Online Library → M. A Calderon → Probability density analysis of ocean ambient and ship noise → online text (page 1 of 3)

Font size

!*-4s

PROBABILITY DENSITY ANALYSIS

OF OCEAN AMBIENT AND SHIP NOISE

M. A. Calderon â€¢ Research and Development Report 1248 â€¢ 5 November 1964

U. S. NAVY ELECTRONICS LABORATORY, SAN DIEGO, CALIFORNIA 92152 . A BUREAU OF SHIPS LABORATORY

THE PROBLEM

Investigate the probability density distribution of the ampli-

tude of ocean ambient noise and ship noise; determine any

differences in the distributions which might lead to the

identification of ship noise masked by a high background-

noise level. Also, determine, by standard statistical

methods, whether the distributions are gaussian or non-

gaussian.

RESULTS

1. Ambient ocean noise was found to have a gaussian

distribution of amplitudes (in the sense that the moments of

the distribution satisfied specific tests) only when the am-

bient noise was relatively clean, i.e., the noise did not

contain high-level ship noise, biological noise, ice noise or

any of the other extraneous noises discussed in the text.

2. The group of ship-noise samples recorded at

close range contained a large number of samples that had a

non-gaussian distribution. However the other types of ex-

traneous noises were found to cause the same kind of de-

viation from a gaussian distribution, so that it was not

possible by these tests to distinguish between a sample

with ship noise and a sample with the other types of ex-

traneous noises (such as biological and ice noise).

MBL/WHOI

D 03D1 00M0503 1

RECOMMENDATIONS

1. Use the method of moments described here if

better accuracy than that given by overlays is desired to

estimate the moments and to determine whether a sample

is gaussian or non-gaussian.

2. In the probability density analysis of a noise

sample, use a range of amplitudes covering at least Â±4

standard deviations; otherwise large errors in the estimates

of the moments will frequently result.

3. In future applications of the PDA, have the output

of the PDA in a digital form rather than a continuous curve,

so that the data will be available in a form more suitable

for the calculation of the moments of the distribution.

ADMINISTRATIVE INFORMATION

Work was performed under SR 004 03 01, Task 8119

(NEL L2-4) by members of the Listening Division. The

report covers work from January 1962 to June 1963 and

was approved for publication 5 November 1964.

The author wishes to express appreciation to the

members of the Listening Division who contributed their

time to perform much of the data processing; W. P. de la

Houssaye who wrote the computer program; and Elaine

Kyle who prepared the data for the computer. Thanks are

also extended to Fred Dickson, who prepared the illustra-

tions, and to G. M. Wenz, who made many helpful suggestions

during the work phase and during the writing of the

manuscript.

CONTENTS

INTRODUCTION. .. page 5

TEST PROGRAM. . . 7

Instrumentation. . . 7

Research Techniques. . . 8

Data Reduction Techniques. . .14

RESULTS... 3

CONCLUSIONS. . .3 7

PD of Ambient Ocean Noise. . .3 7

PD of Ship Noise. . .37

Comparison of Test Methods. ..37

RECOMMENDATIONS. . .38

REFERENCES. ..3 9

APPENDIX A: DESCRIPTION OF THE PDA AND ITS

OPERATION... 41-42

APPENDIX B: DETERMINATION OF NUMBER OF DATA

POINTS OF EACH SAMPLE. . .43-44

TABLES

1 Noise Samples Selected for Analysis, by Location. . . page 11

2 Number of Curves Showing Significant Values of Skewness

and Kurtosis. . . 31

3 Locations of Curves Showing Significant Values of Skewness

and Kurtosis. . . 32

4 Locations of Curves Showing Skewness and Kurtosis at 1

Per Cent Probability Level. . . 33

5 List of Curves for Which Chi-Square Was Computed. . .35

6 Curves Chosen by Normal Curve Overlay as Being Very

Closely Gaussian. . .36

ILLUSTRATIONS

1 Curve of probability density function of a gaussian random

variable. . .-page 6

2 Block diagram of Probability Density Analyzer system. . . 7

3-4 Selected probability density curves, compared with a

normal curve. . . 12-13

5-6 Examples of PD curves obtained by use of overlay method. . . 15-1 7

7 Curve obtained with Probability Density Analyzer. . . 23

8 Normalized cumulative sums of tabulated values. . . 25

9 Curves with positive and negative skewness, computed

with Edgeworth's series. . . 26

10 Curves with positive and negative kurtosis. . . 2 7

11-12 Cumulative probability of noise samples shown in figures

3 and 4. . .28-29

13 Experimental PD curve of random noise showing skewness,

calculated by Edgeworth's series. . .34

Al Theoretical and experimental PD curves of square wave

and sine wave. â€ž .41

Bl Number of times random noise goes into an interval

about x = for PD of 0. 4, vs. cutoff frequency of

low-pass filter. â€ž .4 5

INTRODUCTION

The study reported here was undertaken to investigate

the probability density distribution of the amplitudes of

ocean ambient noise and ship noise with respect to various

bandwidths in several frequency ranges. The question to

be answered was whether ambient noise, without any ship

noise or biological noises, can be considered gaussian, and

whether the presence of ship noise significantly changes the

probability of density distributions. A secondary objective

was to investigate methods of data reduction of the probability

density curves obtained with the B & K Probability Density

Analyzer, using standard statistical tests.

The probability density function, as treated throughout

this report, may be defined as follows.

Lim , .

â„¢. ' Ax. a \

p(x) = Ax-0 _t I (1)

N

N~ CO

where x is a random variable, with its range of values

divided into a large number of continuous intervals Ax.

Measure its instantaneous value a great number of times N.

Let n. be the number of measured values of x in the tth

interval {Ax . ).

The above equation can be rewritten as

Lim . , , A

At . I Ax, ...

p{x) = Ax-* v i (2)

T

T â€” <n

where A t . is the amount of time the signal spends in the

interval A^ and T is the total time of the sample. Equation

2 indicates more clearly how the B & K PDA measures the

probability density function. A more detailed explanation

can be found in reference 1. (See list of references at end

of report. )

The function

p(x) =

exp -Or-*) 2 /2a 3 1

where x is the mean and o~ is the standard deviation, is

illustrated in figure 1,

-3-2-10 12

AMPLITUDE IN STANDARD DEVIATION UNITS

Figure 1 . Curve of the probability density fu no-

tion of a gaussian random variable (normalized to

a unit area under the curve).

TEST PROGRAM

Instrumentation

The equipment used for the investigation is described

below and illustrated in figure 2.

An Ampex Model 350 was used as the record and play-

back recorder. This model has a good low-frequency

response to below 20 c/s.

The filters following the recorder were an Allison

Laboratories Model 2-A (used mainly as a low-pass filter)

and a B & K Band Pass Filter Set, Type 1611.

A Mcintosh amplifier, Model MC30, was used to

raise the signal level to 1 volt rms or greater.

The B & K Probability Density Analyzer, Model 160

(to be referred to as the PDA) was the main piece of equip-

ment and has been primarily designed to obtain the prob-

ability density curves of disturbances that are essentially

random in character. A brief description of the PDA and

its use in this investigation is given in Appendix A. A com-

plete and detailed description of the PDA can be obtained

from the instruction manual. 1

AMPEX 350

PLAYBACK UNIT

ALLISON LABS FILTER

MODEL 2-A

OR

B & K, TYPE 1611

Mcintosh

amplifier

MODEL MC30

B&K

PROBABILITY DENSITY

ANALYZER

MODEL 160

CRO

VAR I PLOTTER

XY RECORDER

COUNTER

Figure 2. Block diagram of Probability Density Analysis system.

An XY recorder by Electronic Associates, Inc., was

used to record the analog X and Y outputs of the PDA.

A cathode ray oscilloscope monitored the signal out-

put of the filter.

The counter used responded to frequencies of at least

10 Mc/s for use with the PDA. The counter can be used in

place of an XY recorder and, in fact, is essential if

measurements are to be made at low probability densities.

Research Techniques

Data which had been recorded for previous ambient-

noise studies were available for this study. These samples

had been recorded on 1 Ofâ€” inch reels of ^-inch tape, at 3f

inches per second, and were from three locations. Two

groups had been made in shallow water - one, about 2 miles

from the western side of an island off the coast of Southern

California, and the other in the Bering Straits. These con-

sisted of short ambient-noise samples recorded at regular

intervals throughout the day, so that one reel covered data

for one day. The third location represented was in deep

water in the North Pacific between Hawaii and Alaska; most

of these samples were of longer duration than the other two

groups, but covered only a few days.

Samples of ship noise were desired, so that their

probability density curves might be compared with those of

"clean" ambient noise. Recordings were made of ships

entering San Diego Harbor, with the sampling made at ap-

proximately the closest point of approach. These included

Navy surface ships, submarines (surfaced), and commercial

ships.

Several factors were considered in choosing the data

samples to be used in this study.

1. "Clean" ambient noise was used to determine

whether the distributions of the amplitudes were gaussian

or near-gaussian according to certain tests which will be

discussed later. Ambient noise was judged to be "clean"

when it was free from ship noise, biological noises, or any

man-made sounds when the sample was monitored. A band-

pass filter and oscilloscope were used to determine whether

60-c/s hum or any other single frequency components were

present in the noise sample.

2. All noise samples should be stationary for their

entire length. When the sample is ambient ocean noise,

this condition will not in general be true. For a noise

sample to be stationary it is necessary for the sample

parameters, the means and the variances, to remain un-

changed as measured from samples taken at different times.

It is possible that no significant difference in the sample

parameters will be found if the time between samples is

short enough. In a previous study 3 it was concluded that

ocean noise is a slowly varying, not a stationary, process.

This conclusion was based on a comparison of samples that

were 3 or more minutes apart. However, no significant

difference was found among the values of some other samples

which were only 3 minutes or less apart. Thus it appears

reasonable to assume that ocean noise is stationary during

a short interval of time (less than 3 minutes).

3. The PDA requires a noise sample of about 30

minutes duration for a complete automatic analysis of the

amplitudes from -3. 00 to +3. 00 standard deviations.

The need for a long noise sample that is stationary

can be satisfied by recording a short noise sample on mag-

netic tape and then making a loop of the tape. A loop length

was selected according to the following requirements.

a. The loop should be short enough so that the noise

could be considered stationary and so that the entire loop

could be analyzed for each amplitude interval. The PDA

(in the particular position used) requires 30 seconds to

sweep a range of amplitudes equal to the window width,

which is 0. 1 times the rms value of the input signal. A

sample length of 7 seconds met all the above requirements

and this gives a loop size of 52. 5 inches, which was

conveniently handled.

b. The recorded noise on the loop should be con-

tinuous, i.e., there should be no blank intervals on the

loop, since a blank interval would change the average rms

value of the recorded noise.

A typical analysis procedure was as follows. A portion

of data was selected for analysis from the recorded data

available. The noise was re-recorded on a loop. The loop

was played back at l\ ips and the analysis proceeded as

indicated by the diagram in figure 2. The filter was set to

the desired bandwidth, and the noise was amplified to 1 volt

rms or greater. The PDA was carefully calibrated and

adjusted just before each analysis. Its input level of noise

was adjusted to 1 volt rms by its potentiometer, thus

normalizing its output.

Probability density of the amplitudes was recorded on

the Y scale of the XY recorder and the amplitude around

which the probability density was measured was on the X

scale. Scale factors were selected to give a deflection of

4 inches on the Y scale for a probability density range of

to 0. 4, and a deflection of 1 inch per standard deviation of

amplitude on the X scale. The automatic sweep time of the

PDA was set at X = -3. 00 standard deviations, and would

automatically sweep through to X = +3. 00 standard deviations,

based on a 1-volt rms input. Total running time was about

30 minutes. This procedure was repeated for each band-

width on every loop analyzed.

Table 1 lists the number of samples analyzed from

each location, the total number of probability density

curves obtained from the samples, and the filter used to

analyze these curves. When the Allison Laboratories filter

was used, the system cutoff frequency at the low end was

about 20 c/s and the upper cutoff frequency was determined

by the filter which was set at 2500, 1500, 1200, 6 00, 300,

or 150 c/s. The B & K filter was used in both the octave

10

and third-octave positions for center band frequencies of

100, 200, 400, 800, and 1600 c/s.

TABLE 1. NOISE SAMPLES SELECTED FOR ANALYSES, BY LOCATION.

(FOR THE BANDWIDTHS USED, SEE ABOVE)

LOCATION

NUMBER OF

NOISE SAMPLES

NUMBER OF P D

CURVES OBTAINED

FILTER USED FOR ANALYSIS OF DATA

SOUTHERN

CALIFORNIA

9

29

8 SAMPLES WITH ALLISON LABS FILTER;

1 SAMPLE WITH ALLI SON LABS AND B & K

BERING STRAITS

9

24

ALLISON LABS

NORTH PACIFIC

8

65

B&K

SAN DIEGO

(SHIP NOISE

IN HARBOR)

9

36

ALLISON LABS

Actual probability density curves of ambient noise

are shown in figures 3 and 4. The large fluctuations in

some of the traces are caused by substantial variations in

the level of the noise sample. Since some of the curves

appeared to be closely gaussian, the methods used to

measure the parameters of the distribution included over-

lays, calculated moments, and cumulative probability

graphs. Tests of significance and the chi-square "good-

ness of fit" tests were used to determine what values of

skewness and kurtosis were improbable at a 5 or 1 per cent

probability level.

11

0.5

0.4

0.3

0.2

0.1

0.5

0.4

0.3

0.2

0.1

DATA TAKEN IN SHALLOW WATER (So. Calif.

i i 1 r

DATA TAKEN IN BERING STRAITS

-2 -1

AMPLITUDE

1 2

IN STANDARD DEVIATION UNITS

Figure 3. Exampl es of some PD curves taken in

shallow water, compared with a normal curve.

12

Â£ o

^Â°-5

CO

<

CO

o

en

0.4

0.3

0.2

0.1

i r

n 1 r~

-DATA TAKEN IN

SAN DIEGO HARBOR (Ship Noise)

-2-10 12

AMPLITUDE IN STANDARD DEVIATION UNITS

Figure 4. Examples of some PD curves taken

both shallow and deep water, compared with a

no rmal curve.

13

Data Reduction Techniques

OVERLAY METHOD

Since it was expected that the probability density-

curves obtained with the PDA would have a gaussian or

nearly gaussian distribution, an overlay with a gaussian

curve was used. The curve had parameters of a mean

equal to zero and a standard deviation equal to one. Figure

5 illustrates the use of this method with two curves, one

judged to be gaussian and the other non-gaussian. Some

probability density curves obtained with the PDA were

judged to be very nearly gaussian.

One disadvantage of the overlay method is that de-

cisions about how well a particular curve compares with

the overlay are purely subjective. Skewness and kurtosis

can be detected, but the magnitudes of these moments cannot

be estimated with accuracy. An extension of the overlay

method which will allow estimates of skewness and kurtosis

is described here.

The extension is an overlay with several curves in-

stead of just one. Each curve has a different set of values

for skewness and kurtosis. The curves are positioned

over the actual probability density curve and the parameters

are estimated by interpolation between the two closest

curves. The curves of the overlay can be computed with

the use of Edge worth's series approximation for nearly

gaussian distributions. 3 The first four terms of this series

are

fix) = h(x)-â€”h 3 (x)+-^hUx)+â€” h e (x) (3)

where h(x) is the normalized gaussian distribution, h (x)

is the nth derivative of h(x), g is the standardized skew-

ness, and g is the standardized kurtosis.

14

-2-10123

AMPLITUDE IN STANDARD DEVIATION UNITS

Figure 5. Examples of two PD curves which

were det ermined to he gauss i an or non- gaus s i an,

using a normal curve as an overlay.

15

Estimates of the skewness and kurtosis can be found

with the above method; but it does not give any indication of

whether these estimates are significantly different from the

expected values, if the sample is taken from a gaussian

distribution. Using the previous overlay, a method can be

developed so that a sample can be accepted or rejected at

any desired level of probability. Basically the method is

to have two of the curves on the overlay plotted so that they

will represent the maximum deviations allowed in the par-

ticular parameter of a sample with (N) points. The method

will be developed for kurtosis, but a similar method can be

used for skewness.

The variance of kurtosis is given by 4

var(o ) = 24/N (4)

for large TV. This holds for a sample taken from a normal

parent population. The standard deviation of kurtosis is

(24/710 2; if the kurtosis is distributed normally, then from

the ratio of a particular value of kurtosis (g s ') and the

standard deviation we can obtain the probability of getting a

value of kurtosis as large or larger than g ' . The ratio is

(24/70 2

The probability of getting a value of kurtosis as large as or

larger than g ' is given by the amount of area under a

normal curve outside the -7? and +7? standard deviations.

A value of 7? = 1. 96 corresponds to a probability level of

(5)

16

5 per cent, or l/20th the total area. A ratio as large as

1.96 may be considered sufficiently improbable and hence

g ' can be assumed to result from a non-gaussian distribution.

The sample would therefore be rejected as coming from a

gaussian distribution. The value of g s ' therefore depends

onf, g ' - 1.96(24///) s. Edgeworth's series would then

be used to compute two curves, one with -g 3 ' (for negative

kurtosis) and one with +g 3 ' (for positive kurtosis). These

curves would represent the limits, at a 5 per cent

probability level, within which a sample of N points would be

considered as coming from a gaussian distribution.

Figure 6 shows two curves as they would appear in

the overlay. These two curves are the limits for a sample

0.5

g' â– +0.50

-3

g' =-0.50

-2-10 12

AMPLITUDE IN STANDARD DEVIATION UNITS

Figure 6. Overlay indicating g 3 ' of +0.50 and of -0.50,

A curve having a value of kurtosis as large or larger

than these values will be non- gauss i an at a 5 per cent

level for a sample of 3 70 points or, e qui val ent 1 y , a

bandwidth of about 55 c/s.

17

of bandwidth about 55 c/s, with N given by the equation N =

6.7/, where / is the bandwidth. The equation is obtained

o o

from Appendix B, using a time constant T = 2.3 seconds.

The overlay method was not used extensively because

of the complexity that comes from considering different

values of N and also different combinations of skewness and

kurtosis in the same sample. A method using computed

moments of the curves is described next; it was felt that

this method would yield accurate values of the mean,

standard deviation, skewness, and kurtosis.

METHOD OF MOMENTS

The method of moments is basically a general method

of forming estimates of the parameters of a distribution by

means of a set of measured sample values. The first few

moments of the actual distribution are calculated and these

are used as estimates of the moments of the parent population.

On the basis of these moments a suitable theoretical dis-

tribution curve is selected. For any particular distribution

curve the moments are functions of the parameters of that

curve. The parameters are determined and tests of sig-

nificance are made on the skewness and kurtosis.

The moments about the origin are defined as 5

m ' = Â£ p.(x)x. (6)

r v i>

i

where p^(x) is the probability that a value selected at ran-

dom from the population will lie in the i-th class. The

variate x with which we are concerned may be discrete or

continuous.

The moment

m 1 ' - E P t ^ x t (7)

18

is defined as the mean value of x, m ' - x.

Another more important set of moments is obtained

by changing the origin to the arithmetic mean. Equation 8

defines the moments about the mean.

m = Â£ p.(x) (x. - x\ (8)

For computing purposes, the relations between the

m and the Tn ' are convenient. Expressing the m in terms

of the rn ' we have the relations

r

m 1 = m 1 ' (9a)

m 2 = m 3 ' - (ot 1 'f (9b)

m 3 = m 3 ' - 3m a 'm 1 ' + 2(% ') 3 (9c)

ff2 4 = ot 4 ' - 4m 3 , m 1 ' + Qm 3 '{m x 'f - 3(m Â± ') 4 (9d)

Grouping errors are negligible, so Sheppard's

corrections are not applied.

- These moments can be expressed in standard units

by the use of a standardized variable z, by dividing the

variable x by s , the standard deviation.

J x

(x-x)

z -

s

X

(10)

The standardized moments are defined by the equations

772

a = â€” â€” , for r = 1, 2, 3, and 4 (11)

r r

s

x

19

a 2

= 1

m a

a 3

s ;

Â«r

a 4

m 4

s

X

The first four standardized moments are

a-L = (12a)

(12b)

(12c)

(I2d)

The third moment, a 3 , is a measure of the skewness of the

distribution. A positive value indicates a distribution with

a longer positive tail than a negative tail.

The fourth standardized moment, a 4 , is a measure

of the kurtosis of the distribution. In some cases it. is a

measure of the "peakedness" of the distribution, though it

is now understood that the length and size of the tails are

very important in this measurement.

For a normal curve the values of a 3 and a 4 will be

and 3, respectively. We redefine the skewness and kur-

tosis as

Q, r Â« 3 (13a)

g 2 = Â« 4 - s (i3b)

so that g is for a normal curve.

It is not very likely that the third and fourth moments

of a random sample will be zero. Depending on the distri-

bution and on the actual sample values, the third and fourth

moments will have some value different from zero. To de-

termine whether this difference is significant, it is neces-

sary to use the variances of the third and fourth moments. 4

varfo, ) = 6N(N-l)(N-2)- 1 (N-l)- 1 (N-3)- 1 (14a)

20

var(Â£ 3 ) - 24A"(^-l) 2 (^-3)- 1 (^-2)- 1 (^-3)- 1 (#-5)- 1 (14b)

For large N use,

var(^) = 6/N (15a)

var(p a ) = 24/27 . (15b)

The hypothesis to be tested is that the data sample is

taken from a gaussian distribution. To test the hypothesis

compare g to (6/N)z and g 3 to (24/#)s (see ref. 5), then

if

> 1. 96 reject the hypothesis at the 5 per cent level

(6 /N)s

> 2. 57 reject the hypothesis at the 1 per cent level.

Similarly, for g r

if > 1.96 reject the hypothesis at the 5 per cent level

(24/tf)*

> 2. 57 reject the hypothesis at the 1 per cent level.

CHI-SQUARE "GOODNESS OF FIT" TEST

The x 3 test will be applied to the hypothesis that a

sample of N individuals forms a random sample from a

population with a given probability distribution. The param-

eters of a distribution are known and are not estimated

from the sample itself. Later a modification will be given

for the situation where the parameters are estimated from

the sample.

21

The quantity*

(F.-Np. f

L -V- (16)

s

is a measure of the deviation of the sample from the ex-

pectation, where F . is the number of observed frequencies

in the tth interval, and Np . is the number of expected fre-

quencies in the ith interval as predicted by the theoretical

distribution. Karl Pearson proved that the above quantity,

in the limit, is the ordinary \ 2 distribution which is now

tabulated in most statistics books.

PROBABILITY DENSITY ANALYSIS

OF OCEAN AMBIENT AND SHIP NOISE

M. A. Calderon â€¢ Research and Development Report 1248 â€¢ 5 November 1964

U. S. NAVY ELECTRONICS LABORATORY, SAN DIEGO, CALIFORNIA 92152 . A BUREAU OF SHIPS LABORATORY

THE PROBLEM

Investigate the probability density distribution of the ampli-

tude of ocean ambient noise and ship noise; determine any

differences in the distributions which might lead to the

identification of ship noise masked by a high background-

noise level. Also, determine, by standard statistical

methods, whether the distributions are gaussian or non-

gaussian.

RESULTS

1. Ambient ocean noise was found to have a gaussian

distribution of amplitudes (in the sense that the moments of

the distribution satisfied specific tests) only when the am-

bient noise was relatively clean, i.e., the noise did not

contain high-level ship noise, biological noise, ice noise or

any of the other extraneous noises discussed in the text.

2. The group of ship-noise samples recorded at

close range contained a large number of samples that had a

non-gaussian distribution. However the other types of ex-

traneous noises were found to cause the same kind of de-

viation from a gaussian distribution, so that it was not

possible by these tests to distinguish between a sample

with ship noise and a sample with the other types of ex-

traneous noises (such as biological and ice noise).

MBL/WHOI

D 03D1 00M0503 1

RECOMMENDATIONS

1. Use the method of moments described here if

better accuracy than that given by overlays is desired to

estimate the moments and to determine whether a sample

is gaussian or non-gaussian.

2. In the probability density analysis of a noise

sample, use a range of amplitudes covering at least Â±4

standard deviations; otherwise large errors in the estimates

of the moments will frequently result.

3. In future applications of the PDA, have the output

of the PDA in a digital form rather than a continuous curve,

so that the data will be available in a form more suitable

for the calculation of the moments of the distribution.

ADMINISTRATIVE INFORMATION

Work was performed under SR 004 03 01, Task 8119

(NEL L2-4) by members of the Listening Division. The

report covers work from January 1962 to June 1963 and

was approved for publication 5 November 1964.

The author wishes to express appreciation to the

members of the Listening Division who contributed their

time to perform much of the data processing; W. P. de la

Houssaye who wrote the computer program; and Elaine

Kyle who prepared the data for the computer. Thanks are

also extended to Fred Dickson, who prepared the illustra-

tions, and to G. M. Wenz, who made many helpful suggestions

during the work phase and during the writing of the

manuscript.

CONTENTS

INTRODUCTION. .. page 5

TEST PROGRAM. . . 7

Instrumentation. . . 7

Research Techniques. . . 8

Data Reduction Techniques. . .14

RESULTS... 3

CONCLUSIONS. . .3 7

PD of Ambient Ocean Noise. . .3 7

PD of Ship Noise. . .37

Comparison of Test Methods. ..37

RECOMMENDATIONS. . .38

REFERENCES. ..3 9

APPENDIX A: DESCRIPTION OF THE PDA AND ITS

OPERATION... 41-42

APPENDIX B: DETERMINATION OF NUMBER OF DATA

POINTS OF EACH SAMPLE. . .43-44

TABLES

1 Noise Samples Selected for Analysis, by Location. . . page 11

2 Number of Curves Showing Significant Values of Skewness

and Kurtosis. . . 31

3 Locations of Curves Showing Significant Values of Skewness

and Kurtosis. . . 32

4 Locations of Curves Showing Skewness and Kurtosis at 1

Per Cent Probability Level. . . 33

5 List of Curves for Which Chi-Square Was Computed. . .35

6 Curves Chosen by Normal Curve Overlay as Being Very

Closely Gaussian. . .36

ILLUSTRATIONS

1 Curve of probability density function of a gaussian random

variable. . .-page 6

2 Block diagram of Probability Density Analyzer system. . . 7

3-4 Selected probability density curves, compared with a

normal curve. . . 12-13

5-6 Examples of PD curves obtained by use of overlay method. . . 15-1 7

7 Curve obtained with Probability Density Analyzer. . . 23

8 Normalized cumulative sums of tabulated values. . . 25

9 Curves with positive and negative skewness, computed

with Edgeworth's series. . . 26

10 Curves with positive and negative kurtosis. . . 2 7

11-12 Cumulative probability of noise samples shown in figures

3 and 4. . .28-29

13 Experimental PD curve of random noise showing skewness,

calculated by Edgeworth's series. . .34

Al Theoretical and experimental PD curves of square wave

and sine wave. â€ž .41

Bl Number of times random noise goes into an interval

about x = for PD of 0. 4, vs. cutoff frequency of

low-pass filter. â€ž .4 5

INTRODUCTION

The study reported here was undertaken to investigate

the probability density distribution of the amplitudes of

ocean ambient noise and ship noise with respect to various

bandwidths in several frequency ranges. The question to

be answered was whether ambient noise, without any ship

noise or biological noises, can be considered gaussian, and

whether the presence of ship noise significantly changes the

probability of density distributions. A secondary objective

was to investigate methods of data reduction of the probability

density curves obtained with the B & K Probability Density

Analyzer, using standard statistical tests.

The probability density function, as treated throughout

this report, may be defined as follows.

Lim , .

â„¢. ' Ax. a \

p(x) = Ax-0 _t I (1)

N

N~ CO

where x is a random variable, with its range of values

divided into a large number of continuous intervals Ax.

Measure its instantaneous value a great number of times N.

Let n. be the number of measured values of x in the tth

interval {Ax . ).

The above equation can be rewritten as

Lim . , , A

At . I Ax, ...

p{x) = Ax-* v i (2)

T

T â€” <n

where A t . is the amount of time the signal spends in the

interval A^ and T is the total time of the sample. Equation

2 indicates more clearly how the B & K PDA measures the

probability density function. A more detailed explanation

can be found in reference 1. (See list of references at end

of report. )

The function

p(x) =

exp -Or-*) 2 /2a 3 1

where x is the mean and o~ is the standard deviation, is

illustrated in figure 1,

-3-2-10 12

AMPLITUDE IN STANDARD DEVIATION UNITS

Figure 1 . Curve of the probability density fu no-

tion of a gaussian random variable (normalized to

a unit area under the curve).

TEST PROGRAM

Instrumentation

The equipment used for the investigation is described

below and illustrated in figure 2.

An Ampex Model 350 was used as the record and play-

back recorder. This model has a good low-frequency

response to below 20 c/s.

The filters following the recorder were an Allison

Laboratories Model 2-A (used mainly as a low-pass filter)

and a B & K Band Pass Filter Set, Type 1611.

A Mcintosh amplifier, Model MC30, was used to

raise the signal level to 1 volt rms or greater.

The B & K Probability Density Analyzer, Model 160

(to be referred to as the PDA) was the main piece of equip-

ment and has been primarily designed to obtain the prob-

ability density curves of disturbances that are essentially

random in character. A brief description of the PDA and

its use in this investigation is given in Appendix A. A com-

plete and detailed description of the PDA can be obtained

from the instruction manual. 1

AMPEX 350

PLAYBACK UNIT

ALLISON LABS FILTER

MODEL 2-A

OR

B & K, TYPE 1611

Mcintosh

amplifier

MODEL MC30

B&K

PROBABILITY DENSITY

ANALYZER

MODEL 160

CRO

VAR I PLOTTER

XY RECORDER

COUNTER

Figure 2. Block diagram of Probability Density Analysis system.

An XY recorder by Electronic Associates, Inc., was

used to record the analog X and Y outputs of the PDA.

A cathode ray oscilloscope monitored the signal out-

put of the filter.

The counter used responded to frequencies of at least

10 Mc/s for use with the PDA. The counter can be used in

place of an XY recorder and, in fact, is essential if

measurements are to be made at low probability densities.

Research Techniques

Data which had been recorded for previous ambient-

noise studies were available for this study. These samples

had been recorded on 1 Ofâ€” inch reels of ^-inch tape, at 3f

inches per second, and were from three locations. Two

groups had been made in shallow water - one, about 2 miles

from the western side of an island off the coast of Southern

California, and the other in the Bering Straits. These con-

sisted of short ambient-noise samples recorded at regular

intervals throughout the day, so that one reel covered data

for one day. The third location represented was in deep

water in the North Pacific between Hawaii and Alaska; most

of these samples were of longer duration than the other two

groups, but covered only a few days.

Samples of ship noise were desired, so that their

probability density curves might be compared with those of

"clean" ambient noise. Recordings were made of ships

entering San Diego Harbor, with the sampling made at ap-

proximately the closest point of approach. These included

Navy surface ships, submarines (surfaced), and commercial

ships.

Several factors were considered in choosing the data

samples to be used in this study.

1. "Clean" ambient noise was used to determine

whether the distributions of the amplitudes were gaussian

or near-gaussian according to certain tests which will be

discussed later. Ambient noise was judged to be "clean"

when it was free from ship noise, biological noises, or any

man-made sounds when the sample was monitored. A band-

pass filter and oscilloscope were used to determine whether

60-c/s hum or any other single frequency components were

present in the noise sample.

2. All noise samples should be stationary for their

entire length. When the sample is ambient ocean noise,

this condition will not in general be true. For a noise

sample to be stationary it is necessary for the sample

parameters, the means and the variances, to remain un-

changed as measured from samples taken at different times.

It is possible that no significant difference in the sample

parameters will be found if the time between samples is

short enough. In a previous study 3 it was concluded that

ocean noise is a slowly varying, not a stationary, process.

This conclusion was based on a comparison of samples that

were 3 or more minutes apart. However, no significant

difference was found among the values of some other samples

which were only 3 minutes or less apart. Thus it appears

reasonable to assume that ocean noise is stationary during

a short interval of time (less than 3 minutes).

3. The PDA requires a noise sample of about 30

minutes duration for a complete automatic analysis of the

amplitudes from -3. 00 to +3. 00 standard deviations.

The need for a long noise sample that is stationary

can be satisfied by recording a short noise sample on mag-

netic tape and then making a loop of the tape. A loop length

was selected according to the following requirements.

a. The loop should be short enough so that the noise

could be considered stationary and so that the entire loop

could be analyzed for each amplitude interval. The PDA

(in the particular position used) requires 30 seconds to

sweep a range of amplitudes equal to the window width,

which is 0. 1 times the rms value of the input signal. A

sample length of 7 seconds met all the above requirements

and this gives a loop size of 52. 5 inches, which was

conveniently handled.

b. The recorded noise on the loop should be con-

tinuous, i.e., there should be no blank intervals on the

loop, since a blank interval would change the average rms

value of the recorded noise.

A typical analysis procedure was as follows. A portion

of data was selected for analysis from the recorded data

available. The noise was re-recorded on a loop. The loop

was played back at l\ ips and the analysis proceeded as

indicated by the diagram in figure 2. The filter was set to

the desired bandwidth, and the noise was amplified to 1 volt

rms or greater. The PDA was carefully calibrated and

adjusted just before each analysis. Its input level of noise

was adjusted to 1 volt rms by its potentiometer, thus

normalizing its output.

Probability density of the amplitudes was recorded on

the Y scale of the XY recorder and the amplitude around

which the probability density was measured was on the X

scale. Scale factors were selected to give a deflection of

4 inches on the Y scale for a probability density range of

to 0. 4, and a deflection of 1 inch per standard deviation of

amplitude on the X scale. The automatic sweep time of the

PDA was set at X = -3. 00 standard deviations, and would

automatically sweep through to X = +3. 00 standard deviations,

based on a 1-volt rms input. Total running time was about

30 minutes. This procedure was repeated for each band-

width on every loop analyzed.

Table 1 lists the number of samples analyzed from

each location, the total number of probability density

curves obtained from the samples, and the filter used to

analyze these curves. When the Allison Laboratories filter

was used, the system cutoff frequency at the low end was

about 20 c/s and the upper cutoff frequency was determined

by the filter which was set at 2500, 1500, 1200, 6 00, 300,

or 150 c/s. The B & K filter was used in both the octave

10

and third-octave positions for center band frequencies of

100, 200, 400, 800, and 1600 c/s.

TABLE 1. NOISE SAMPLES SELECTED FOR ANALYSES, BY LOCATION.

(FOR THE BANDWIDTHS USED, SEE ABOVE)

LOCATION

NUMBER OF

NOISE SAMPLES

NUMBER OF P D

CURVES OBTAINED

FILTER USED FOR ANALYSIS OF DATA

SOUTHERN

CALIFORNIA

9

29

8 SAMPLES WITH ALLISON LABS FILTER;

1 SAMPLE WITH ALLI SON LABS AND B & K

BERING STRAITS

9

24

ALLISON LABS

NORTH PACIFIC

8

65

B&K

SAN DIEGO

(SHIP NOISE

IN HARBOR)

9

36

ALLISON LABS

Actual probability density curves of ambient noise

are shown in figures 3 and 4. The large fluctuations in

some of the traces are caused by substantial variations in

the level of the noise sample. Since some of the curves

appeared to be closely gaussian, the methods used to

measure the parameters of the distribution included over-

lays, calculated moments, and cumulative probability

graphs. Tests of significance and the chi-square "good-

ness of fit" tests were used to determine what values of

skewness and kurtosis were improbable at a 5 or 1 per cent

probability level.

11

0.5

0.4

0.3

0.2

0.1

0.5

0.4

0.3

0.2

0.1

DATA TAKEN IN SHALLOW WATER (So. Calif.

i i 1 r

DATA TAKEN IN BERING STRAITS

-2 -1

AMPLITUDE

1 2

IN STANDARD DEVIATION UNITS

Figure 3. Exampl es of some PD curves taken in

shallow water, compared with a normal curve.

12

Â£ o

^Â°-5

CO

<

CO

o

en

0.4

0.3

0.2

0.1

i r

n 1 r~

-DATA TAKEN IN

SAN DIEGO HARBOR (Ship Noise)

-2-10 12

AMPLITUDE IN STANDARD DEVIATION UNITS

Figure 4. Examples of some PD curves taken

both shallow and deep water, compared with a

no rmal curve.

13

Data Reduction Techniques

OVERLAY METHOD

Since it was expected that the probability density-

curves obtained with the PDA would have a gaussian or

nearly gaussian distribution, an overlay with a gaussian

curve was used. The curve had parameters of a mean

equal to zero and a standard deviation equal to one. Figure

5 illustrates the use of this method with two curves, one

judged to be gaussian and the other non-gaussian. Some

probability density curves obtained with the PDA were

judged to be very nearly gaussian.

One disadvantage of the overlay method is that de-

cisions about how well a particular curve compares with

the overlay are purely subjective. Skewness and kurtosis

can be detected, but the magnitudes of these moments cannot

be estimated with accuracy. An extension of the overlay

method which will allow estimates of skewness and kurtosis

is described here.

The extension is an overlay with several curves in-

stead of just one. Each curve has a different set of values

for skewness and kurtosis. The curves are positioned

over the actual probability density curve and the parameters

are estimated by interpolation between the two closest

curves. The curves of the overlay can be computed with

the use of Edge worth's series approximation for nearly

gaussian distributions. 3 The first four terms of this series

are

fix) = h(x)-â€”h 3 (x)+-^hUx)+â€” h e (x) (3)

where h(x) is the normalized gaussian distribution, h (x)

is the nth derivative of h(x), g is the standardized skew-

ness, and g is the standardized kurtosis.

14

-2-10123

AMPLITUDE IN STANDARD DEVIATION UNITS

Figure 5. Examples of two PD curves which

were det ermined to he gauss i an or non- gaus s i an,

using a normal curve as an overlay.

15

Estimates of the skewness and kurtosis can be found

with the above method; but it does not give any indication of

whether these estimates are significantly different from the

expected values, if the sample is taken from a gaussian

distribution. Using the previous overlay, a method can be

developed so that a sample can be accepted or rejected at

any desired level of probability. Basically the method is

to have two of the curves on the overlay plotted so that they

will represent the maximum deviations allowed in the par-

ticular parameter of a sample with (N) points. The method

will be developed for kurtosis, but a similar method can be

used for skewness.

The variance of kurtosis is given by 4

var(o ) = 24/N (4)

for large TV. This holds for a sample taken from a normal

parent population. The standard deviation of kurtosis is

(24/710 2; if the kurtosis is distributed normally, then from

the ratio of a particular value of kurtosis (g s ') and the

standard deviation we can obtain the probability of getting a

value of kurtosis as large or larger than g ' . The ratio is

(24/70 2

The probability of getting a value of kurtosis as large as or

larger than g ' is given by the amount of area under a

normal curve outside the -7? and +7? standard deviations.

A value of 7? = 1. 96 corresponds to a probability level of

(5)

16

5 per cent, or l/20th the total area. A ratio as large as

1.96 may be considered sufficiently improbable and hence

g ' can be assumed to result from a non-gaussian distribution.

The sample would therefore be rejected as coming from a

gaussian distribution. The value of g s ' therefore depends

onf, g ' - 1.96(24///) s. Edgeworth's series would then

be used to compute two curves, one with -g 3 ' (for negative

kurtosis) and one with +g 3 ' (for positive kurtosis). These

curves would represent the limits, at a 5 per cent

probability level, within which a sample of N points would be

considered as coming from a gaussian distribution.

Figure 6 shows two curves as they would appear in

the overlay. These two curves are the limits for a sample

0.5

g' â– +0.50

-3

g' =-0.50

-2-10 12

AMPLITUDE IN STANDARD DEVIATION UNITS

Figure 6. Overlay indicating g 3 ' of +0.50 and of -0.50,

A curve having a value of kurtosis as large or larger

than these values will be non- gauss i an at a 5 per cent

level for a sample of 3 70 points or, e qui val ent 1 y , a

bandwidth of about 55 c/s.

17

of bandwidth about 55 c/s, with N given by the equation N =

6.7/, where / is the bandwidth. The equation is obtained

o o

from Appendix B, using a time constant T = 2.3 seconds.

The overlay method was not used extensively because

of the complexity that comes from considering different

values of N and also different combinations of skewness and

kurtosis in the same sample. A method using computed

moments of the curves is described next; it was felt that

this method would yield accurate values of the mean,

standard deviation, skewness, and kurtosis.

METHOD OF MOMENTS

The method of moments is basically a general method

of forming estimates of the parameters of a distribution by

means of a set of measured sample values. The first few

moments of the actual distribution are calculated and these

are used as estimates of the moments of the parent population.

On the basis of these moments a suitable theoretical dis-

tribution curve is selected. For any particular distribution

curve the moments are functions of the parameters of that

curve. The parameters are determined and tests of sig-

nificance are made on the skewness and kurtosis.

The moments about the origin are defined as 5

m ' = Â£ p.(x)x. (6)

r v i>

i

where p^(x) is the probability that a value selected at ran-

dom from the population will lie in the i-th class. The

variate x with which we are concerned may be discrete or

continuous.

The moment

m 1 ' - E P t ^ x t (7)

18

is defined as the mean value of x, m ' - x.

Another more important set of moments is obtained

by changing the origin to the arithmetic mean. Equation 8

defines the moments about the mean.

m = Â£ p.(x) (x. - x\ (8)

For computing purposes, the relations between the

m and the Tn ' are convenient. Expressing the m in terms

of the rn ' we have the relations

r

m 1 = m 1 ' (9a)

m 2 = m 3 ' - (ot 1 'f (9b)

m 3 = m 3 ' - 3m a 'm 1 ' + 2(% ') 3 (9c)

ff2 4 = ot 4 ' - 4m 3 , m 1 ' + Qm 3 '{m x 'f - 3(m Â± ') 4 (9d)

Grouping errors are negligible, so Sheppard's

corrections are not applied.

- These moments can be expressed in standard units

by the use of a standardized variable z, by dividing the

variable x by s , the standard deviation.

J x

(x-x)

z -

s

X

(10)

The standardized moments are defined by the equations

772

a = â€” â€” , for r = 1, 2, 3, and 4 (11)

r r

s

x

19

a 2

= 1

m a

a 3

s ;

Â«r

a 4

m 4

s

X

The first four standardized moments are

a-L = (12a)

(12b)

(12c)

(I2d)

The third moment, a 3 , is a measure of the skewness of the

distribution. A positive value indicates a distribution with

a longer positive tail than a negative tail.

The fourth standardized moment, a 4 , is a measure

of the kurtosis of the distribution. In some cases it. is a

measure of the "peakedness" of the distribution, though it

is now understood that the length and size of the tails are

very important in this measurement.

For a normal curve the values of a 3 and a 4 will be

and 3, respectively. We redefine the skewness and kur-

tosis as

Q, r Â« 3 (13a)

g 2 = Â« 4 - s (i3b)

so that g is for a normal curve.

It is not very likely that the third and fourth moments

of a random sample will be zero. Depending on the distri-

bution and on the actual sample values, the third and fourth

moments will have some value different from zero. To de-

termine whether this difference is significant, it is neces-

sary to use the variances of the third and fourth moments. 4

varfo, ) = 6N(N-l)(N-2)- 1 (N-l)- 1 (N-3)- 1 (14a)

20

var(Â£ 3 ) - 24A"(^-l) 2 (^-3)- 1 (^-2)- 1 (^-3)- 1 (#-5)- 1 (14b)

For large N use,

var(^) = 6/N (15a)

var(p a ) = 24/27 . (15b)

The hypothesis to be tested is that the data sample is

taken from a gaussian distribution. To test the hypothesis

compare g to (6/N)z and g 3 to (24/#)s (see ref. 5), then

if

> 1. 96 reject the hypothesis at the 5 per cent level

(6 /N)s

> 2. 57 reject the hypothesis at the 1 per cent level.

Similarly, for g r

if > 1.96 reject the hypothesis at the 5 per cent level

(24/tf)*

> 2. 57 reject the hypothesis at the 1 per cent level.

CHI-SQUARE "GOODNESS OF FIT" TEST

The x 3 test will be applied to the hypothesis that a

sample of N individuals forms a random sample from a

population with a given probability distribution. The param-

eters of a distribution are known and are not estimated

from the sample itself. Later a modification will be given

for the situation where the parameters are estimated from

the sample.

21

The quantity*

(F.-Np. f

L -V- (16)

s

is a measure of the deviation of the sample from the ex-

pectation, where F . is the number of observed frequencies

in the tth interval, and Np . is the number of expected fre-

quencies in the ith interval as predicted by the theoretical

distribution. Karl Pearson proved that the above quantity,

in the limit, is the ordinary \ 2 distribution which is now

tabulated in most statistics books.

Online Library → M. A Calderon → Probability density analysis of ocean ambient and ship noise → online text (page 1 of 3)