crossfix errors of directional acoustic sensors in a ... · npsarchive 1966 dorsey,a. ^:jmm 1...

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CROSSFiX ERROKS OF DIRECTIONALACOliS'fIC SENSORS IN A MEDIUM WITH A CONSTANT

HORIZO'NTAL VELOCITY GRADIENT

ARTMUR G0I1.MAM DOI^SEY, JR.:^

'HUATE SCHOOL

DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOlMONTRREY CA 93S;43-f^101

CROSSFIX EEIRORS OF DIRECTIONAL ACOUSTIC SENSORSIN A MEDIUM WITH A CONSTANT HORIZONTAL

VELOCITY GRADIENT

by

Arthur Gorman Dorsey, Jr.

Lieutenant Commander, ^ited States NavyB.E., Johns Hopkins University, 1953

Submitted in partial fulfillmentfor the degree of

MASTER OF SCIENCE IN OPERATIONS RESEARCH

from the

UNITED STATES NAVAL POSTGRADUATE SCHOOLOctober 1966

^ ABSTRACT

An investigation was made of the magnitude of the bearing and dis-

tance errors which result from two passive directional acoustic sensors

obtaining a crossfix on a target in a medium with a constant horizontal

velocity gradient. Results are tabulated in Appendix III. For the con-

ditions considered, the errors were found to be small in relation to the

target range but became appreciable in absolute value at very large tar-

get ranges. However, with a knowledge of the horizontal velocity gradient,

sensor separation, sensor axis orientation, apparent target range and

apparent target bearing the errors could be predicted. Further, by

proper sensor orientation, the errors could be minimized.

TABLE OF CONTENTS

Section Page

1. Introduction 7

2. Theoretical Background 8

3. The Model 15

4. Results 20

5. Conclusions and Recommendations 44

6. Bibliography 50

7. Appendix I - The Computer Program 51

8. Appendix II - Flow Chart for Program 57

9. Appendix III - Tabulated Results 58

LIST OF TABLES

Table Page

1. Effect of Velocity Gradient (VGEIAD = .0001) 58

2. Effect of Velocity Gradient (VGRAD = .00025) 59

3. Effect of Velocity Gradient (VGRAD = .0005) 60

4. Effect of Sensor Separation (SS = 1 km.) 61





9. Effect of Target Range ( RAPP = 1 km.) 66

10. Effect of Target Range (RAPP = 3 km.) 67

11. Effect of Target Range (RAPP =100 km.) 68

12. Effect of Target Range (RAPP = 1,000 km.) 69

13. Effect of Target Range (RAPP = 2,000 km.) 70

14. Effect of Sensor Axis Orientation (OMEGA = 0°) 71










24. Effects at Reduced Range Equal to 1.25 (OMEGA = 90°) 81

LIST OF TABI£S (CONTINUED)

Table Page

25. Effects at Reduced Range "Equal to .5 (OMEGA = 0°) 82

26. Effects at Reduced Range Equal to .5 (OMEGA = 90°) 83

27. Effects at Reduced Range Equal to .2 (OMEGA =0°) 84

28. Effects at Reduced Range Equal to .2 (OMEGA = 90°) 85

LIST OF ILLUSTRATIONS

Figure Page

1- General Ray Path Geometry 11

2. Constant Gradient Ray Path Geometry 11

3. Computer Program Geometry 16

k. Computer Program Geometry (Sensor Axis Rotated) 17

5. Flow Chart for Computer Program 19

6. Effect of Velocity Gradient 23

7. Effect of Sensor Separation 25

8. Effect of Sensor Separation 26

9a. Effect of Target Range on Absolute Error (Short Range) 28

9b. Effect of Target Range on Relative Error (Short Range) 28

10. Effect of Target Range on Relative Error (Long Range) 29

11. Effect of Sensor Axis Orientation 31

12a. Effect of Rotation of Sensor Axis 32

12b. Effect of Rotation of Sensor Axis 33

13. Actual Target Locations 35

14. Effect of Sensor Separation at Constant Reduced Range 36

15. Relative Errors at Constant Reduced Range 37

16a. Target Bearing Corrections 39

16b. Target Bearing Corrections 40

17a. Effect of Sensor Axis Orientation (Reduced Range = .5) 42

17b. Effect of Sensor Axis Orientation (Reduced Range = -2) 42

18. Effects at Reduced Ranges Less Than 1.0 43

19a. Target Bearing Corrections (Reduced Range = .5) 46

19b. Target Bearing Corrections (Reduced Range = .5) 47

20a. Target Bearing Corrections (Reduced Range = .2) 48

20b. Target Bearing Corrections (Reduced Range = .2) 49

1. Introduction .

Straight-line propagation of sound is to be expected only if the

velocity of propagation is constant throughout the medium. Because of

the heterogeneous nature of the ocean, this condition is usually not met

and marked deviations from straight line propogation have been observed

to occur.

The principle factors that influence the velocity of sound in sea

water and therefore result in refraction of the sound ray are temperature,

salinity, and depth. Variations in salinity are of importance near the

mouths of large rivers vAiere appreciable amounts of fresh water run into

the sea, in the vicinity of large ocean currents such as the Gulf Stream,

and in water close to the surface where rain and evaporation will have a

maximum effect. However, by far the most important factors influencing

variations in the speed of sound are differences in temperature and depth

of the points through which the sound ray passes.

A considerable amount of work has been done investigating the re-

fraction of sound rays caused by variations of temperature with depth.

The aim of these studies has been to increase the detection capability

of sensors by placing them in such a manner that "shadow zones" are either

eliminated or caused to be as far from the sensors as possible. Further,

the studies have investigated ways to make optimum use of sound "chan-

nels" caused by having a layer of water in which the sound velocity is

lower than that both above and below it.

However, once a detection has been made by two directional acoustic

sensors and the resulting fix plotted, very little has been done to de-

termine by how much this apparent fix is displaced from the actual target

location due to horizontal temperature gradients. Although these gradients

are normally small in relation to the vertical ones, there are certain

areas near ocean currents, such as the Gulf Stream, where they do become

appreciable and thus could possibly cause significant bearing errors in

crossfixes. Further, as the range at which sensors can detect a target

is increased through technical improvements and making use of "channel-

ing", bearing errors could become significant.

One investigation of the fixing errors which could result from hori-

zontal temperature gradients considered the case where there was a sharp

vertical boundary between two bodies of water having different sound

propagation velocities. 6 The next logical condition to consider is a

constant velocity gradient. Therefore an investigation of the effect of

a constant gradient is presented in the ensuing sections of this thesis.

2. Theoretical Background .

Sound energy takes the form of disturbances of the pressure and den-

sity of some medium. Therefore, the basic relationships between impressed

forces and the resulting changes in pressure and density are applicable

to the theory of sound propagation in water. However, the solution of

the differential equations applicable to pressure transmission phenomena

requires a knowledge of certain initial and boundary conditions. This

knowledge is almost always lacking for sound transmission in sea water.

Fortunately, it is possible to discuss some of the important characteris-

tics of sound propagation in sea water without considering the frequency

and wave structure of the sound. For such discussions the wave can be

thought of as a beam or ray, and the discussion is concerned only with

the geometry of the situation.

The ray concept is based on the assumption that sound energy is

transmitted along curved paths, called rays, which are straight lines in

all parts of the medium where the velocity of sound is a constant, and

which curve according to certain rules in parts where the velocity of

sound is changing. However, in making use of the ray path approach to

the solution of sound propagation problems it must be kept in mind that

certain assumptions are implicit in this technique- If the conditions

imposed by these assumptions do not apply, the use of the method is not

valid. These assumptions are:

(a) The direction of the ray must not change much over a distance

of one wave length; that is, the radius of curvature of the rays must

be large in relation to the wavelength.

(b) The velocity of sound must not change appreciably over the dis-

tance of one wavelength. 7

The simplest continuous velocity gradient which can be considered

is one which is a constant. This particular gradient has the interesting

property of resulting in a sound ray path which is the arc of a circle.

This fact is easily shown by a consideration of the following:

Assume that the velocity of sound, V, at a point (x, y), is a linear

function of the distance of the point from the origin in the x direction;

i.e. V = V + gx, where V and V are the velocities of sound at someP o p o

point P and the origin respectively, and g is the horizontal velocity

gradient, a constant. Figure 1 illustrates the situation under consider-

ation.

In the diagram the velocity gradient is in the horizontal (x) direc-

tion. The gradient at any point P with coordinates (x, y) is, by

definition,

dx

By integrating this expression and specifying that the x distances shall

be measured from a vertical line through the points where the velocity

is zero it is possible to avoid the introduction of a constant of inte-

gration and obtain the expression

(2-1) X = JL.g

Then, recalling Snell's Law

where 9 is the angle measured from the boundary of the medium (an iso-

velocity line in this case) to the ray under consideration and K is the

Snell's Law constant for this ray, equations (2-1) and (2-2) can be com-

bined by eliminating V to obtain

(2-3) X = -|- cos 9g

Differentiating this expression yields

(2-4) dx = — sin 9 d 9g

But, from elementary analyticel geometry dx = tan 9 dy (note that in this

situation angles are measured clockwise from the y axis), thus (2-4) be-

comes

(2-5) dy = — cos 9 d 9g

Now integrating (2-5) and specifying that vertical distances, y,

are to be measured from the horizontal line through the point at which

9=0 yields

(2-6) y = — sin 9g

By squaring (2-3) and (2-6), then adding them, the following result

10

= g

=

Figure 1. General Ray Path Geometry

(0,0)

Figure 2« Constant Gradient Ray Path Geometry

11

is obtained:

(2-7) x^ + y2 = (_K_)2

g

This is easily recognized as the equation of a circle of radius .

g

The circle is centered at the origin specified in the foregoing paragraphs

as being the intersection of a horizontal line through the point where

9=0 and a vertical line through the points where V = 0.

Since the Snell's Law constant, K, at any point is the ratio of the

velocity of sound at that point to the cosine of the angle of inclination

of the ray at that point, it is readily seen that the radius of the circle

Vis actually £-

, i.e. the ratio of the velocity of sound at that" g cos 9p '

point to the gradient times the cosine of the angle of inclination of the

ray at that point. Thus, if the velocity and angle of inclination of the

ray are known at some point, the coordinates and angle of inclination of

the ray are completely specified for the total path of the ray.

By making use of the fact that a constant velocity gradient results

in rays that are arcs of circles some useful relationships can be ob-

tained for tracing the path of a sound ray through the water. Consider

a circle of radius R with the points (x, , y,) and (x2, y^) as in Figure

2. By geometry,

X, = R - R cos e-j^ = R(l - cos 9,)

X2 = R - R cos 9 = R(l - cos 92)

Therefore

(2-8) AX = X2 - x-j^ = R(l - cos 92) - R(l - cos 6^^)

= R(cos 01 - cos 02)

Now, for a linear velocity gradient, g, the velocity, V2 , at a point,

(x , y^) , may be found from the following relationship:

12

V^ = V^ + gAx

where V is the velocity at the point, (x , y,)-

Thus

(2-9) AX =

g

Equating (2-8) and (2-9) yields

(2-10)

Solving this expression for R,

(2-11) R =

R(cos 9 - cos Q^) =

V2 - Vi

V - V^2 ^1

But, from Snell's Law,

g(cos 9-, - cos 62)

Vi(2-12) cos en = h- cos 9

V2

Substituting (2-12) into (2-11) yields

(2-13) R =V2 - Vi V2 - V^

1 1

g(^p- cos 92 - cos 02) S ^°^ ^2 ^v~- 1)

Similarly.

But, from (2-8)

V2 - V,

^1 - Vog COS e^ (^^) g cos 02

(2-14) AY = 72 - Yi = R ^in ©2 - R sin 9, = R(sin e.-sin 9 )

= R 2 sin i (02 - 0^ cos j (9^ + 9^)]

R =AX

COS 9 - cos 9

13

Thus, substituting this into (2-14)

y = ^ cos e.'^-'cos 9, > b ''^ ¥'2 - «l) -^ ¥^2 ^ V]

= r A^ I

I -2 sin i(e, + e„) sin 1(9, - 9J J

2 1 2 2 J- 2

[2 sin i (92 - 9^) cos i (02 + 9^)]

(0

^ AX [-sin J (9^ - 92) cos j (9^ + 92)]

[-sin i(9^ + 92) sin 1(9^ - e^)]

Simplifying this expression by means of a trigonometric identity yields

^1 + ®?(2-15) y = Ax cot (

—

i-)

Equation (2-15) is the basic equation used in the model described in

section 3 which is used to trace the sound rays through the water.

It should be noted that the minimum radius of the circle, as compu-

ted from equation (2-13), describing the path of a sound ray is obtained

when cos 9^ and the gradient, g, are both a maximum. This means that for

the model described in section 3 the minimum radius will occur when 9 = 0*^

and the velocity gradient is 0.0005 meter/sec per meter. This radius will

be — = 3 X 10° meters. Since the wavelength of a 200 cycle/sec

signal (approximately the frequency of a low frequency noise source on a

medium speed submarine \j^\) is about 7.5 meters it is readily seen that

the conditions of assumption (a), that the radius of curvature must be

large in relation to the wavelength, are met and the use of the ray path

method is valid. Further, the conditions of assumption (b) , that the

velocity of sound does not change appreciably over one wavelength, are

fulfilled since the velocity only changes by 7.5 x .0005 = 2.5 x 10"^ =^1500

14

.000257o in a distance of one wavelength.

3. The Model .

In modeling the problem under investigation the following assumptions

were made:

a. A detection had been made by two directional acoustic sensors and

the apparent fix plotted.

b- There was no change in horizontal gradient with depth, therefore

the amount of bending of the ray was the same at any depth and the prob-

lem could be reduced to one of finding the ray path in a horizontal plane

only. (It is realized that the horizontal gradient does, in actuality,

change slightly with depth, thus all results are approximations. How-

ever, to treat the problem in three dimensions would require an amount of

time exceeding that available for the conduct of the present investiga-

tion.)

c. The velocity gradient was a constant in one direction and zero

in a direction perpendicular to this one. This allowed the use of the

circle approximation to the path of the sound rays.

d. The directional acoustic sensor had no bearing error. To add

this to the problem would obscure the effect of the temperature gradient.

Figure 3 is an illustration of the situation studied. An apparent

target was located at XAPP , YAPP, the apparent fix obtained from sensors

A and B located at XI, Yl , and XXI, YYl respectively. The bearings of

the apparent target from sensors A and B were BRGA and BRGB respectively.

To relate the fix to some common reference point, an "origin" was chosen

at the midpoint of a line joining the two sensors. The bearing and range

of the apparent target from this point were BRGP and RAPP. However, due

to the horizontal velocity gradient, some bending of the sound rays

15

BRGAXI,Yl

BRGBXXI,YYl

BRGT

ERROR

XACT, YACT

Flgura 3* Computer Program Geometry

16

y

OMEGA

Sensor Axis

'^^~l OMEGA J/ TAU J/ BRGT P\

• /XAPP, YAPP

0)/ y/' \ ERR

/ // \/ /V/ ^^'^^^vm:,'^

/ /3^^^/ z:^^^^^^

/•

ALPHA

~r^v^^^/x^ /BRGA

XI, Yl

GAMMA

BRGGPHI

BRGP

^-S.^^^ /<C /^DELTA

•

BETA /y^^^-^---^

brgbT/ """^""^"^---..^^^^

XXI,YYY « ^

Figure 4. Computer Program Geometry (Sensor Axis Rotated)

17

occurred and the actual target was located at XACT, YACT. The bearing

and range of the actual target from the origin were BRGG and RACT. The

distance from the apparent target to the actual target was the ERROR and

the bearing of the actual target from the apparent target was called BRGT.

RNGERR was the difference between the actual and apparent ranges.

If the sensors had always been located along an isovelocity line,

Figure 3, with the bearings and distances shown, would be sufficient to

illustrate the geometry of the problem. However, the inclination, by an

angle OMEGA, of an axis through the two sensors with respect to an iso-

velocity line resulted in a need to define angles with respect to an

isovelocity line. By selecting the x and y axes perpendicular to and

along an isovelocity line respectively it was possible to make the best

use of the fact that the velocity gradient was a constant in one direc-

tion and zero in the other. These angles measured with respect to the

isovelocity line were assigned Greek names corresponding to the letter

appearing after the prefix BRG; i.e. BRGA, the bearing with respect to an

axis through the two sensors, became ALPHA, the bearing with respect to

an isovelocity line. The Greek letter bearings equal OMEGA plus the

appropriate BRG. Figure 4 illustrates the problem when the sensor axis

is rotated with respect to the isovelocity line.

Making use of the relationships developed in section 2 relating the

sound ray path to the arc of a circle, a computer program was written to

trace the actual path of the sound ray through the water. Thus, given

initial bearings of the apparent target from two sensors, it calculated

the actual target position from the point of intersection of the actual

ray paths. The program is included in Appendix I.

The "mechanics" of the program are illustrated in Figure 5, a flow

START

I. —^ READ INPui UAiA

oc «» n SS^ _ STOP

COMPUTESENSOR POSITION

' '

PRINTINPUT DATA, OUTPUT TITLES

U

SET APPARENT TGT. POSIT.

V

SET STARTING VELOCITY& COORDINATES OF RAYS.COMPUTE INTERSECTION PT.OF RAYS

U

COMPUTE ERRORSAND

PRINT RESULTS

V

INCREMENT BRGPB RGP < 350°

'

BRGP > 35)0°

TNCREMRNf RAPP

RAPP> ]RAPPMA.K. RAPP ^ RAPPMAX

BRGP = Apparent Target BearingRAPP = Apparent Target RangeRAPPMAX = Maximum RAPP to be ConsideredSS = Sensor Separation

Figure 5. Flow Chart for Program

19

chart for the program. For the reader interested in greater detail a more

detailed block diagram may be found in Appendix II. The Roman numerals

in the blocks of Appendix II correspond to the numerals in the left hand

margin of the program in Appendix I.

For a given velocity gradient and sensor separation an apparent tar-

get position was assumed. From this position the apparent bearings and

distance of the target from the origin and the sensors was calculated.

Then the actual position of the target was determined by calculating the

actual ray paths through the water and computing their point of intersec-

tion. Finally, the errors were calculated. A new apparent target posi-

tion was then assumed and the calculations repeated for this new position.

Apparent target positions were assumed for each 10 degrees of bearing

from an axis through the sensors (except for the and 180 degree posi-

tions where no fix is possible) and for various apparent ranges to a

maximum range of five times the sensor separation. Further, as stated

earlier, the angle of inclination of the axis through the sensors to the

isovelocity line was varied in 10 degree increments from to 90 degrees

for selected sensor separation and apparent range combinations.

4. Results .

To evolve a means for estimating the crossfix errors resulting from

horizontal temperature gradients, the effect of the following parameters

on the errors was studied using the model described in section 3:

(a) target bearing

(b) magnitude of velocity gradient

(c) sensor separation

(d) target apparent range

(e) sensor axis orientation

20

The range of values over which the parameters were varied was as

follows

:

(a) target bearing (BRGP) : 10° increments from 10° to 350° relative

to the sensor axis except for 180° where no fix was possible due to target

alignment with the sensor axis.

b) velocity gradient (VGRAD): .0001 to .0005 meter/second per meter.

These were found to be the approximate maximum velocity gradients which

exist over long (about 100 km.) distances in the deep sound channel and

over short (about 40 km.) distances near the surface at the Gulf Stream.

(c) sensor separation (SS) : 1 km. to 400 km. These were considered

to be the minimum and maximum feasible for obtaining a crossfix.

(d) target apparent range (RAPP) : 1 km. to 2,000 km. but never less

than .1 nor more than 5 times the sensor separation since beyond these

values the target bearings from the sensors become almost equal.

(e) sensor axis orientation (OMEGA): sensor axis 0° to 90° from the

isovelocity line in 10° increments.

As might be expected, the errors were found to be small in relation to

the target distance from the sensors but to be appreciable in absolute

value for targets at long ranges from the sensors or in areas where a

large velocity gradient existed. It was found that the errors were pre-

dictable and that they could be minimized by proper sensor orientation.

The results are summarized in sections 4.1 through 4.8 and depicted

graphically in Figures 6 through 20. Although all of the computer out-

put data used in plotting the graphs for the figures is not included,

samples of the data which illustrate particular effects are included in

Appendix III. Also, it should be noted that the abbreviations used in

21

the figures are defined at the beginning of the computer program in Appen-

dix I.

4.

1

Effect of Target Bearing .

Figure 6 illustrates the influence of target bearing (BRGP) on the

magnitude of the crossfix errors. As the figure and the data in Tables

1, 2 and 3 show, for a given velocity gradient condition the error was

very much influenced by target bearing from the sensor axis when the sen-

sor axis was oriented along an isovelocity line. However, when the sen-

sor axis was at 90° to the isovelocity line, i.e. along the velocity

gradient line where OMEGA = 90, target bearing had almost no effect on the

error. The influence of sensor axis orientation is discussed further in

section 4.5. For the case where the sensor axis was along an isovelocity

line the errors were seen to increase rapidly as the target bearings ap-

proached the sensor axis. As discussed in section 2, this was due to the

fact that the radius of curvature of the sound ray approaches its minimum

value as the angle which the ray makes with the isovelocity line approaches

zero.

4.2 Effect of Velocity Gradient .

Figure 6 also illustrates the influence of velocity gradient (VGRAD)

on the magnitude of the cross-fix errors. As the figure and the data in

Tables 1, 2 and 3 show, the errors were a linear function of the velocity

gradient, i.e. the velocity gradient of .0005 m/sec per meter resulted in

errors 5 times larger than those for a gradient of .0001. Although Fig-

ure 6 depicts the particular case of a 2 km. sensor separation and 10 km.

target range, the effect which it illustrates was found to be the same

for all sensor separation and target range combinations examined. The

plot for the 90° rotation of the sensor axis from an isovelocity line

22

200

^

Q

»

160 -

120 -

80

^0

OMEGA =

40, ,

1

1.

""

. III 1 j1

i

'

1

.....,_ .,

hr1 i XCL 4 -1 1 i

i

i ^ 1 11 M 4 1 :

i

1I _I^!_L _.

T 1 J 1 J^ 1 1

^ ij f- -!1

\

4L^J L .' . ! ...L ! 1

1!^1 ~i ;

:y~r'' ':'-'-^--' :

1 1' .-^_U_ —

i

—'—'

'

'

''

\

I ..

ij-l-i.S.^ 1 l._i.".L;. ... .

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1 1 1 -'J LL .. ' • L' •'

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:-"^"i

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"*i

90 180 270

OMEGA = 90

360 BRGP

VGRAD = .0005 Reference:VGRAD = .00025 Tables 1,2,3VGRAD = .0001

SS = 2 km, RAPP = 10 km.

Figure 6. Effect of Velocity Gradient

23

illustrates the linear relationship between velocity gradient and cross-

fix errors even more clearly than the plot for the 0° rotation of the

sensor axis.

4. 3 Effect of Sensor Separation .

The effect of sensor separation (SS) on crossfix errors is not as

simply described as the effect of velocity and target bearing. As shown

in Figures 7 and 8 and by the data in Tables 4, 5, 6, 7, and 8 it was

found that for any given target range, sensor separation had much less

effect on the errors than target bearing. Also, the minimum error due to

target bearing generally increased and the maximum error generally de-

creased as the sensor separation was increased. However, when the sensor

separation was twice the apparent target range the maximum error due to

target bearing was at a minimum value and the minimum error was at a max-

imum value, thus there was the least dependence of error on target bear-

ing. When the sensor separation was increased to values more than twice

the apparent target range the minimum error continued to increase and the

maximum error now increased also. The effect of sensor separation on

crossfix errors when the sensor separation was greater than twice the

target range is discussed further in section 4.8.

An examination of the data in the figures and the tables revealed

no simple relationship between the magnitude of the error and the sensor

separation per se. However, as will be discussed in section 4.6, it was

possible to formulate an estimating relationship for errors based on the

ratio of target range to sensor separation. Also, the large variation in

error for the case where the sensors were 10 km. apart and oriented 90°

to the isovelocity line will be discussed in section 4-8.

24

50

Q

40 -

30

20

10

OMEGA =

10 T

1

^^:: -.:

_L_.w ' ' '- -!-i-|-f x-fx

1 -^ -"TT;

rV^.-1 h<K - -

t

-'/' ..^^1 1 '

7^-^L1

—rp^-^'^.

1^ "•

<1 1 ^^

> VU-4- y !T .

j"'U'" L"T" '\

/ -vi/ 4 L 1 '

. .,j.

!'A-

1 .'1

,- i 1 / 1

1 ''

«' 1 1

90 180

OMEGA = 90

270 360 BRGP

SS = 1 km,

SS = 5 km.— SS = 10 km.

RAPP = 5 km. , VGRAD = .0005

Reference

:

Tables 4,5,6

Figvire 7. Effect of Sensor Separation

25

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! m\w ' ' ^r- -+-j-^ ^^"^ \^ ! ] '

-

i

^^ ' ^ •i

' . ; i '

•

!

^^^.=s>î

1 1 M ,^^~—^'^:

,1 ;

,

1i

'! M i

i

: M i i

H .M1

1 n 1-1 i-;^- |M-

i 1 M1

OMEGA =

20 r

90 180

OÊGA = 90

270

f

P(»fPTPnop •

MM» •—" -— SS 400 km. Tables 7,8RAPP = 500 km., VGRAD = .0001

360 BRGP

Figure 8. Effect of Sensor Separation

26

4.4 Effect of Target Range .

Figures 9 and 10 show the dependence of crossfix errors on target

range (RAPP) . Samples of the data used to plot the graphs may be found

in Tables 4, 8-12, and 13. Figure 9a shows the absolute error as a func-

tion of target bearing for three target ranges and a sensor separation of

1 km. Figure 9b shows the relative error (the ratio of the error to the

target range) for the same target and sensor positions. Figure 10 shows

the relative error as a function of target bearing for different sensor

separation and velocity gradient conditions.

In examining the data it may be noted that there was no simple

linear relationship which related the absolute error at one target range

to another, thus interpolation is difficult. However, by plotting rela-

tive error versus target bearing, as in Figure 9b, it may be seen that

the relative error was a linear function of target range.

As Figure 10 illustrates, when the sensors were on an isovelocity

line, this relationship was found to be generally true for extreme sen-

sor separations and target ranges also. However, for target ranges less

than one-half the sensor separation and for target bearings less than 45°

from the sensor axis the relationship did not hold. As was mentioned in

section 4.3, the case when the target was less than one-half the sensor

separation will be discussed in section 4.8. At target bearings less than

45° from the sensor axis there was less than a 10° difference between the

target bearings from the two sensors, thus it is unlikely that a "fix"

would be considered from these two bearings due to the large errors which

would be induced by small differences in the measurement of either bearing.

When the sensor axis was at 90° to the isovelocity line, the relative er-

ror was a linear function of target range for all target bearings, but as

27

60

COOS

Q

»

40 r

20

1

l_ l_l.' . .

- -"- - ~;i—J— !.- ._L.T i-

1 ;'

LZTi..-i \.. .

1 11

i 1 1

t 1

1 J_1

1

] 1 1

1

'' ~4^I|-

t, 1 ' 1

*1

i

^1 i

"

1 V \

'

. ..;,:. _J. - —1 '•

. _ .._..

-W-~ /- -- \ - -

-t-^-i

1 \ /' A 11 1

1 1

1 :\

III ]1 j—

1—V -f-J--(-i-^-i-

1 1In 1 i

' l 1 1\

'i 1 r i i

i : i

' 1

fli-i:-''/

i V ' ' I—I—

'

/// r {

\ 'Ti- ~\-\--\-

/ / » !\ . __ 1: I 1 -.,/ ! / t * 1 \ [

t^ri ! 1

1 v-^- V-J i-J - >^^ ^^''

!>:_,i_.

^''-?_—_ — —'

i

90 180 270 360 BRGP

Figvire 9a. Effect of Target Range onAbsolute Error (Short Range)

1.2

CMIO

06

270 360 BRGP

Figure 9b. Effect of Target Range onRelative Error (Short Range)

RAP? = 5 km.

RAPP = 3 km.. RAPP = 1 km.

SS = 1 km. , VGRAD = .0005, OMEGA =

References

:

Tables 4,9,10

28

9

.6 .

.4

.2 -

!' 111 '

; !

" rVj ;'

I 4 U \ t :

.

1 1 !

-+- r- r--t-r^.. , .1

\! 1

1 . 1.\ \ \

\

'\ ;-

!

' '

1

1

~Hij

'\\.|,. „^-. .

±ix::ii:t:"

'

j i1

'1

i:. 1 -

1 1

1

1 ! 1 1 X-XL1 Mm _:.,

'!

' 1 5r!- 1 M ! 1

1

i !1

':

1

' 'Ml 11

[

Ill II 1' ^ ! 1 i

Mil;

1 '

\

ill' 11

'i t L j

1 1 1i '

'

1 M 11 11 1

1j

1 ! 1 4- iirI

'

1

'

!

+; :_ii±-. I11;! 1 M

\ 1 1

I

i 1 ' i

i

1 |.'

\ 1 i1

1 j / i1 Ill 1 1

i /-I\ 1 _^ U- / ^-L_' I ' 1

ii

11

i 1

/ t 1 '

1'

1

'

1 i , /

\ '

i'

ii /U -:. I

' 1! ! 1 —1-ij- 1

1' \ .i '

! 1^ '-^

1

\\ 'Ml 1

\ '

i'

1 i

'

1

\' ''

'1

' i'

,

' \ ' '1 1 ! 1 1 M 1

1 1

\ \ ;i ^ i

i^y /4-.ji \ '

'

i

''

1 !

\ A 1 1

\ ; . :

^^ \: ;1 1

~i1 '/ i:

1

\ \1 1 i #

Iv \ '1 / /I r \ M '

'

1 / ^U J

'VA ' '

V^ 7 7 i\A, M 1

1'

I' / i 'i

1 ' ' i 1 /\ _ \ / 1 /

/ J * ',v ^ y t /

'•k \ JS»^L ' 'i > ^ x 1 /* VU- \ -y^ >

1

i

>.'•! ' I

,V S

-<^-<^4t-' -V.

^: _4-^^_„ ,.

. 1

1 i

OMEGA =

—

r

—r- !1

. -ff i j

1M

. 1

Ml1

1

1 1 .

L_ Mi'MM '11]1 '

i M i

i

'

1

1

11

•1

1 ! 1 I 4-M 1

1

!

t1

" ^

1

1

'

! 1

! 1' '

\ \

,1

: li i

1'

.i-'-*ir

i'

~*~MH^I^Bf~

-*^-. t-^x •*«•1 1

i—^-9m -.^^H

*.»H^

90 180

OMEGA = 90

270 360 BRGP

RAPP = 2,000 km.

RAPP = 1,000 km.

RAPP = 500 km.

RAPP = 100 km.

Reference:Tables 8,11,12,13

SS = 400 km., VGRAD = .0001

Figure 10, Effect of Target Range onRelative Error (Long Range)

29

in the case v^en the sensor axis was along an isovelocity line, when the

target range was less than one-half the sensor separation the linear re-

lationship did not exist.

4.5 Effect of Pensor Axis Orientation .

As was noted in Figures 7, 8 and 10, the orientation (OMEGA) of the

sensor axis with respect to the velocity gradient line had a very large

effect on the magnitude of the errors. In general, errors were minimized

when the sensor axis was aligned with the velocity gradient line. Fur-

ther, the errors were practically constant for all target bearings when

the sensor axis was aligned with the velocity gradient line. The only

exceptions to this occurred when the target range was less than one-half

the sensor separation and this case is discussed in section 4.8.

Figures 11, 12 and 13 illustrate the effect of rotation of the sen-

sor axis from alignment with an isovelocity line (OMEGA equals 0°) to

alignment with the velocity gradient line (OMEGA equals 96). The data

used to construct the graphs may be found in Tables 14 through 23 of Ap-

pendix III.

Figure 11 is a histogram showing the maximum and minimum errors, de-

pending upon target bearing, for targets at an apparent range of 6 km.

from sensors 2 km. apart. As the graph shows, by rotating the sensor

axis from 0° to 90° the maximum errors were reduced by a factor of 10 to

a value approximately equal to the minimum error. Figures 12a and 12b

further illustrate the changes in the errors brought about by rotating

the sensor axis.

Figure 13 depicts the actual and apparent target locations when the

sensor axis was aligned with an isovelocity line and when the sensor axis

was aligned with the velocity gradient line. It should be noted that the

graph is not to an absolute scale, since on such a scale even the maximum

30

^

O

H

**^ J 1 -^"r-J-:.-! ^"1

1 i

J ^_ii p -1

..: . .

J J ' i

1 1 1 1 ^ "'1 H

1

"'\ '^ 1 • '

1 1 4 _,^ -• ._ - U_ <

j•' ! •

'

^ 1 1

1' ' 1

j

1

1 — _i\

i

r^c\-' ^ "

i1 1

1 1\

'•

00 V II!

' '

'1 ;

'

1' '

1

I I 1

j1

j1 .4 }_

11 1

1

. u.... 11- -^ ^_ _|- t_4 ,

1'

1—'—' 1—' —

*

hjr\ !

1

1 1;

1 1 ' 'X . . X - X_J_ 1 ;

1 i

1"

'1

I -P i-1 ' ;

1'

!

'! 1

\ 1 ^ ^^| 1 ; 1

'^ '_|_J_ u _J L.._ 1 J i T '

i

i ,

'1

.1

1 i

1

1

i 1

1 I'

]i

1

i 'n- 1 r

1 1 "l! 1 1

i'

1,

1

1

Exi: -zM: d4 :5=t-r4-L^ '

< 1 ;

1

1

4+ -H+-ii: I X J. ._. ._ ;- ......

-1 ;

20J... . U - _|- -|_U - _ul -j_i_^_ ._i_.L

i^-^L^.

^ •

ii H / : ri .

' . . . • ''

MM1

1 11

"

. _i .__ ^. ,. _ 1 1 1

j J; :

.' - 11.: ...

i-j

—

[^• • k I

'; '

!

., .1.,, J

n -^-fF+fft-=4^- .T _ri'\ ; , 1 1 1

!1

1i

J

i1

\-i

30 60 90

OMEGA - DEGREES

SS = 2 km., RAPP = 6 km.

VGRAD = .0005Reference

:

Tables 14-23

Figure 11. Effect of Sensor Axis Orientation

31

sH

Q

H

60

40

20

60

40

20

\

'- -r--1

[ 1M

- 4 U -^ - -- --1 - - |- -- 4-jj

!-

- - -----i T^. _.. .' _._. i. . .M. |. . -|i''

:||:::=:::::z:::::: :::::::::::::_:::-:

1_i V /' 1 ' Mill:TEF:-i:: ::4::id / ''-\^_^ -

"^--M

"

!['

|fIfffî]/ii \p|fî^| ; ::

IlA^^^4|::-y;4::;::j\^ ::;::; ]/.I:::

""'''31"^-1,1 i -tl --, -f-

liJ f^ ^' i

OMEGA =

"p-L - -- -p- -'P-li -P—i--Lp] ±Ltt—L^_ ij^p ..

11

i

•

' / 1 !

1"

1

11 M

1! . ' 1

'

!-^': ::E}:3+ ::-.:::: |;': : :.__;:::-::I{: :;4,::r- ,;/!

J

:| :t :f j ::i .J:|H- J:-.. ,1 Wm,M' :" / :

w'^M.î^^:\.[.,4^

Jy,^^

I-.,

j ^

j4| .

,^ ._| 4. .^ .__ 1^.^1

_ .|^... -rjipi-^

-x'

90 180 270 360 BRGP

OMEGA. = 30

SS = 2 km., RAPP = 6 km. Reference

:

VGRAD = .0005 Tables 14-23

Figure 12a. Effect of Rotation of Sensor Axis

32

40 ^

20 -

1 . . - ---r TT 1 - 1 :i

1 !-;

r : 1M1

:

/;

11

" ' "

! : /;

1/

'

]

ill . ._._.-...

i \

^ -1

- I

1 \ i J ' 11 ,X<^.^J V

1 _..L J., !

_ rr— "fTi'

L i i'Tl - ;- n iTTri rij-rri-i irrrnM 1 ''

,

OMEGA = 60

CO

E-i

§

20

1 1 11 L 1

! i 1

1 . .. 1 1,

_, .

^

:.:i;::[I:|:ffi'::::P^^ .4- 1;-

1 I -i^^

—

/h~"f "1

.- J- ---]\j

^Filitriii:::

\ '/ I /— -[-'- -^^^^- ---1--\ — J^r ' " ' '

'\ ',

.' V ij i "^

1^1$:? :^:i\ji+ _^- T _l__ I

^-^T; i

1^

jj j:i :[: :_^.i

'•

'1" ^ 1,K[ 1 .

1 M 1., < r 1 —

U

OMEGA = 70

»20 "

1 1 i 11

_j J 1 .

\.\

/iM 1 J 4i;-

" 1 r. L tt-^- 1 1 X]\

lJJ-: 1

] : _.t:, ":...t/l..

1

1

1 1 J ol -ttp -! l/-^r r' "1

i"" ~^'

T i 1/f ' ^n^ 'v/lT

1

1

_ 1

- \-'\-v _'

1.!

'' »'-^^ 1\

i [! ^^>.>^ I 1 i 1

\! ,1 j.xvr^j- 1

1

Vv'

'

' -^ ^v Jf^^ii'l

1

~^1

. J4-V->^ .Ix,.!.—I'll. i

' ''

1

1 1 i~ t ^'

r.^ i" •r \

OMEGA = 80

201

t,t;:

1

1

1 -71 j_ t_

1 1

1 1--

i

iTl-i ;

: !

1 i1

— ^ -

1!•.

][

:

1i

1 !

! I 1 . .

!'

1' '

\

.

II1 1 !

:

iM 1 11

1

V_ 1 r"

-'tl '\ 1" r-\ — — j-.

.J '1 _L_ I

1 "1

1 ^.! [ 1 ' 1\

90 180 270 360 BRGP

OMEGA = 90

SS = 2 km., RAPP = 6 km. Reference:VGRAD = .0005 Tables 14-23

Figure 12b. Effect of Rotation of Sensor Axis

33

error of 68 m. in 6000 m. could not be seen. However, the relative magni-

tudes of the errors are correct. As Figure 13 and the data in Tables 14

through 23 show, the actual target was closer than the apparent target

when the sensors were in a lower velocity medium than the target and fur-

ther away when the sensors were in a higher velocity medium. When the

sensors were oriented along the velocity gradient line, however, this re-

lationship was reversed.

4. 6 Comparison at Constant Reduced Range .

Reduced range (REDRNG) was defined to be the ratio of the apparent

target range to the sensor separation. This term was introduced to facil-

itate estimation of errors for any sensor separation and target range when

the errors at a different sensor separation and target range are known.

Figure 14 illustrates the linear relationship between relative errors at

one sensor separation and those at another when compared at the same re-

duced range.

Figure 15 depicts relative error as a function of reduced range for

various target bearings and a velocity gradient of .0005 m/sec per meter.

As an example of the usefulness of such a plot, consider the following:

For a target located at a range of 5 km. on a 30° bearing from the

midpoint of a line connecting two sensors 1 km. apart. Figure 15 shows a

relative error of about .003 (REDRNG = 5., BRGP = 30). For a target lo-

cated at a range of 500 km. on a 30° bearing from the midpoint of a line

connecting two sensors 100 km. apart (at a reduced range of 5.) a predic-

ted relative error would therefore be .3 (.003 times 100) for a gradient

of .0005. Thus, using the fact (from section 4.2) that relative errors

vary linearly with velocity gradient, for a gradient of .0001 a predicted

relative error would be .06. The absolute error would then be estimated

34

SS = 2 km. , RAPP = 6 km.

VGRAD = .0005Reference:Tables 14 & 23

Figvire 13. Actual Target Locations

35

.05

OMEGA =

.01I'll

-r-t-r-l- _. „4.j_.,;„_ till1 1

1 1__'

'1

...U.--i-"*"",

•

! 1

1 '—H-—! 1

—f-j

--f

. — -.1—1—1—^.X . 1 1

! 1'

i !—J—._

j1 1 i_^

—1— '

—

'—f

—

1 ! i i

_!_J_

. 1 -4-1—

^

:H-+-I-

! 1 ! ; .. '1 I'II

\ 1 \

' ' '

Lb s B k B—K, C ..-_L,_X- •» » t-^

~~~ -

- r. r- 1i^=-r

9Q 180

OMEGA = 90

270 360 BRGP

— SS = 20 km.— - SS = 10 km.— SS = 5 km.- - SS = 2 km.

SS = 1 km.

REDRNG =5.0VGRAD = .00025

Figure 14. Effect of Sensor Separation atConstant Reduced Range

36

10

COI

o

s

8 -

4 -

'1 1

1 1 1 ! 1 1 1

^r

''..'. -{44-

^t^F'' '

1 1

^. ...f 4 ^ -|.-p-

- - -

1 1' 1 1

11

.. _L_L -r/p|-- -|-

1

p- -| [-L -!

i1l^ . ^_ ' -,_i_L^... -

1'

'^y^' '

1

^ 1 I 'L/t'

1i>r

[

t

/^1-~|- -^- t "1 r^ II .1 1

"i/i 1

i1 1

11

1 !1 '

' / -J IJ_L4_1

4~i y 1 L '_4_,1 X 1

i

1 1^^ ^_J ^ 'r J-><^li4- '

i

'1-f-^ 4 h n ry 1 rrr- 1

I

1[

i'

i

1! 1 1 ; .

1 1 l/^1

1ill

J - +- -k/p -{ rh1 1

j ,

! i P 1

~^~^ n 7/^11 ^' i

i

! ! _._ 1 . L/f J 1 . __, L.Li

'

'1 j/^

1

\ f1

1 _1

'' X ' '

'

H-lriN-iF-"!

iX I 1 1 1

lUl-ir^'x1 f

-^y1

r -] M- '+" ^ rij

Ll 4 1

' '

'

I'M u —J V^-r -j H" ' ""- " " ^''^'- ^ ^- ^ ^-^-- 1 i ^,^^^

_;.l._|jr-_L rl ^^1

r h-^''^ i' ' 1 1'

A'^^'-^^-rrill >^

i ^<^''i 1 '^ ' '

^^-^-^ ^Xi '1 T^'

J\

i

t[ jT'^

\'1—r^"/n rr^"^HH~!<tTri—K-J^rrJl- -|-i—

r

'

1

' '1

f > <\

f Ij

(A::^^^^_jrf» r i i -•<^îi

111 1!

! I'll î !

1

i

' _-*«*1*! i 1 !

Ill 1 '1 j

1 ,'

. _' i ^„»» '

i1 ! i 1

i1 :

' !^

1

i _ 1

- - "

%,i&ri^^"iJ I Li— '

I: _|j. '„' — '— ^ "' ^^

j4_^„M^_U ^^_r5^*^^^_j. ~-i---j-ill' '

\'

T- "^ 1 1"î

j 1 M 1 !.1 I

' lA ' J_L^

'

; 1: r \~

, i i1 1 1 M '

! :i II MM 1 ! i

'' '

. 1

REDRNG

BRGP = 10,170,190,350BRGP = 20,160,200,3i*-0

BRGP = 30,150,210,330BRGP = 60,120,240,300BRGP = 90,270"

SS = 1 km.

VGRAD = .0005

Figure 15. Relative Errors at Constant Reduced Range

37

to be 30 km. (.06 times 500). This compares favorably with the actual

error of 28,151 m. found in Table 7 for this situation.

Over a wide range of sensor separations, target ranges and velocity

gradients considered the error estimates obtained in this manner were al-

ways within 14% of the value computed by the model described in section 3.

The unusual behavior of the curves in Figure 15 below reduced ranges of .5

will be discussed in section 4.8.

4.7 Effect on Bearing Error .

For targets located such that there was at least a 10° bearing differ-

ence between the bearings from the two sensors, the maximum error in bear-

ing from the origin on the sensor axis was 2.2° (Table 13). Except for

the extreme sensor separations and target ranges these bearing errors were

less than 1, thus a graphic portrayal of these would not be meaningful.

However, it was noted that the bearing of the actual target from the appar-

ent target position was a function of target bearing from the origin and

was independent of velocity gradient, sensor separation and apparent tar-

get range except when the range was less than the sensor separation. This

case will be considered in section 4.8.

Figure 16 illustrates the relationship between the bearing of the

actual target from the apparent target (BRGT) and the bearing of the ap-

parent target from the origin (BRGP) . Four figures have been drawn to

illustrate the change in this relationship as the sensor axis was rotated

from alignment with an isovelocity line to alignment with the velocity

gradient line. As the figures show, the actual target was always toward

the lower velocity medium from the apparent target position except for a

sensor axis orientation along the velocity gradient line.

38

oQ

360 "/J 1 .XX..

' /\ ' "If

"i i'' '"

1"' *:tt /.f

-—

"

_^vi-;i±:t[^:1X Ar"J ' i

.

P / ^ ' / M f'

270 .- -j- - X<— ->—<y1 i

-i-^-f -^l-^J-1 J^ J" 1 "T"

1-.- ^^ ..._-- ,

,^^ 11 1

'1

i / i1 X 1

III X 1"^ !

*

_ -^^ —

L yf L Ml ! M./ 1 J / J\ -iO-i- !.-4/

1 f\"~i III '

!

/' 1 y i ' _j ^_X.L4_Lji.

\ /f "_ii z±- ~-L -^ 4- X r.: ; ; rp

i L'

^ 1III ill

1 1\ \ _i_L.{._ a-iX

1 SO 'i i_ —1— 1

1

1

Ml 1 !

I r+ H1

ill I ..

-j l- -t-i

1 1 1 i '

i1

^ 1

1

r hi *^1 ; i ! 1

{

i1 1

i1 1

i

1- n r ' -^ - X- "

i i

i

' ! ! i

1 i . 1 i 1 :

ill! 1

^ \i {Mi 1

90 »• II. i'!; v '

' ; 1 1 : . 1 fc

OMEGA =

360 r

270

180

90

'

L X ^M 1' 1 : 1

ii

i J '-J ;

_j_.

^_,

.

..

i !

' M 1

M 1i

:

1 1

t-4-^-i i

i Xi-^/\

H -1! 1

!

' J_ /1

1 \ 1 / !~t1

' !

r ;

\

1 L ' / '1

' '' : 1 t

1

/ ^• 1

11 '.-

' 'MM .1 1 r / '

'/\ '' L x' !

/ttxi1-^I+X- '/ '

~^ "tXIX M 1

f~ M ;

ii ' / M^ ;

i

, y1

! M i_i _j_i :_ /

j

i :

1 ;/

1• :

!

'1 Tt \ \ \

'

1' /^l

i

' '

JJXl L 1' ' i/* -^ -L • '

,' ! 1 . .^/i '

uX_ !

j^^.,-^- LM '

1 ' . ' J^^ i

'

' M ;

! ' ^TSs.^^ «i*-^^^ > •: !

1 1

'""1 1

1

'1

/"^ , 1

'

1 '

1 1 j/^j i

' / ' :~^X ^1!

i

'' /' 1

r 1 'j

i 1 'i ! ! i

M1

J /_["

l~ P" / ' M '

1!

i 1 ; Ml/ 1 i 1

( 1 / 1 M 'ill1 1

f\ i

•1 1 ; 1 1

: : 1 11,:1

1 11 r ~"

1 1\

\-

\ . ! H 1'MM

•III Mli I

1i Mi' 1 1

j_ . , 1

1 1 !'Mm i 1 M

1

'''

1 1

_i-.-t- -| |- nX r 1 ill1

.

1 1

' i1 M

1'

;

-, M ^ 1 1

1I

1 j_ i_l_ ;!

1'<

;

1

I1

1

( 1 ! ii M i 1

-rrxf1 i -, i M1

''xx : j ! 1 1 i -X^' L

i 1 1

1

1

1 ' 1 MM ' r 1 1 ! 1

1 1

f-.- i- L 1

i ._ 1

i' '

1; 1

1 1 '

1 1

1 -|M±M ' 1

1 ! .11 1 1

I ! 1

1 1

1 1 1 i

....

90 180

OMEGA = 30

270 360 BRGP

SS = 400 km.

RAPP = 500 km.

VGRAD = .0001

Reference

:

Tables 8,24

Figure 16a. Target Bearing Corrections

39

360

CO

270

180

90 »•

r ,±:: di± :l):1; : J: . ;i:r ±.±„ 1, .l 4_i _1

ifi-i

H:-;:V.::t -::::--- /- .

^ -t. ._.. 1:..^ 1

'

/i4:[:.x±ri:::tb:;:.j7'dp

y n -

J / i n' / ^1 / i i ' ! 1

.'\ i 1

:r_:-:::.::;.:-:-7:::-::::=::i::.:-:-^-v: 1 X

- irri

:::^_l_:::^^::zr.:::::::4^::-;jjXI 1

-Jj-

— ,— —i-.

1

—

!

--^-H—\^ - -'--y—hf-pf- ""^"h -ffVj^-- 1

1

"1"1- -LMl 1III 1

_. .].__ -_-Lj..i

! 1 1 11

1

1

~ '

i

'''' '

1I - \ i

u- -i_ 1

\

L L i

_i.j. - '-I-i i;-

,

11

1

1 t ""XJj:"J , , , L,.

i i_1 1 'III |^ 1

-X1 n i

1

OMEGA = 60

360 : : L .i

1

— -Hr-h- --h--j H-- 11

' -

'! 1 L . *— ; -

1 1 1 uL-i H1'

'

, '

I

1 1 I

^ .„ ..

:" G^ ^i 1—1 ! 1"*

1\

11 1 1

.J-_ L_^j

,.J.., ^_L_^ _. MM julj \\ \ \ XXX1 n^ / '

1 JXLL1- :l .. -A

^ ~T / 7 ' ^

r 1 in 1 '' /' 1 \,

!

i

' i

• / i

«,^ X..H- X- j 1 / i 1

L ^_._jj -4.... ;1 / ' M

270 •1 I/I ' \r\ n it I 1 i i

ill;n ^ / j -1 X -ix 1

1 1 Mi'-'"^1 1

'/' ' 1 { 1

1

', / i1

! ] 1 ! / 1 I 1 1 /1 1

~T /!i

1 1i

1 ! ( ' / 1

,

'

i ii

'1 '

~r r"i X"" ' / 'i -iJ-i '.

i 11 / i

1 11 ! 1 1 1

^' / 1 1 Ml

;

i1 ._ :/- LUL 1 - 1 J- J _ - 4-.; ' L / '

'

1 ' /ii

1'

1 i1

' '1

i

1

/'<' 1 t

1

._ -L-|_c 44--/ XLj 44^

rf!"!-.

: 11

/ 1

180 "-^'A\/' i

T/r.Ld-X--[--4—

-

[/ .Xi !

1 1 / '! , 1

1 ; 1 y 1 1 J_l , /. 1- trr-i'/ ' - - Tt 1 1

_l ' i'

[- - 1! /j 1 1 1 ) 1

±x -ix ii'7 1 '"!1

--- -;X44'- -| A T"

'

'/1 i_L/ X-4 ^ -p---p- „. -,

p" n„ X^ i1

1' _I I ! 1 ! !

90 .-.l-.:-;/l J 1 ^ "iXCX .L ! i i_r 1 ; i 1 1 ! 1 1 !

i / 1 1 1 ^ 1 ' 1 '

'1 1

1 / ; 1 IIIi ' ' '

1 I / 1

1

1 / 1 1II 1 ' ) ;

i 1 >I

'/ ' - ^L- ^ /—I K -X 'j

1 Ml;/ i

1

'i'

! tili

11

'/1

' 'y 1

'

, J ;1

/! ' / 1 1 1

i1

i1 : , 1

/ 1

' _^^ !/ ! 1Mil III

, i • 1

/ j -1- itX M-j/i-. II M 1I

1

f 1

M-T- H /p1

1

1 ! ! !i i

'

1 1 1

!i

'

'1

11 1

^1- 1 .11, , !.! I i 1

.' MM .1.1 1

90 180

OMEGA.

J70 360 BRGP

= 90

Figure 16b. Target Bearing Corrections

40

4.8 Effects When Target Range is Less Than Sensor Separation .

In the foregoing sections numerous references have been made to the

fact that when the target range was less than the sensor separation, and

particularly when the target range was less than one-half the sensor separa-

tion, many of the estimating relationships were no longer valid.

As may be seen from Figures 7 and 15, when the sensor axis was aligned

with an isovelocity line, the effect of target bearing on the distance er-

rors was the same for reduced range values less than 0.5 as it was for

those above 0.5. However, when the sensor axis was rotated to alignment

with the velocity gradient line, instead of the errors being almost con-

stant the errors were greatest at those target bearings where they had been

the least when the sensor axis was along an isovelocity line. This may be

seen in Figure 7 and by a comparison of the data in Table 25 with that in

Table 26.

Figures 17a and 17b illustrate the change in the absolute errors as

the sensor axis was rotated. As may be seen in Figure 17a, the absolute

errors decreased as the axis was rotated but when the reduced range equaled

0.5, the variation in error due to target bearing remained constant. This

is contrasted with the behavior, as indicated by Figures 11 and 17b, of

the error for values of reduced range both above and below 0.5 where the

variation in the error as well as its absolute value decreased when the

sensor axis was rotated to alignment with the velocity gradient line.

Figure 18 depicts absolute distance errors as a function of reduced

range for reduced ranges from 0.1 to 1. As the figure shows, the maximum

errors increased in value as the reduced range increased or decreased from

a value of 0.5 when the sensor axis was along an isovelocity line but de-

creased uniformly as the reduced range decreased when the sensor axis was

41

T-t

:-L

1. L,

I I

I

iH-t

Mi

1

1

I- ..T

-U

^.i'Ti

1 I

30 60OMEGA - DEGREES

'^90

SS = 400 km., RAPP = 200 km.

VGRAD = .0001Reference

:

Tables 25,26

00Pi

I

0iQ

H

Figure 17a. Effect of Sensor Axis Orientation(Reduced Range = .5)

16

12

fl 11 -ill Ll. I

- 1 1 ! i1

, i ;

i

~I*"

1 1' HT-

i «p1

11

1Mt

i1 i

1 _ ljl,.]_; jl)l.,_*

1' "if

!1 _[ L!jJ..

i '

- -^ .|- T 1'"1 1

1

'

1

'

1 1 ,

'—1 a_5h::: __.jj

J. 1 i

""Lr""'

1

1T^ r'~ 1

111 1'

1 'l " ' "' '" i1

I CT- -^y± I 1

•

1

1rr* .. J-l- .J-.! I

'.

11 1

j

^ . t

j

; 1

{j

1

: 1

i I 1

! 1 :

1 ,

i

i1 1 M ! ! 1 1 i 1 ! 1 •.

1 ,1

'i

m j_ l_L l-L '

,•[1 :

11 ' ' t

i; 1

1: M :

i 1 ' :1

• '1 i

1 !

, ;

f T i ' 1,

'1

t_[ : 1

1' ' J- -—

;

1, : ,

i

1!

11

!i

1 1 1 1

1j 4-

1

J 1 Ti ir I'M '111

'II. t 1 1 '1!

1i

•

! 1'it;

'1 1 '

i

1 '

1

!1

i

i i"'

' 1•

^• , ,

1

;1 1 1

1

1 i i -^ 'til 1 ,

'

'

i ;

i1 1

i i M I'll1

'

1

1 1 . _Ll uL. '

1 1

"11 r T 1 1

11

I 1 Ml' ,'''

11

1

Ill Li_[

j1

: „.-!-

1 '

"1i_

11

1i

1

T1^ :'l

1

M '

'

i . i I

14-4- 1 1 i i Mil: ^ilH _j

1

1

i-r-f- Th_r 1 i

ji

11 1 1 I

1

•

1 i 1 1•

'

1;

1 ''

j

1'

' 11 1

i 1i

1 ;

1Pi

„X _i —uLj. —

1

:!

1

; !

1 1'

'

i1 1

"'1

1

1

1 1 -U-4Ill, 1

1;

; I

_- n 1 . _^ .- -:-'*1

, 1 1'

1

'' '

•

1 n i

1

1

i

i ! i_'

• , j

1 Ti i

^ ' - J1 •1 L I —1'

'

ll1

1 1'

,—UJ_ h-rr-'

1

L ;,

J

30 60

OMEGA - DEGREES

SS = 400 km., RAPP = 80 km.

VGRAD = .0001

90

Reference

:

Tables 27,28

Figure 17b. Effect of Sensor Axis Orientation(Reduced Range = .2)

42

60 r

o

oOS

50

40 r

30

20

10

'"'" "'-

"i" "

' t ""f"' tTT''-

j-i-i-t-

iTtr

'

' :'

( 11

-{ X -!--- -_-|--4 (----J

'H- -t ii'

' ' (- W X -1 \~ '-+•- -r-f

'

Tilt---J— -[ 1 - t1

j! [I j

1 1t M r 1

1 1 1

1{

I

'1 '

! ' ' 1 i . ,

1'

1 -4- XMil if 1

1" M M 1 1 f; 1 1

^! !

1

"111 1 1

ji

1M i 1^rix u;

'1 !

-j..-l,J-J1 ' ' '

TT nirx1 [ 1 -

I

': I

"

' M i ! 1 i : 1

_.-4 i-J- -X—-Xm^ ._^L_lm__ _|^_„4__1 i i~!

--^ [——j-

Mill XI ' 11 ' 1 1 1

' ' !il^I

i 1 11 1 1 M 1 1

M M. 1

,

1 1i 1

1 1 i MM i 1i

1

_rj-i-r1

1' !_[""1 i

T M 1

1i 1

' - _ . L L L 1

1 M ''

' M1 .M !

^ '1

'III1 1 M 1 ' M

1 , . 1 1 1 1M

.

.-M_M -_^| . ^_-- -1 ! U-_--jJ_l_i .,.L|..L

Jh^.

,_.-...

1

-^ X .. — t JJ L- i M...XLi

! 1 ! "^ T '

' i ' I i ji! '

; :

X-!- tM- 4-lXi X=Xr'X^" '"H"'

Xl-rr-XXTM M 1 ; 1 1

"'1 Xl A 1

:

,! ^ ''

! 1 ! 1 j . _.i

! -1 1 ! I'M1 1

i

iX-i--J—;—_ L

^.:-4_;.-:-1 ,1 ,

M ;i 1

1

...| 1 i! III i ,

1* ^ . 1• •

1 .•

Mil-rX-h• • i ' ' 1 Ml

mm - " '. ! i 1 i

I'M!1,1., , 1

!

' ', .i, ;.,X ' '; 1 1 X '

'

, i XOMEGA =

10 r

»-

-r-1

1

'-frh 1!

1 j

i i

i'

1

"~i—r~i—1

"

—J1 1

- l-t !

1 1

-_Lu.-.-!- 1

' -f{-!-}- •

--

^

J"

1

11

I 1

a-LJ-:.

III• " * '— * |-

' ' 1'

r^* -!Ill"

XX":M . .I-H-'-

^i li..

. r 1 1 1

1

1

1 y

-.—

T

1

; 1

1 1 L-lX- X "^ _i_Xi-

.2

SS = 400 km.

VGRAD = .0001

,4 .6

OMEGA. = 90

.8 1.0 REDRNG

Reference:Tables 25,26,27,28

Figure 18. Effects at Reduced Ranges Less Than 1.0

43

along the velocity gradient line.

Figures 19 and 20 illustrate the effect of apparent target bearing

from the origin (BRGP) on the bearing of the actual target from the appar-

ent target (BRGT) . The figures also illustrate how the bearing correction

varied with the sensor axis orientation. A comparison of these figures

and the data in Tables 24-28 with Figure 16 and the data in Table 8 reveals

that as the reduced range was lowered from a value greater than one (Fig-

ure 16) through 0.5 (Figure 17) to 0.3 (Figure 18) the relationship between

the bearing from the apparent to the actual target and the bearing from the

origin to the apparent target reversed.

5. Conclusions and Recommendations .

As a result of this investigation the following conclusions may be

drawn regarding the crossfix errors of two directional acoustic sensors

in a medium with a constant horizontal velocity gradient:

a. Although the distance errors are small relative to the target

range, they can become appreciable in absolute value at very large target

ranges

.

b. For gradients within the values studied, the distance errors are

a linear function of the velocity gradient.

c. The distance errors are sensitive to the ratio of the apparent

target range to the sensor separation. If two directional sensors are

used to obtain crossfixes and it is known from past experience that a

large proportion of their detections occur at a certain range, they can

be placed so as to minimize the crossfix error.

d. The distance errors are sensitive to target bearing and sensor

axis orientation. By placing the sensors along the velocity gradient line,

distance errors due to target bearing can be minimized.

44

e. The distance errors are predictable, i.e. the errors can be esti-

mated from a knowledge of the horizontal velocity gradient, the sensor

separation, the sensor axis orientation, the apparent target range and the

apparent target bearing.

f

.

The bearing from the apparent to the actual target is independent

of velocity gradient, sensor separation, and apparent target range. It is

only a function of target bearing and sensor axis orientation, thus it is

also predictable.

Since the model used to formulate these conclusions did not take into

account the variations in the path of the sound ray due to vertical velocity

gradients or horizontal gradients other than linear ones, the investiga-

tion of errors using a more sophisticated three-dimensional model is recom-

mended. Such an investigation should, of course, determine the sensitivity

of the errors to bottom conditions such as slope, since a sloping bottom

will also result in deflection of the sound ray from its original course.

Further, since some directional sensor systems depend upon the time phase

relationship between a pulse received by two sensors, the errors caused by

the actual paths being of different lengths than the apparent paths may be

entirely different from those calculated by the model used in this investi-

gation. Therefore, an investigation of this problem should be considered.

45

360 r

CO

o

270 i

180 •

90

..Fj^ll^lfipp-^ri^f:^^-i'_..ii' r.-,'. ' .'.'zir ,r/ z.''. ..rc'TZZT .ir i.zi "

^ ..-t'-1 -i . .

.1 ..p^^. j.

__J_ ^ ^^v'd !

" -^-i \ i

::,::t :j^::^.^^J^::..: |:E^:r4^-^

-f-j r^r^h^—

^"i""" ""^

1^""

" """t^' 1 11 1

! 1 .

'

i 1i

l-' •

I JJ^_- -+-^ _1 .i;'l '1 '

''

.... -p.,...-i

1 ! !

i1

'1

di-ix•--i- -4-^L

___|:^=Ft^- -^- ii±^--H=I-f -=^ - -ffr1 i ! 1 ! 1

'\~" \' '\ '''"i" \'\

\\

i r 1 1 ' 1 ' ! 1 1 ! 1 - • 1 1 1

11 1 i 1 - L i_J.-II 1 ' f

i

'

iU_l i-

1 "I ] ti r '1 'i }n-t r-

1 1i

. . 1 1 , , , . .1 ! ,1 1

-,! .1 . 11

1j

1 i 11

'i

i

'

.l-l-4-r-M-tU- -.-1-1 -^—^-r- - - \\--^ AVA- -t44-,h-i-I-]-l! ! i

OMEGA =

360

270

180

90 *•

360 BRGP

SS = 400 km.

RAPP = 200 km.

VGRAD = .0001

Reference

:

Table 25

Figure 19a. Target Bearirig Corrections(Reduced Range = .5)

46

360 p

270

180

•.. ..I'...." X.-

_Jj_|.. ._| _ ....

1 1

1

::.L±|:

1

1'

1 11

i4".i_4

\-\ v-\-

_,.„r.„_ -_."._„-.

:-:i--:1-

-

1 .......

.

t 1

\

P± ^+_/. lb :"'- "-.-j-^- i

. 1- +H+•.i^^4::i

^^d^<r t..

"1\:r

"

- 4-_. ^ • .î ''

1 ^1 ^^ T 1 \1 T i ^r '

1 Y 1/^ 1 \

j -LLL-i -J- -iv^- 1 1t

1\ .

1 j^ i\ 1 -

t-p-1 1

V y^ '

j

\j^ ^ i^^ 1 1

1^Wi"î !

1

j j 1 ; -U[- - 1 1

i 1^j

1

-J J—1--4lThT!-- -14714=-i-^l --I- Ui- --^4-1 I.I.I i

OMEGA = 60

360

w

C5

I

270

180

90

i-r .'

i 1 1 1 4 4 111 1 1

1

i

i1

1

1 11 1

I _[_J ! i

1; i

i ! 1 ' 1'

1 i 11 • :

1 1 !

- -[-4-p -\—L .-j-44-.- i-H"! i !

i

j_\ L1 . ;.. .

1 1 44 4 '

1 ' 1 1 ! 4 ' 1 :

1

1i i

, !'

1

-XII 1

i, 1 : i

:

i

1

''

1

' 1 "i i

il'li ; i' 1 i

1 > ! 1 i 4 •

"

' >'4 4 'yi

1 11 1 i J/ i • in4 1 1 H

1: '

y^\ 1

1 1 1 1 j y\ i1 '

i i

^

' y -

1i

y\ 1 ' 4 j i 44-'-y 1 1.

/ \ 1-14' 1 iX i i y^ 4 1 i 1-yZ t 1

1 1 y \ \ 1 1 / 1 :

' 1 I'll1 III/ i 1 1 ! : 1 ! 1 .

'1

1

41 J yf ! 11 !

'1

^' y\

'i

! :"I !

' ^F] 1 1 1

14 1/ ' -i

,i ! i

1 ^1

1 1 \ \ \ y\ :1 ! 1 !

1 1 y 1 I'lli Vi ' '

1' 1 1

' ' y\ 1

:

: 1i

1/ ! ! 1 1 i1 >

''^1 1 Il'l 1 ! 1 r

yi' 1 \ \ \/ \

i1 1 1

i 4!y \ Mi 1 iX' i

! 1'11

Y 1i ! !

1 y 1 1i

1 1 I'll1 x^

iI ill 1

/^ \i i '

i;

1-

Xi 1! ! iX 4 1 '

If-'i

1 1

/î

' 1:1, I'M1

1

I

! 11 i 1 4;

' i •

! i' i .

i'

1

1 { ' 1

I

\'•

'

1 1 1 1 ! 'i

1

i

1 !

1 1 '

' 4 Mil 1 1j

1'

11 1 4 ;

1

i

]^n 4|

1

yr- p i i 1 1

1 ! 1 4 i ''

1 '

i1

II 1 ^1 ! ;i

; 1

'

i1

I 1

i.

1

Ij' 4" H 1

! 1 ! 1 >

: 1 i 1 ' I

4 1^

i :

r„ 4 i

' 1

.'

'

' :1

1

90 180 270

OMEGA = 90

SS = 400 km. Reference;RAPP = 200 km. Table 26VGRAD = .0001

360 BRGP

Figure 19b. Target Bearing Corrections(Reduced Range = .5)

47

360

270

180

90

f>|^;H: 4pJ:K ^[|^<Fh|::fe:t t'tl1

i ^s. 1

i

1 I i 1 ! ^^^' '1MM i M i 1

—t' ^V' ' ' -i —l^S. ' '

1 11 ' LJ

£V --{- - --. - ^--f- .L--\a-j^- ..|_ __.Lr

i 1 M 1

T'lXI~i-t 4-S it.. - JJ^^ i|i:A ... -

"i

—

--i .. 1 '^L.,1 i .

f \ ' II \i '---'r -M-

..i-piiiiiiS.iii._=^:iiiiiiii-+-ii-iA^ t -.IT ^V .. _, ._ ..^

i

1 ' \ ' ' ' ' \1 \^ MM 1

M \ 1 ,

-i-b-H„y II lNii X ..T 1{_ -p£p-—i-

j^v^

11

n \ |-^

1 '^v ' ' '

'^

1

i i

'' n V 1

jI j 1 \

1 ' J J '

1 11 1 i

i i 1:1 1 ;

4„L.4.1

: M .hi ^..

^_1 1 1 .. l.._,.-L , , . 1 __.. ! ... J . 1 , L M ' '1 ^

'

1 n I r ni

1 1 11^ ^

.- 4_ 11. n .

4 _a_ _, .- .. . -X 1 |- U_j _|_ _^ _|_ 11 (., |_

J.

L. U-t-1 t 1

i i i

1M

1

* !_1 '^1

.L ^ _+_ 1 ,... _i_.. ! L^ J rri 1.U.^.^ __ ^ .LJ4 u ^ -^-4-J X '-

L 1 .1 li.. 1 iiiii. . .1 II.. Ill I44-;OMEGA =

CO

Of

360

270

180 •

90 •

360 BRGP

SS = 400 km.

RAPP = 80 km.

VGRAD = .0001

Reference:Table 27

Figure 20a. Target Bearing Corrections(Reduced Range = .2)

48

360

CO

I

270 -14

180 -

OMEGA = 60

360

270

180

90

: 1i 1

1

1

! ; !

' ' ' i

'

1'

t , .

ITJ-r-l-

'I M

1 . 1..LL.Ii L I

.--L-i

J.-I- '.!_1 1 1

!1

~1 Pi 1

''

'

1• 1

j

1'"T ["!

j!

1

1 1

~t~1 / TuJ1

11 ..L- J ^..f

'

1

' ill!1 i M 1 1 r"

i

r ; 1 ]1 i

1

1

1 i

11

ii

•

1 ^ •

1 i

i -J- 1 1-4 -

'

! 1'

\ 1 1 I'D- [~n"

:

-1—1 --];

1

i

~\"\i

r 1'

1

-U' 1 L 1

1 1

; 1 1 i i ! ' ' ! 1 . .

: r^ T" *"1

1 U l4- {'1

1

11 -, _,_Ln 1

.Illill r L A. iV i 1

i- '

-:

: 1\

"'1

ii

r i^ r" ri" '

1! .

'

'. 1

'1

1 1 _ ' 1 L 1 1 ! |_1Jj_ l1. L Lj_! l.

!

''

Ti1 »^ 1 1 I. - LIJ L

; i

__^^

-L'' - -h

.^^^ '

-H LJ , 1 !, =_ i_i_

^**--— .i-—<"'"'' i 1I

1 1

.-' T^l II

^-^T"! iT !!"

11

1

_!_,. __4_1

.1

:

1 1 i 1i , 1 1

1i

i

1 J 3X- i 1 11

1

i

-4-p{-]_' 1

1 !

1

•I

1 ' i i1

""! 1 '

1 i

"1i

t 1 ; ; I'll ' ! 1 '

1 ,1

i 1 i1 ! 1 1

i\

i

' ^1

'

1 1 -V A\ 1

1 1 1 1 > 1 i -f1 1 <

1 1' i

90 180 270 360 BRGP

SS = 400 km.

RAPP = 80 km.

VGRAD = .0001

OMEGA = 90

Reference:Table 28

Figxire 20b. Target Bearing Corrections(Reduced Range = .2)

49

BIBLIOGRAPHY

1. Ayres, E.; Woff, P.M.; Carstenson, L. ; and Ayres , H.C. A Ray TracingProgram for a Digital Computer . Fleet Numerical Weather Facility,Monterey, California. June 1966.

2. Hal ley, R. Physics of Passive Sonar . U.S. Navy Electronics Labora-tory, San Diego, California. June 1962.

3. Horton, J.W. Fundamentals of Sonar Vol. 1 . U.S. Navy UnderwaterSound Laboratory. September 1950.

k. Klnsler, L.E. and Frey, A.R. Fundamentals of Acoustics, 2nd ed . JohnWiley and Sons, 1962.

5. Newmann, G. and Plerson, W. Principles of Physical Oceanography .

Prentice-Hall, 1966.

6. Plgman, J.W. Horizontal Gradient Effects on Directional AcousticSensors . Thesis, U.S. Naval Postgraduate School, May 19 66.

7. U.S. Office of Scientific Research and Development, National DefenseResearch Committee. Physics of Sound in the Sea . Research AnalysisGroup, Committee on Undersea Warfare, National Research Council, 1951.

50

APPENDIX I

-COOP».A DORSEY, S/1 S/2S » 3 5000 , 2

.

-FTN,L»E.PROGRAM HRZGRDl

C THIS IS A PROGRAM TO COMPUTE THE BEARING AND DISTANCEC ERRORS IN A SONAR FIX WHICH RESULT FROM HORIZONTALC TEMPERATURE GRADIENTS IN THE WATER.C THIS PROGRAM IS WRITTEN IN FTN-63 FOR A CDC 1604 COMPUTER.C THE Y AXIS IS AN ISOVELOCITY LINE WHERE THE VELOCITY IS VO.C THE ORIGIN (X=0,Y=0) IS A POINT MIDWAY BETWEEN THE TWOC SENSORS AND IS THE INTERSECTION OF THE Y AXIS AND A LINEC THROUGH THE TWO SENSORS.C SENSOR A IS LOCATED AT XI, Yl AND SENSOR B IS AT XXI, YYl.C ALL BEARINGS WITH THE PREFIX BRG ARE RELATIVE TO AN AXISC THROUGH THE TWO SENSORS.C ALL BEARINGS DESIGNATED BY GREEK LETTERS ARE RELATIVE TOC THE Y AXIS.C ALL RANGES AND DISTANCES ARE IN METERS.C ANGLES IS A SUBROUTINE TO CALCULATE ANGLES FROM TO 360C DEGREES USING TANGENTS.C BRGA IS THE BEARING FROM A TO THE APPARENT TARGET.C BRGB IS THE BEARING FROM B TO THE APPARENT TARGET.C BRGG IS THE BEARING FROM THE ORIGIN TO THE ACTUAL TARGET.C BRGP IS THE BEARING FROM THE ORIGIN TO THE APPARENT TARGETC BRGT IS THE BEARING FROM THE APPARENT TO THE ACTUAL TARGET.C ERROR IS THE DISTANCE BETWEEN APPARENT AND ACTUAL POSITION.C RACT IS THE ACTUAL RANGE OF THE TARGET FROM THE ORIGIN.C RAPP IS THE APPARENT RANGE OF THE TARGET FROM THE ORIGIN.C RAPPING IS THE RANGE INCREMENT.C RAPPMAX IS THE MAXIMUM APPARENT RANGE CONSIDERED.C REDRNG IS APPARENT RANGE DIVIDED BY SENSOR SEPARATION.C RELERR IS THE RELATIVE DISTANCE ERROR ( ERROR/RAPP )

.

C RNGERR IS THE DIFFERENCE BETWEEN RACT AND RAPP.C RPD CONVERTS ANGLES IN DEGREES TO ANGLES IN RADIANS.C (RADIANS=DEGREES TIMES .01745329252 RADIANS PER DEGREE)C SS IS SENSOR SEPARATION.C VO IS THE VELOCITY AT X=0, VGRAD TS THE VELOCITY GRADIENT.C XACT,YACT IS ACTUAL TARGET POSITION.C XAPP,YAPP IS APPARENT TARGET POSITION.C XINC IS THE DISTANCE INCREMENT IN THE X DIRECTION.C YDIFF IS THE DESIRED PRECISION IN CLOSENESS OF THEC COMPUTED AND ACTUAL POINTS OF. INTERSECTION OF TWO RAYS.C ALPHA IS THE BEARING OF THE APPARENT TARGET FROM SENSOR A.C BETA IS THE BEARING OF THE APPARENT TARGET FROM SENSOR B.C DELTA IS THE DIFFERENCE BETWEEN GAMMA AND PHI.C GAMMA IS THE BEARING FROM THE ORIGIN TO THE ACTUAL TARGET.C OMEGA IS THE ANGLE BETWEEN THE Y AXIS AND A LINE THROUGHC THE TWO SENSORS. IT CAN VARY FROM TO +90 DEGREES.C PHI IS THE BEARING FROM THE ORIGIN TO THE APPARENT TARGET.C PI IS 3.1415926536C TAU IS THE BEARING FROM THE APPARENT TO THE ACTUAL TARGET.

51

II

III

IV

APPENDIX I

cC READ IN DATA.C

100 RPD = ,01745329252 $ P I =3

.

1415926536110 READ 111, VO,VGRAD,YDIFF111 FORMAT(3F10.0)120 READ 121» SS,RAPP,RAPPMAX,RAPPINC,XINC121 FORMAT (5F10.0)PROGRAM STOPS WHEN SENSOR SEPARATION IS NEGATIVE.122 IF (SS) 900,900,125125 OMEGA=0.0132 OMEGAR=OMEGA*RPD140 Y1=SS/2*C0SF(0MEGAR) $ YYl=-Yl141 X1=SS/2*SINF(0MEGAR) $ XXl=-Xl170 REDRNG=RAPP/SS183 PRINT 184, OMEGA, SS,RAPP,REDRNG,VO,VGRAD,YDIFF184 FORMAT( 1H1,16(/) ,5X,8HOMEGA = F3.0,9X,5HSS = F6.0,9X,

17HRAPP = F7.0,9X,9HREDRNG = F4. 3 , / /I 5X , 5HV0 = ^6.1,212X,8HVGRAD = F7. 5 , 12X ,8HYDI FF = F3.1,//)

185 PRINT 186186 FORMAT (2X,4HBRGP4X,4HR ACT 6X,2HV26X,4HBRGA4X,4HBRGB5X,

15HERR0R4X,4HBRGT3X,5HDELTA3X,6HRNGERR3X,6HRELERR)Tc . . .

200 DO 600 1=1,35201 IF(I-18) 205,520,205ROUTINE TO SET APPARENT TARGET POSITION205 BRGP = 10. *I206 ICOUNT=0 /210 PHI=OMEGA+BRGPTEST TO ENSURE THAT PHI IS LESS THAN 360 DEG.211 IF(PHI-360. )213,213,212212 PHI=PHI-360,

VI C ROUTINE TO COMPUTE RELATIVE BEARING OF TARGET FROM SENSORS213 PHIR=PHI*RPD

XAPP=RAPP^SINF(PHIR) $ YAPP=RAPP*COSF ( PHI R

)

K=l $ CALL ANGLES(X1,Y1,XAPP,YAPP,ALPHA,K)ALPHAR=ALPHA*RPD $BRGA=ALPHA-OMEGAIFtBRGA) 223,225,225BRGA=BRGA+360.K=2 $ CALL ANGLES(XX1,YY1,XAPP, YAPP, BETA, K)

BETAR=BETA*RPD $BRGB=BETA-OMEGAIF(BRGB) 228,230,230BRGB=BRGB+360.

ROUTINE TO TRACE RAYS AND COMPUTE ACTUAL TARGET POSITION

XA=X1 $ YA=Y1 . $ XB=XX1 $ YB=YY1IF(OMEGA) 232,299,232VA=V0+VGRAD*X1 $ VB=V0+VGRAD*XX1ALPHAR1=ALPHAR $ BETAR1=BETAR

TEST TO SEE IF XI IS LESS THAN XAPP IN ABSOLUTE VALUE233 IF(ABSF(XAPP)-ABSF(X1) ) 240,280,280

52

APPENDIX I

C ROUTINE TO COMPUTE XACT.YACT WHEN XI IS GREATER THAN XAPPC IN ABSOLUTE VALUE.

240 RATI0A =C0SF(ALPHAR1)/VA $ RAT I OB=COSF ( BETARl ) /VBX2»XX1 $ XXINC«XINC

241 X2=X2+XXINC $ V2=V0+VGRAD»X2IF(ABSF(V2*RATI0A) .GT.l.) 257,242

242 IF(ABSF(V2*RATI0B) .GT.l. ) 257,24324 3 ALPHAR2=2.*PI-AC0SF(V2*RATI0A)

BETAR2=AC0SF(V2*RATI0B)DXA=XA-X2 $ DXB=X2-XB

NOTE SIGN OF DXA IS ALWAYS +.

DYA=DXA/TANF( ( ALPHAR 1+ALPHAR2 ) /2.)DYB=DXB/TANF( ( 8ETAR 1+SETAR2 ) /2 .

)

IX Y2A=YA-DYA $ Y2B=Y3+DYBC TEST FOR INTERSECTION OF RAYSC CHECK TO SEE IF CRITICAL ANGLE LIMITED,

IC0UNT=IC0UNT+1 $ I F (I COUNT

,

lT. 200 ) 244,530244 IF( 180.-BRGA) 245»520,255245 IF(Y2A-Y2B) 246,258,241246 IF( (Y2B-Y2A) .LT.YDIFF) 258,257255 IF{Y2B-Y2A) 256,258,241256 IF( (Y2A-Y2B). LT.YDIFF) 258,257 ^

257 X2 = X2-XXINC $ XXI NC=. 1*XX I NC S GO TO 241

ji 258 XACT=X2 $ YACT=Y2A $ GO TO 400

C ROUTINE TO GET XB EQUAL TO XA WHEN ABSOLUTE VALUE OF XI ISC LESS THAN ABSOLUTE VALUE OF XAPP

280 IF(PHI-180. )281,281,291 .

Ac^

C CALCULATION OF STARTING POINTS OF RAYS WHEN PHI IS LT. 180281 DX=XA-XB

BETAR1=AC0SF( VA*COSF( BETAR ) /VB)DY=DX/TANF( ( BETAR+BETAR 1 ) /2 .

)

285 YB=YB+DY $ XB=XA 5 V1=V0+VGRAD*X

A

ALPHA1=ALPHA $ BETA1=BETAR 1 /RPDC AT THIS POINT BOTH RAYS HAVE THE SAME X COORDINATE.

286 GO TO 300

C CALCULATION OF STARTING POINTS OF RAYS WHEN PHI IS GT. 180291 DX=XA-XBNOTt THAT SIGN OF OX IS ALWAYS +.

ALPHAR1=2.*PI-AC0SF(VB*C0SF(ALPHAR)/VA)XII DY=DX/TANF( (ALPHAR+ALPHARl)/2. )

295 YA=YA-DY $ XA=XB $ V1=V0+VGRAD*XBALPHA1=ALPHAR1/RPD $ BETA1=8ETA

AT THIS POINT BOTH RAYS HAVE THE SAME X COORDINATE.296 GO TO 300

k 299 V1=V0 $ ALPHAR1=ALPHAR $ BETARl=BETARVA=V0+VGRAD*X1 $ VB=V0+VGRAD*XX1ALPHA1=ALPHA $ BETAi=BETA

CC AT THIS POINT THE RAYS ARE DISTANCE STEPPED IN THE + OR -XC DIRECTION (DEPENDING UPON PHI) UNTIL THFY INTERSECT.

53

XI

V'

XIIIif

APPENDIX I

XIV

XV

T±ACc

XVI

c

XVII

300 DX=0.0 $ Y2A=YA $ Y2B=YB301 RATI0A=C0SF(ALPHAR1)/V1 $ RATIOB=COSF( BETARl ) /Vl

XXINC=XINC303 IF(PHI.LT.180. )306»336

307

308309

313

CALCULATION OF XACT.YACT306 DX=DX+XXINC

V2=V1+VGRAD*DXIF(ABSF(V2*RATI0A) .GT.l.)IF(ABSF(V2*RATI0B) .GT.l.)ALPHAR2 = ACOSF(V2-»-RATIOA)DYA=DX/TANF( ( ALPHAR 1+ALPHAR2 ) /2 .

)

DYB=DX/TANF( { BETARl +BETAR2 ) /2 .

)

Y2A=YA+DYA $ Y2B=YB+DYBIF(BRGA) 325,314,314

TEST FOR INTERSECTION OF RAYS314 IF( 180,-BRGA) 325,520,315

IF{Y2A-Y2B) 316,328,306IF( (Y2B-Y2A) .LT.YDIFF) 328,32?IF{Y2B-Y2 A) 326,328,306IF( (Y2A-Y2B) .LT , YD I FF ) 32 8 , 327DX=DX-XXINC $ XXINC=.1*XXINCXACT=XA+DX $ YACT=Y2A

ACc

XVIII

WHEN PHI IS LESS THAN 180 DEG.

327,308327,309

$ BFTAR2=AC0SF(V2*RATI0B)

315316325326327328 $

GOGO

TOTO

306400

CALCULATION OF336 DX=DX+XXINNOTE THAT SIGN337 V2=V1-VGRA

IF(ABSF(V2338 IF(ABSF(V2339 ALPHAR2=2.

BETAR2=2.*DYA=DX/TANDYB=DX/TAN

NOTE THAT SINCDYA AND DYB W

343 Y2A=YA~DYATEST FOR INTER344 IF(180.-BR345 IF{Y2A-Y2B346 IF((Y2B-Y2355 IF(Y2B-Y2A356 IF( {Y2A-Y2357 DX=DX-XXIN358 XACT=XB-DX

XACT,YACT WHEN PHI IS GREATER THAN 180 DEG.COF DX IS ALWAYS +. /'

D*DX*RATIOA) .GT.l. ) 357,338*RATIOB) .GT.l. ) 357,339*PI-AC0SF(V2*RATI0A)PI-AC0SF{V2*RATI0B)F( ( ALPHA R1+ALPHAR2 )/2.

)

F( (BETARl+BETAR2)/2. )

E SIGN OF DX WAS KEPT + THROUGHOUT, SIGN OFILL BE THE OPPOSITE OF WHAT THEY SHOULD BE,

$ Y2B =YB-DYB -^

SECTION OF RAYSGA) 345,520,355) 346,358,336A). LT.YDIFF) 358,357) 356,358,336B).LT.YDIFF)358,357C $ XXINC=.1*XXINC $ GO TO 336

$ YACT=Y2A $ GO TO 400

ROUTINES FOR CALCULATION OF ERRORS

400 XERR=XACT-XAPP $ YERR=YACT-YAPP405 ERROR=SQRTF(XERR*XERR + YERR*YERR)410 RACT = SQRTF(XACT*XACT + YACT*YACT

)

415 RNGERR = RAPP ~ RACT

54

APPENDIX I

XVIII

Xc

SUBTRACT RNGERR FROM RAPP TO GET RACT.416 RELERR=ERROR/RAPPZ IS A DUMMY VARIABLE TO ENABLE USE OF SUBROUTINE ANGLES,420 Z=0.0

ANGLES

(

Z,Z»XACT,Y ACT, GAMMA, K)- GAMMAFROM PHI TO GET GAMMA.ANGLES (Z,Z,XERR,YERR,TAU,K)OMEGA

421 K=3 $ CALL422 DELTA = PHISUBTRACT ERRORS425 K=4 $ CALL427 BRGT = TAU •

428 IF(BRGT)429,500,500429 BRGT=BRGT+360.

CALCULATIONS COMPLETE, PRINT RESULTS.

XIX

XX

XXI

V

c

c

c

c

1<

ccc

c

500 PRINT 510, BRGP, RACT, V2,BRGA,BRGB, ERROR, BRGT,DELTA,1RNGERR,RELERR

510 FORMAT ( 2X , F3 .0 ,2X , F7. ,4X ,F6. 1 ,3X ,2 ( F5. 1 ,3X ) , F7. ,

13X,F5.1,3X,F4.1,2X,F8.0,3X,F6,4)515 GO TO 600520 PHI=PHI+10. $ PRINT 521, PHI521 FORMAT(F5.0,24X,28HIMPOSSIBLE FIX(BRGP=180 DEG))

GO TO 600530 PRINT 531, PHI531 FORMAT ( F5. ,24X, 22HCR I TICAL ANGLE LIMITED)600 CONTINUE

INCREMENT RAPP AND GET NEW RESULTS./

610 RAPP=RAPP+RAPPINCTEST TO SEE IF RAPP HAS EXCEEDED RAPPMAX. IF YES, READNEW DATA, IF NOT, GET RESULTS FOR NEW RAPP.

620 IF(RAPP-RAPPMAX) 170,170,120900 CONTINUE

END

SUBROUTINE TO CALCULATE ANGLES FROM TO 360 DEGREESUSING TANGENTS.

SUBROUTINE ANGLES ( A , B ,C , D , ANGLE , K

)

700 RPD = .01745329252701 IF(D-B) 710,705,710705 IF(C-A)706,738,707706 ANGLE=270. $ GO TO 740707 ANGLE=90. $ GO TO 740710 ANGLER= ATANF ( ( C-A ) / ( D-B )

)

715 IF( (C-A)/(D-B) )716, 718,717716 IF(C-A)720,730,722717 IF(C-A) 722,736,740718 IF(D-B) 730,738,736720 ANGLE=ANGLE+360. $ GO TO 740722 ANGLE=ANGLE+180. $ GO TO 740730 ANGLE=180. $ GO TO 740736 ANGLE=0. $ GO TO 740

$ ANGLE=ANGLER/RPD

55

APPENDIX I

738 GO TO(750,760»740,770) <740 RETURN750 PRINT 751 $ GO TO 740751 FORMAT( 35Xil8HTARGET AT SENSOR A)

760 PRINT 761 $ GO TO 740761 FORMAT(35X,18HTARGET AT SENSOR B)

770 PRINT 771 $ GO TO 740

771 FORMAT { 35X , 19HTARGET AT XAPP,YAPP)END

ENDFINIS

EXECUTE.1500 .0001 .1

100000 100000 500000 100000 10000

200000 100000 1000000 lOOOOO 10000400000 100000 2000000 200000 10000-999. -999. -999. -999. -999.

M.

APPENDIX II

(J Read in Data:VO, VGRAD, YDIFF^

Q-*(ii

V

Read in Data

SS, RAPP, RAPPMAX, RAPPING, XINGD

III

If

SS >

NoXXI Stop

Yes

IV Gompute Sensor Position

f 3 Y-W""^ Print Input Data, Print Output Headings J

VISet Apparent Target P

Gompute Target BearingsOS it. \ngs J

Set Starting x—n,^

Goordinates :/ 1j

and Velocity x—

^

ABSF(XAPP)-ABSF(X1) <

Galculate Y Goordinate

of Ray B When X = XXI

Step X Toward XI Until

Rays Intersect at XACT,

YAGT

If

PHI > 180

Yes

XIGalculate Y Goordinate

of Ray 3 When X = XIXII

Galculate Y Goordinate

of Ray A When X = XXI

XIV Set Starting Velocityand Goordinates

No If

PHI > 180

Yes

XVIstep X in +X

Direction until Rays

Intersect at XAGT, YACTXVII

^ XVIII Compute Errors

step X in -X

Direction until Rays

Intersect at XAGT, YACT

i

cXIX

Yes

rYes

XXIncrement RAPP

if RAPP > RAPP

Print Results

2:3

If XAGT, YAGT have beenFound for all PHI

No

57

s

APPENDIX III

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59

APPENDIX III

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60

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85

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center 20Cameron StationAlexandria, Virginia 22 314

2. Library 2

U.S. Naval Postgraduate SchoolMonterey, California

3. COMASWFORLANT 1

Commander Antisubmarine Warfare Force,U.S. Atlantic FleetNorfolk, Virginia 23511

4

.

COMASWFORPAC 1

Commander Antisubmarine Warfare Force, PacificFPO San Francisco, California 96610

5

.

COMOPTEVFOR 1

Commander Operational Test and Evaluation ForceNaval BaseNorfolk, Virginia 23511

Attn: Lt. Russell Blatt, USN

6. Officer in Charge 1

Fleet Numerical Weather FacilityNbnterey, California 93940

7. Prof. William P. Cunningham (Thesis Advisor) 1

Department of PhysicsU.S. Naval Postgraduate SchoolMonterey, California

8. LCDR Jack R. Harris, USN 1

Naval Air Development Center JohnsvilleWarminster, Pennsylvania 18974

9. LCDR Arthur G. Dorsey, Jr., USN 1

P.O. Box 2344

U.S. Naval Postgraduate SchoolMonterey, California 93940

86

UNCLASSIFIEDSecurity Classification

DOCUMENT CONTROL DATA - R&D(Security claaaHlcmllon ot Ullm, body o/ abatract and indexing annotation muat ba antarad whan tha ovarmll raport la ctmaaillad)

1. ORIGINATING ACTIVITY (Corporate author)

U.S. Naval Postgraduate SchoolMonterey, California 93940

2a. RKÔNT SCCUNtTY C LASairiCATION

UNCLASSIFIEDZb SROUP

3. REPORT TITLE

CROSSFIX ERRORS OF DIRECTIONAL ACOUSTIC SENSORS IN A MEDIUM WITH ACONSTANT HORIZONTAL VELOCITY GRADIENT

4 DESCRIPTIVE NOTES (Type ol raport and Inctualve datea)

Thesis, M.S., October 19665 AUTHORfS; (Laat name, tirat name, initial)

DORSEY, Arthur Gorman, Jr.

6. REPORT DATE

October 1966

?« TOTAL NO. OF PASES

88

7b. NO. OF RKFS

7

8a. CONTRACT OR GRANT NO.

b. PROJECT NO.

9a. ORIGINATOR'S REPORT NUMBERfSJ

(]{JvÎâf hJ^sAAly^^9b. OTHER REPORT NoCS; (Any othat rtvmbaam that mmy b* maatinaa

thia raport)

10. AVAILABILITY/LIMITATION NOTICES

nrtnrn mn-/ nht'm'n_i(irnpi" nn nf—|-1iiii iiiiiiil Pi mil nriP

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

13- ABSTRACT

An investigation was made of the magnitude of the bearing and distanceerrors which result from two passive directional acoustic sensors obtain-ing a crossfix on a target in a medium with a constant horizontal velocitygradient. Results are tabulated in Appendix III. For the conditionsconsidered, the errors were found to be small in relation to the targetrange but became appreciable in absolute value at very large target ranges.However, with a knowledge of the horizontal velocity gradient, sensorseparation, sensor axis orientation, apparent target range and apparenttarget bearing the errors could be predicted. Further, by proper sensororientation, the errors could be minimized.

DD FORM1 JAN e4 1473 UNCLASSIFIED

Security Classification

UNCLASSIFIEDSecurity Classification

u-KEY WORDS

LINK A LINK B

ROLELINK C

ROLE

Acoustic SensorsFix ErrorHorizontal GradientSonobuoys

INSTRUCTIONS

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DD FORM1 JAN 64 1473 (BACK) UNCLASSIFIED

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