasonable to expect that this
pattern will also hold for other ships.

3. The frequency distribution patterns of the pitch angles measured on the USCGC

CASCO show a correlation with those of the wave height observations.

4. The Pearson Type I Curve may be fitted to frequency distributions of wave heights.

ACKNOWLEDGMENTS

The studies made in conjunction with this report were done under the supervision and
guidance of Mr. N.H. Jasper. Members of Statistical Engineering Laboratory of the National
Bureau of Standards, namely Mr. I. Richard Savage, Mr. Marvin Zelen, and Dr. Edgar King
made suggestions as to treatment and presentation of data. The wave height data presented
in various figures was obtained with the cooperation of members of the Division of Oceano-
graphy, U.S. Hydrographic Office and Aerology Branch, CNO. Finally, the author is indebted
to Dr. George Suzuki for review of the statistical mathematics in this report and the suggestion

of Appendix 2.

REFERENCES

1. TMB ltr $29/12 (732:AWM:cae) to CNO (Aerology Branch) of 30 July 1958.

2. ‘*A Manual of Marine Meterological Observations,’’ U.S. Weather Bureau Circular M,
8th Ed (1950).

3. U.S. Hydrographic Office ltr Code 541-LBB/rvg of 10 July 1953.

4. Ehring, H., ‘‘Kennzeichnung des Gemessenen Seegangs auf Grund der Haufigkeitsver-

teilung von Wellenhoehe, Wellenlaenge und Steilheit,’’ T.B. 4 (1940), pp 152-155 (Translation:
U.S. Hydrographic Office, S10 Wave Project, Report 54, Contract Nobs 2490).

5. Jasper, N.H., ‘‘Study of the Strain and Motions of the USCGC CASCO at Sea,’’? TMB
Report 781 (May 1953).

6. Harney, L.A. et al, ‘‘A Statistical Study of Wave Conditions at Four Open-Sea Locali-
ties in the North Pacific Ocean,’? NACA Technical Note 1493 (Jan 1949).

7. ‘Waves in the North Pacific Ocean,’’ U.S. Hydrographic Office Jan-Dec, H.O. Misc
11, 117-1.

14

8, Longuet-Higgins, M.S., ‘‘On the Statistical Distribution of Heights of Sea Waves,”’
Journal of Marine Research, Vol XI, No. 3 (1952).

9. Arkin, H., and Colton, R.R., ‘‘An Outline of Statistical Methods,” 4th Ed., Barnes and
Noble, Inc., New York, p. 116 and p. 75.

10. Birnbaum, Z.W., ‘‘Numerical Tabulation of the Distribution of Kolmogorov’s Statistic
for Finite Sample Size,’’ Journal of the American Statistical Association, Vol. 47, No. 259,
p. 425 (Sep 1952).

11. Elderton, W.P., ‘‘Frequency Curves and Correlation,’’ 4th Ed., Harren Press, Washing-
ton, D.C. (1953).

12. Sverdrup, H.U., and Munk, W.H., “Wind, Sea and Swell: Theory of Relations for Fore-
casting’’, H.O. Publication No 601.

15

APPENDIX 1
PEARSON-TYPE DISTRIBUTION CURVES

A set of curves that may be fitted to different frequency distributions was compiled by
the statistician Karl Pearson. The theoretical derivation and calculations necessary for
fitting these curves are described in Reference 11. The curve type which best fits a frequency
distribution may be identified from criteria calculated on the basis of the values of the
moments p;.

The steps for identifying and computing the constants for, the fitting of the curve shown
in Figure 6 are given here. The numerical values are those for the frequency distribution of
the Weather Bureau data for the 414-yr period, Figure 1.

The moments measured about the mean value of the distribution are:

py, -= 2 -0 [3]
x f 2

Hy =~ = 18.68 [4]
wie

By =—— = 40.27 [5]
See

hq =< = 593.6 [6]

where z represents the deviation of the actual value f from the mean,
f is the frequency, and
N is the total frequency of the sample.

The criteria 8,, 8,, and K computed from the preceding moments are:

ue
B, =— = 0.6336 [7]
BS
By 2 = Bale [8]
2
Ho

B,(B, +3)

eS 103090 [9]
4(46, be 38) (2B, 7 38, Te 6)

Since the criterion K is negative, it identifies Pearson’s Main Type I Curve as the most

suitable one. This curve is defined by the equation

y= (1+=] 2 [10]

1

16

where y, is the ordinate at the mode,”
z is the distance from the mode, and
Taieyue2
are computed from the equations which follow and

a 2

pp &a

m, ™
pad yet [11]
lap
First a parameter 7 must be evaluated
6(2,- -l
pm peel aie Ha [12]

(6 + 38, - 28,)

a8 - ollie (B, (r + 2)? + 16(r + 1)} = 22.71 [13]

[14]

When y, is positive, m, is the root corresponding to the plus sign; if , is negative, m, is the
root corresponding to the minus Sign.

For our numerical example

m, = 0.441
mM, = 3.493
Finally m m
igo Mo S My 2 : P(m, +m, + 2) [15]
i arita, m+m, U(m, + 1)0(m, +1)

(m, + my) 1

Tables of the gamma function are given in the reference 11. With the use of logarithms, y,
was computed to be 1404, and the mode of the distribution was found to be at 3.99 ft.
Equation [10] becomes:

0. b
pole (t-—2 el ae ae [16]
2.55 20.2

This gives the frequency distribution in terms of a class interval of unit length. Therefore
for a class interval of 1.6 ft, the frequency would be 2171. Since the probability density
distribution was desired, y was divided by N.

*The mode is the most frequent or common value; it will correspond to the maximum ordinate of the frequency
distribution.

17

APPENDIX 2
DURATION OF SAMPLE

Assume in the first approach that for the specific locale indicated, one of the wave-

height distributions has a mean

X = 7.76 ft
and
o, = 5.07 ft
Then, by standard statistical procedure, the sample size necessary to obtain a sample mean
which differs from the true mean by no more than 5 percent with a confidence coefficient of

90 percent can be obtained by solving for n in the equation

ee

= [17]
0.05 X

vn
where + & is the particular abscissa on the ‘‘¢’’ distribution with n defined such that the area

under the ‘‘¢’’ distribution between * & is 90 percent. By substitution

pe [SOO age
ORC ||

that is approximately 467 indcpendent and random observations are necessary.

Weather observations are characterized by the lack of independence in successive ob-
servations when the time interval between observations is relatively short. The duration of
interval necessary to insure independence cannot be determined. If one independent obser-
vation can be obtained every 10 days, then by the above calculations, more than 13 years are
necessary to obtain a sample fulfilling the stipulated conditions. If 7 days are sufficient,
then about 9 years are necessary.

As another approach, suppose that the means of the wave heights obtained for each
year (Table 1, page 10) represent independent observations. In this treatment the statistical
“‘population’’ is the totality of these independent observations. Basing the following compu-
tations on the observed mean values, it is found that n = 7. This implies that 7 years of rather

extensive observations are necessary to fulfill the conditions imposed.

Problem:
Find the number of samples which are required to make
|2-1) <0.0% [18]
x;

with a probability of 0.90, where ¥ is the mean of the population and X, is the mean of the

ith sample.

18

Procedure:
1. Find the n means of n distributions, XeP Mop nee

2. Assume the mean of the population to equal the average of these means and compute

xX ap Ae) #P 000 ah [19]

3. Assume that the standard deviation of the means of n yearly distributions is equal to

the standard deviation of the population of means of yearly distributions and compute

_ Vc
7 - Vad [20]

where d is the distance of the mean of the sample from the mean of the population.

4, The mean of any one distribution of c yearly means may take on a range of values

X is the mean value of the population,
o is the standard deviation of the population,
aa = Ska [21] xe is a sample of c means of the population,
é k is taken from a student’s ‘‘z’’ table at probability level
0.90 and (c — 1) degrees of freedom, and
c is the number of distributions required; in other words

the number of independent observations.

5. From Equations [18] and [21] obtain

ko

Ve

< 0.05 |X,|

Thus, solving for /c, gives
k
“ [22]

> =
vez 0.05.

t
6. Values of c are assumed until Equation [22] is satisfied.

Example:
Assume each Weather Bureau yearly distribution to represent one sample. Then, from

Figure 1 (with n = 4) se
X, = 7.36 (year 1949)

P< |

7 = 1.50 (year 1950)
X, = 8.45 (year 1951)
x, = 7.40 (year 1952)

19

Therefore
Xx - 30.71 _ 769
4
a -\/2 d? [23]
= It
o =] /0-10 + OBE 0.591 + 0.08 _ 9 xo
and

0.52 i ’
ec > 14" _— 1.4% from Equati 22,
Ve> aap (7.36) guaonie

when x, is used.
As a first approximation, assume c = 5. The value &(¢) from the ‘‘2’’ table at a 0.90

probability level and four degrees of freedom is 2.13

V5 > 1.4 - 2.18

As a second approximation, assume c = 6

V7 = 1.4 - 2.02
Therefore c = 7, indicating that measurements would have to be made over a period of 7 yr to
establish a distribution pattern which would be valid at a probability level of 0.90 such that

Equation [19] is satisfied, providing that no year-to-year bias exists in the data.

"

~ ht" TTA Bers <0

Sy pereeeray acer ,

(ESTA ORE}
ET ute 7

i eet

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