Algorithms For Confidence Circles and Ellipses · 2017-05-16 · NOAA Technical Report NOS 107 C&GS 3 Algorithms For Confidence Circles and Ellipses Wayne E. Hoover Charting and Geodetic

NOAA Technical Report NOS 107 C&GS 3

Algorithms For Confidence Circles and Ellipses Wayne E. Hoover

Charting and Geodetic Services Rockville, MD September 1984

U. S. DEPARTMENT OF COMMERCE Mdodm Bddrlgl, SocmtUy

National Oceenlc and Atmospheric Admlnlstrstlon Anthony J. Wio, Acthg Admhrshator

National Ocean Sew Paul M. Wdfl. Assistant ~ i s t m t w .

NOAA Technical Report NOS 607 C&GS 3

Algorithms For Confidence Circles and Ellipses Rockville, MD September 1984

U. S. DEPARTMENT OF COMMERCE National Oceanlc and Atmospheric Administration National Ocean Service

ERRATA

Hoover, h y n e E., "Algorithas for Confidence Circles m d Ellipses b'

Urshington D.C. , Nrtionrl Ocernic and Atmospheric Administr~tiOlle M O M Technical Report NOS 107 C&GS 3, (1984) pp 1-29.

l e Page 8, In the equation for p(R), Insert a closing brrcltct between the . superscript 2 and the drdy,

2. Page 13, Equrtian 7b incerrcctly reads!

p = fL + T1 + T2 + T31c/n

me correct formula ( f o r the trapesoidal'rulc)' Is:

p z k T l + T2 + T31c/a

3. Page 25, Line 19 erroneously contains "trots" the correct spelling is nerrorsn.

3-27-86 UAYNE E. HOOVER

ALtiORITHMS FOR CONFIDENCE CIRCLES AND ELLIPSES

Abstract

I n many hydrographic surveylng, navlgatlon, and pos l t lon loca t lon systems

the observed pos l t ion i s defined as t h e in te rsec t ion o f two l i n e s o f position,

each o f which may be i n error. This paper gives algorithms with new stopplng

c r i t e r i a f o r the determination o f t he p robab i l i t y that the t r u e pos i t i on T

l i e s within a c i r c l e o f given radius centered a t t he observed pos i t ion 0, and

conversely, the determination o f the radius o f a c i r c l e C w i th center 0 such

t h a t the p robab i l i t y i s p t h a t T l i e s within C. I n e i the r case, the c i r c l e

centered a t 0 i s ca l led a confidence c i rc le .

Confidence e l l i pses a r e a l so considered and a re shown t o be superior t o

confidence c i r c l e s since they provide the same p robab i l i t y o f loca t ion but

general ly over a s i g n i f i c a n t l y smaller region.

It i s assumed that the er ro rs associated with the l i n e s o f pos l t i on ray

be approximated by a nonorthogonal b i va r ia te dependent Gaussian d i s t r i b u t i o n

where the errors are measured orthogonally t o t h e l i n e s ob position. The

a l g o r i t h m given are s t ra ight forward and easy t o inplement on a microcomputer.

Biographical Sketch o f Wayne E. Hoover

Wayne E. Hoover i s a systems analyst w i t h the National Oceanic and

Atmospheric Administration i n Woods Hole, Massachusetts, and a lso i s an

adjunct professor o f mathematics a t Cape Cod C o m n i t y College i n West

Barnstable.

S t a t e Uni vers i ty . I n 1977 he received h i s Ph.D. i n nunerical analysis from Michigan

-i-

ALGORITHMS FOR CONFIDENCE CIRCLES AND ELLIPSES

TABLE OF CONTENTS

ABSTRACT

TABLE OF CONTENTS

ILLUSTRATIONS

TABLES

KEY TO SYMBOLS

1 INTRODUCTION

2 MATHEMATICAL CONSIDERATIONS

2.1 Geometry 2.2 Assurrptions 2.3 Transformation t o an Orthogonal System 2.4 The Error E l l i pse 2.5 Confidence E l l ipses 2.6 Confidence Circ les 2.7 Numerical Quadrature

3 ALGORITHMS

3.1 Algor i thn 1 f o r p(R) 3.2 Algor i thn 2 f o r R(p)

4 NUMERICAL EXAWLES

4.1 Example 1 4.2 Example 2 4.3 Example 3 4.4 Exanple 4

5 EQUIVALENT FORMJLAS FOR THE ERROR ELLIPSE

5.1 Special Cases

6 APPLICATION TO LORAN-C

7 CONCLUSIONS AND RECOMMENDATIONS

APPENDIX: CIRCULAR ERROR PROBABILITIES

REFERENCES

Page

i

li

iii

iii

i v

1

2

2 3

11

12 14

14

14 18 18 22

23

23

25

25

27

28

-i i-


ILLUSTRATIONS

Figure 1 Nonorthogonal Coordinate System

Figure 2 Orthogonal Coordinate System

Figure 3 The Error E l l i pse

Figure 4 E l l i p t i c a l Scale Factor vs. Probab i l i t y

Figure 5 The Error E l l i pse .. .

Figure 6 Confidence C i rc les w i t h a1 = 2, o2 = 1, a = 30",

Figure 7 Radius o f the 95% Confidence C i r c l e when

andpli = 0

u1 = u = 1 and p l Z - 0 2

TABLES

Table 1

Table 2

Table 3

Parameters associated wi th al = 2, o2 = l r a = 30'.

Parameters associated wi th al = a2 = 1 and p12 = 0

Areas o f 95% Confidence C i rc les and E l l i pses

andpI2 = 0

when o1 = a2 = 1 and p12 = 0

Paqe

2

4

6

8

15

17

2 1

16

89

20

-iii-

C

C

CEP

E

. g i

h

K

K i

k

n

0

P

ALGORITHMS FOR CONFIUENCE CIRCLES AND ELLIPSES

KEY TO SYMBOLS

Confidence c i r c l e

Rat io o f smaller t o la rger standard deviation: o/ax

Circu la r e r r o r probable = radius o f t he 50% confidence c i r c l e

Probab i l i t y e r r o r estimate

2.7182818284590452. . . Integrand

Aux i l ia ry funct ion

In tegrat ion step s i ze

Ratio: R/ax

i t h i t e r a t e o f K

E l l i p t i c a l scale fac to r

Length o f semimajor ax is o f confidence e l l i p s e

Length o f semiminor ax is o f confidence e l l i p s e

F i r s t l i n e o f pos i t i on

Second l i n e o f pos i t i on

Line o f pos i t i on

Pos i t i ve i nteger

Observed pos i t ion

Probabi 1 i ty associ ated w i th confidence c i r c l e o r e l l i p s e

Probab i l i t y as a func t ion o f radius o f confidence c i r c l e

C i r cu l a r e r r o r probabi 1 i ty

Radius o f confidence c i r c l e

Page

8

8

16

11, 13

7 .

9

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10, 13

8

14

7

7

7

2

2

1

10, 13

1

7, a, 13

a, 13

9

8, 14

- 1 V -

ALGORITHHS FOR CONFIDENCE CIRCLES AND ELLIPSES

X

Y

a

8

e

n

912

p Y

O X

O Y

0

1 dRMS

2dRMS

Radius o f Confidence c i r c l e

True pos i t ion

End point trapezoidal sun

as a funct ion o f p robab i l l t y

I n t e r i o r end point trapezoidal sum

Centroid or midpoint sun

F I r s t nonorthogonal ax is

Second nonorthogonal ax is

Auxi l iary function

F i r s t orthogonal axis

Second orthogonal ax1 s . Angle o f crossing, 0 < a < n 2cAI

(K/2c?

Orientat ion o f semlmajor ax is o f e r r o r e l l i p s e w i t h respect t o L,

3.1415926535897932.

Correlat ion coe f f i c i en t i n ul-u2 coordinate system

Transformed co r re la t i on coe f f i c i en t

Standard deviat ion associated w i t h L,

Standard deviat ion associated w i t h L,

Length o f semimajor axis o f e r r o r e l l i p s e

Length o f semiminor ax is o f e r r o r e l l i p s e

Variable o f in tegrat ion

Radial or root mean square e r r o r

Upper bound f o r radius o f 95% confidence c i r c l e

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1.0 INTRODUCTION

Hydrographic surveyors, navigators, and others concesned with pos i t i on

loca t ion have t r a d i t i o n a l l y determined t h e i r pos i t i on by means of two

in te rsec t ing l i n e s o f pos i t ion (LOPS). The LOPs may be derived from c e l e s t i a l

observations, t r i l a t e r a t i o n , LORAN signals, s a t e l l i t e signals, etc.

Two questions important t o pos i t i on locators are the following: (1) What

i s the probab i l i t y tha t the t r u e pos i t i on T, which i s generally unknkn, 1s

located R un i t s o r less from the observed pos i t i on 0; and conversely, (2) Nhat

i s the radius o f the c i r c l e C centered a t 0 such t h a t the p robab i l i t y i s p

t h a t T l i e s w i th in C.

'. I n e i t h e r case, a c i r c l e o f radius R which i s centered a t the observed

pos i t i on 0 i s ca l l ed a confidence c i r c le .

uncertainty o r c i r c l e o f equivalent p robab i l i t y .

It i s a lso ca l l ed a c i r c l e o f

This paper w i l l ou t l i ne the mathematical aspects o f these problems and

then give new algorithms f o r t h e f r solution. The algorithms are s t ra igh t -

forward, e f f i c i e n t , and read i l y implemented on a microcomputer.

Also, mention w i l l be made o f confidence e l l i p s e s which are ac tua l l y arch

easier t o ca lcu late than confidence c i r c les ; moreover, they are superior t o

confidence c i r c l e s since they provide t h e same p robab i l i t y o f loca t ion over a

generally s i g n i f i c a n t l y smaller area.

F ina l l y , several nunerical examples i l l u s t r a t i n g the appl icat ion o f t h e

a lgo r i thms w i 11 be presented.

-1-

ALGORITHMS FOR CONFIDENCE CIRCLES AND ELLIPSES . .

2.0 MATHEMATICAL CONSIDERATIONS

. 2.1 Geometry

Designate the two l l n e s of pos i t ion by 1, and L,, respectively, and l e t

a, 0 < a < n, be the crossing angle from L, measured i n a pos i t i ve O r

counterclockwlse d i rec t l on t o L,. Let 0 denote the In te rsec t ion of the

LOPS. Thus 0 represents the observed or measured posltlon.

.

Uefine the nonorthogonal u1-u2 coordinate system such tha t u1 and u2

In tersect at 0, u, I s perpendicular t o L,, u2 i s perpendicular t o L,, and t h e

pos i t i ve angle from u1 t o up i s n + 0. This geometry fo l lows tha t o f Swanson

[9] and i s I l l u s t r a t e d i n FIgure 1.

"I

"2

Figure 1 Nonorthogonal Coordinate System

Note that Burt, Kaplan, and Keenly, et a1 141 and also 'Bowcitch [2] use a

d i f f e r e n t geometry by reversing the d i rec t ion of the u2-axis. I n t h i s case,

the pos i t i ve angle from u1 t o up equals a. Moreover, t h i s changes the sign o f

t h e cor re la t ion coe f f i c i en t plp .

- 2-


2.2 Assumptions

Throughout t h i s paper we w i l l assune the following:

I n a small region G containing 0, t h e ear th i s f l a t and the two LOPs are s t ra igh t l ines.

The errors i n the measurements which determine Ill and L, are

normally d i s t r i bu ted random var iables wi th cor re la t ion coe f f i c i en t

zero means, and standard deviat ions o1 and 02D respectively, p12 * where u i s measured along ui which is perpendicular t o Li . The b i va r ia te e r r o r d i s t r i b u t i o n i s constant throughout the region G.

Thus i t i s assumed t h a t the er ro rs i n the measurements o f the LOPs, which

may o r may not be correlated, may be approximated by a nonorthogonal b i va r ia te

dependent Gaussian d is t r ibu t ion .

This paper applies only t o those pos i t i on loca t ion systems f o r which the

It can be above three assumptions provide the basis f o r a v a l i d e r r o r model.

a sizeable task t o decide whether t h i s model i s appropriate f o r a spec i f i c

pos i t ion loca t ion system.

2.3 Transformation t o an Orthoqonal System

Now transform the nonorthogonal ul-% system t o an orthogonal x-y

Cartesian coordinate system centered a t 0 and or iented such t h a t the angle

from 4 t o the pos i t i ve x-axis i s given by 8. Following convention, a

pos i t i ve angle i s measured i n a counterclockwise direct ion. These coordinate

systems are i l l u s t r a t e d i n Figure 2.

The transformation i s given by

5 = xsin(8 ) + ycos(8 )

- 3 -


= xsin(a - e) - ycos(a - e). "2

The angle 8, which i s defined i n the next section, i s chosen so tha t the

transformed variables are s tochast ica l ly independent.

Y

0 0

Figure 2 Orthogonal Coordinate System

2.4 The Er ror E l l i p s e

I n order t o determine the radius R(p) o f the confidence c i r c l e C, or the

probab i l i t y p(R) associated with C, i t i s necessary t o f i r s t ca lcu late the

parameters o f the error e l l ipse , namely, the lengths o f t he semimajor and

semiminor axes and t h e i r o r ien ta t ion w i th respect t o a coordinate System.

Def in ing the anc i l l a ry var iables 2

a1 = olsin(2a) + 2p12alu2sin(a)

a2 = a1cos(2a) 2 + ~ P ~ ~ ~ ~ U ~ C O S ( ~ ) + a2 2

- 4 -

ALtiORITHMS FOR CONFIDENCE CIRCLES AND ELLIPSES

2 5 = 2s in (a)

and using the transformation given i n Section 2.3, i t can be shown tha t the

given nonorthogonal standard dev ia t ions a1 and a2 are transformed t o ox and

respectively, i n the orthogonal x-y Cartesian coordinate system, where O Y S

u = [(a, - q,)/a51 4- Y

and a x ’ L o holds for a l l v a l i d values o f the input variables: a, 20,

a2 2 0, 0 < a < It, and -1 < p12 < 1. The e r r o r e l l i p s e i s the e l l i p s e wi th

center 0, semimajor ax is a x which coincides w i th the pos i t i ve x-axis, and

semiminor ax is oy which coincides wi th the pos i t i ve y-axis.

Y

The or ientat ion o f the e r r o r e l l i p s e i s calculated from

tan (a) = a,/%.

Note tha t t h i s ca lcu lat ion must be performed so tha t 0 i s obtained i n the

proper quadrant. This can be achieved with the a i d o f the double argument

arctangent function o r the rectangular-to-polar function. Thus, -n/2 < 0 <

n/2, where0 i s the angle from L, t o the pos i t i ve x-axis. As before, a

pos i t i ve angle represents a counterclockwise direct ion.

ALtiOHITHMS FOR CONFIDENCE CIRCLES AND ELLIPSES

Y

"2 0

Figure 3 The Error E l l i p s e

The error e l l i p s e with parameters ox, u and 8 i s i l l u s t r a t e d i n Flgure 3.

XY

Y ' I n the x-y coordinate system, the cor re la t ion p

var iables i s zero.

between t h e transformed

- 6 -


2 a5 Confidence E l 1 ipses

A confidence e l l i p s e i s an e l l i p s e which i s concentric t o the e r ro r

e l l i p s e and which has parameters tu,, by, and 8 ; k i s ca l led the e l l i p t i c a l

scale factor.

Since u and uy represent the standard deviations o f s tochast ica l ly

independent random variables, the addi t ion theorem f o r the chi-square

d i s t r i b u t i o n may be used t o show t h a t the probab i l i t y associated w i th a

confidence e l l i p s e i s given by

Conversely, the semimajor ku, and semiminor ka axes o f a confidence

e l l i p s e having specif ied p robab i l i t y p may be calculated from ax, uy, and Y

L k = [ -2* ln( l - p)J2m

Thus the e r ro r e l l i p s e i s a confidence e l l i p s e w i th e l l i p t i c a l scale

fac to r k = 1 and p robab i l i t y approximately p = 0.3935. The 50X and 95%

confidence e l l i pses have e l l i p t i c a l scale factors approximately l a 1 7 7 4 and

2.4477, respectively.

Figure 4 contains a graph of the e l l i p t i c a l scale factor as a function

o f probabi 15 tym

- 7 -


ELLIPTICAL ERROR PROBABILITY p

Figure 4 E l l i p t i c a l Scale Factor vs. P r o b a b i l i t y

2.6 ConfIdence C i rc les

Let C denote a confidence c i rc le , x2 + y2 = R2, which is centered a t 0

and which has pos i t ive radius R. Then the probab i l i ty p = p(R) tha t t h e true

posi t ion f l i e s wl th in a confidence c i r c l e C i s

Def ining the aux i l ia ry parameters

K = R/ax

c = o / a Y X

-8-

ALGORITHMS FOR CONFIDENCE CIRCLES AND ELRIPSES

8 = 2 c h

y -- ( K / ~ c ) ~

and the functions

i t can be shown tha t t h i s double i n teg ra l over the c i r c l e C can be reduced t o

the s ing le d e f i n i t e i n teg ra l

a

The value o f t h i s i n teg ra l provides the so lu t ion t o questian (1) stated i n the

introduct ion.

2.7 Numerical Quadrature

Values of p(K,c) have been tabulated and are given i n the Appendix.

However, i n order t o use such a table, double i n te rpo la t i on i s required. For

values more precise than those given i n the table, the In teg ra l p(K,c) must be

evaluated numerically since i t apparently cannot be expressed i n closed

form. For d e f i n i t e i n teg ra l s o f the type p(K,c), F e t t i s [6] has shwn t ha t if

- 9 -


a s u f f i c i e n t l y small step s ize i s chosen, the trapezoidal r u l e provides an

estimate f o r p(R) w i th a r b i t r a r i l y small error..

Uhenever the trapezoidal r u l e i s e f fec t i ve l y employed, Frame [7] suggests

t h a t l i nea r combinations o f the ru le w i th d i f f e r e n t step sizes w i l l provide

addi t ional estimates t o the d e f i n i t e in tegra l with only a m i n i m a l increase i n

computational e f fo r t . Such a nmer i ca l quadrature fomula i s the f i f t h -o rde r

der iva t ive corrected Simpson's ru le [lo] with step s i ze h = (b - a)/n:

2 h n

i = l + 16 1 f(a+ih-h/2)] - 60 [ f ' (b ) - f'(a)].

Since the integrand, f ( 4 ) = [eyw(+) - l ] / w ( o ) , which i s required f o r the

ca lcu lat ion of p(R), i s per iod ic w i th per iod a, i s synmetric about n, and has

continuous f i r s t der ivat ive, f ' ( 4 ) vanishes a t the end points o f the i n te rva l

o f in tegrat ion, 4 = 0 and 4 = n. Therefore, f o r the d e f i n i t e i n teg ra l under

consideration, the der iva t ive corrected Simpson's r u l e i s a l i n e a r combination

of trapezoidal sums w i th step sizes h = n /n and h = n/(2n).

The so lut ion t o question (1) stated i n the in t roduc t ion may now be

obtained by applying the trapezoidal r u l e w i t h step s i z e n/(2n) t o p(K,c),

construct ing from appropriate trapezoidal suns the der iva t ive corrected

Simpson's value, and then using the absolute value o f the d i f fe rence

- 10-


t o estimate the maximum absolute error for the former calculat ion n h k h i s

taken as an approximation t o the p r o b a b i l i l t y p(R).

The technique o f employing the f i f t h -o rde r der iva t ive corrected Sinpson's

r u l e i n order t o estimate the error, E, i n the nunerical quadrature i s

believed t o be new and i s more e f f i c i e n t than the customary repeated ha lv ing

the step size u n t i l a s u f f i c i e n t l y small d i f ference i s obtained, since the

t r a d i t i o n a l technique uses a quadrature formula o f the same order t o

approximate the error.

t i m e i s more c r i t i c a l than on la rger computer systems.

This technique Is well suited t o microcomputers where

The required inputs f o r the ca lcu la t ion o f t he p robab i l i t y p(R) a re ul,

u2 a a , pI2 R, and n. The value o f n i s chosen so that the e r ro r estimate E

i s s u f f i c i e n t l y small. I n most p rac t i ca l appl icat ions (i.e., K - 4 and e - >

0 - l ) , a value of n = 20 w i l l resu l t i n a t leas t seven d i g i t accuracy f o r p(R).

Question (2) stated i n the in t roduct ion may naw be solved by i t e r a t i n g

I n pract ice, on the radius R(p) u n t i l the desired p robab i l i t y i s obtained.

t he i t e r a t i o n i s ac tua l l y on the aux i l i a ry parameter K - R/ux. For values o f

p(R) less than 0.9999999, K assumes values between zero and 5.7.

These considerations provide an ou t l i ne o f the theore t ica l foundation f o r

the two algorithms given i n the next sect ion f o r the ca lcu la t ion o f p(R) and

R(p) associated with confidence c i rc les.

. The ca lcu lat ions required f o r the parameters o f a confidence e l l i p s e are

straightforward and have been given i n Sections 2.4 and 2.5.

3.0 ALGOR I THMS

The f i r s t a lgor i thm solves question (1) and i s a lso referenced by the

second algorithm. The ca lcu la t ion o f 8 i n step two i s an opt ional ca lcu la t ion

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since i t i s not required f o r the computation of t he p robab i l i t y p(R)

associated with a confidence c i rc le . The input parameters are ul, u2, a, p l P B

and R.

The second a l g o r i t h solves question (2) and i s based on the secant

method. Note t h a t the i t e r a t i o n i s ac tua l l y pe r fo rmd on K which i s re la ted

t o the radius o f a confidence c i r c l e by K = R/u,. The input parameters are

01 # 42 # a s 012 s and P.

3.1 Algorithm 1 f o r p(R)

2 1. 5 = ulsin(2a) + 2plZala2sin(a)

2 2 9 = ulcos(h) + 2pl2ala2cos(a) + u2

2 % = % i n (a)

8 arctan(al/a2) (Note: use arctan (y,x) or P-R function)

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ALGORITHMS FOR CONFIDENCE CIRCLES AND ELLIPSES /

3. K = R / a ,

5. Select a posit ive fnteger n (e.g., n - 4)

w(+) = (c2 - l)cos($t) - (c2 + 1)

TI = f(0) + f(n)

n-1 T2 = 1- f ( i h )

1 =1

7. E = [[TI + 2(T2 - T3)]c/nI2/6

8. I f E i s suff ic ient ly small (e.g., E < loo5), accept p = p(R).

Otherwise, select a larger value f o r n and repeat steps 6 through 8.

-13-


3.2 Algorithm 2 f o r R(p)

1.. Perform steps 1 and 2 o f algorithm 1.

2. Set 1 = 1 and select appropriate s t a r t i n g

values for the secant method (e.g. KO = 0.1, K, = 3.9. go = 0.08 - p,

and g, = 1.0).

3. Using the value o f

Ki - Ki-l Ki+l = Ki - g i g5 - giO1

where g1 = p i - p f o r 1 > 1, perform steps 4 through 8 o f algor i thm 1

t o obtain probability pit1.

4. If gi* i s s u f f i c i e n t l y small (e.g., lg i t l l < IO-'), s e t

R = R(p) =uxKi% and stop. Otherwise, repeat steps 3 and 4 w i t h

i replaced by I + 1.

4.0 NUMERICAL EXAWLES

The fo l lowing examples i l l u s t r a t e the app l ica t ion o f the two algorithms

presented i n the previous section.

4.1 Example 1

A navigator reports the ship's pos i t ion a t 41'46' N and 50'14' W.

.Assuming the angle o f crossing between the two LOPS i s a = 30'. there are no

systematic errors, and the random er rors i n the two nonorthogonal d i rec t ions

are normally and independently d i s t r i bu ted w i t h standard dev iat ions a, = 2 nm

andu2 = 1 nm, calculate the parameters o f the e r r o r e l l i p s e and the r a d i i o f

-14-


t he confidence c i r c l e s f o r the various p robab i l i t i es ind icated i n Table 1.

Also compute the sizes, areas, and the probab i l i t i es associated wi th the ldRMS

and 2dRMS c i r c l e s where ldRMS = ox + or and 2dRMS = 2*1dRUS. The term ldRMS

I s also ca l led rad ia l er ror or root mean square error.

2 2 2

Use algorlttm 1 t o calculate the parameters of the error ellipse: ox =

4.3778 nm, uY - 0.9137 nm, and e = 24.5533'. The resu l t i ng er ro r e l l l pse i s

shown i n Figure 5. . .

Continuing with algor i thm 1, set n - 7 and compute ZdRMS = 11.9443 nm,

p(2dRHS) = 0.9579, and E = 4.3*10-' where E i s an estimate of t he maximum

absolute er ror i n p(R). S im i la r l y ca lcu late the values fo r the ldRMS c i r c l e

as indicated i n Table 1.

9 24.55

WRYS 4.47 D4RYS 8.94

a,& .2l

Y

X

0

Figure 3 The Error E l l i p s e

-15 -

ALGORITHMS FOR CONFIDENCE CiRCLES AND ELLIPSES

Next, use algorithm 2 t o ca lcu late the remaining values l i s t e d I n Table

1. For reference, the ldRMS and 2dRMS values computed from a l g o r i t b 1 are

included i n Table 1. See Figure 6 f o r a p l o t o f t he radius o f the confidence

c i r c l e as a nonlinear funct ion o f probabi l i ty . Note tha t 2dRMS i s an upper

bound f o r the radius o f the 95% c i rc le . The c i r c u l a r probable e r ro r o r

c i r c u l a r e r ro r probable, CEP, i s the radius o f t he 50% c i rc le .

Use the e l l i p t i c a l scale fac to r k = 2.4477 t o ca lcu late the semimajor and

semiminor axes o f the 95% e l l pse, 10.7158 nm and 2.2365 nm, respectively.

The area o f the 95% e l l i p s e i s 75.3 nm . Since the radius o f the 952 c i r c l e

i s 8.6302 nm, the area o f the 95% c i r c l e i s 234.0 nm? Thus the area o f t he

2

95% c i r c l e i s 211% l a rger than the area o f t he 95% e l l i p s e and ye t both

provide the same confidence f o r pos i t i on location.

Table 1 Parameters assoc ia ted w i th u1 8 2, a2 8 1, a = 30°, and p,, = 0

Probabi 1 i t y P

-01 .lo a50 .68218

.75 a90 .95 .95786

e99 a999 e9999 a99999

Rad1 us R

0 . 2846 0.9565 3.1033 4.4721*

5.1216 7.2604 8.6302 8.944 3**

11.3144 14.4349 17.0573 19.3592

Area A

0.3 2.9 30.3 62.8

165.6 234 . 0 251.3

402.2 654.6 914.1 1177.4

82.4

n

1 1 3 4

5 6 7 7

8 9 10 10

Error Bound E

1.3E-13 1 9E-7 6.2E-8 1.8E-7

i.a~-8 3.4E-7 2.5E-7 4.3E-7

6.1E-7 4.OE-7 1.OE-7 1 .1E-7

* l&MS ** 2cRMS

-16-

ALtiOKITHPS FOR CONFIOENCE CIRCLES AND ELLIPSES

Figure 6 Confidence C i rc les w i t h a1 = 2 , o2 = 1, a = 30°, and p12 = 0

- 1 7 -


4.2 Example 2

Consider a pos i t ion loca t ion system where u1 = u2 = 1 u n i t and the angle

o f crossings var ies between 0 and n. For various values o f Q and assuming

= 0, Table 2 gives the parameters o f t he e r ro r e l l i p s e and the sizes, 912

areas, and p robab i l i t i es associated wi th the 95% and 2cRMS confidence

c i rc les. Table 3 gives the areas o f these 95% c i r c l e s and e l l i pses as a

funct ion of a. Figure 7 shows a p l o t o f the radius o f the 95% c i r c l e as a

funct ion o f a . 4.3 Example 3

(See Bowditch [e].) Assuming ul = 15 m, u2 = 20 m, a = SOo, and p12 = 0,

determine the p robab i l i t y o f loca t ion w i t h i n a c i r c l e o f radius R = 30 m.

Set n = 2 i n algor i thm 1 and obta in ox = 29.8895 m, uy = 13.1023 m, 8 =

15.7733', and p(30 m) = 0.6175. The e r r o r estimate i s E = 1.3*10°7 whi le the

actual er ror i s 1.0*10 . -8

Set n = 3 i n algor i thm 2 and compute the radius o f the.95% c i r c l e : R =

60.2437 m wi th E = 1.4*106. Also, using n = 5, the radius o f the 99.9%

c i r c l e i s found t o be R = 99.3274 m w i t h E = 8.1*10°9.

The parameters o f the 95% e l l i p s e a r e ko, = 73.1620 m, kuy = 32.0712 m, 2 and0 = 15.7733'. The area o f the 95% c i r c l e , 11401.8 m i s 55% greater than

the area o f the 95% confidence e l l i pse , 7371.4 m . 2

-18 -

Table 2 Parameters associated with a1 - a2 - 1 and pl i - 0

Q X e

0.05 (-.05) 810.2848 0.5 (-.50) 81.0295. 2.5 (-2.5) 16.2108 5 (-5) 8.1131

10 (-10) 4.0721 15 (-15) 2.7321 20 (-20) 2.0674 25 (-25) 1.6732

a Radius

a c-ay/ax 2dRMS Y

.70711 .00087 1620.5702 -70713 .0087 162.0652 .rot78 ,0417 32.4526 m7098 m0875 16.2883

.7180 .1763 8.2698

.7321 .2679 5.6569

.7525 -3640 4.4003

.7802 A663 3.6922

0.1 (179.9 1 (179) 5 (175) 10 (170)

, i Prob Radius Area ' p(R-2dRWS) R(p-0.95) R(p-0.95)

-95450 1588.1292 7923 581.2 .95451 158.8165 79 239.4 .95465 3 1.7805 3 173.0

796.0 095511 15.9174

.95693 8 a140 201.8

.95986 5.4069 91.8

.96375 4.1280 53.5 -96833 3.3867 36.0

26.9 -97316 2.9266 67753 2.6458 22.0 .98059 2.4950 19.6 .98168 2.4477 18.8

20 (160) 30 (150) 40 (140) 50 (130)

2mHS 9-

1.0199 1.0205 1.0211 1,0233

1.0319 1.0462 1.0660 1.0902

1.1160 1.1376 1.1511 1.1555

60 (120) 70 (110)

90 80 (100)

i! iti%i 1 1.41421 .E165 1 S774 -1 3.2660 1.2328 ,0632 -7002 3.0099

2.8721 1.1001 .9231 '-8391 45 1.oOoo 1.0000 1~000 2.8284

8250 600.6 82 514.3 3 308.6 833.5

214.9 100.5 60.8 42.8

33.5 28.5 25.9 25.1

I I 1


~~

0.1 (179.9) 1 (179) 5 (175) 10 (170)

Table 3 Areas o f 95% Confidence Circles and Ell ipses when aL = a2 = 1 and pI2 = 0

810.2848 81.0295 16.2108 8.1131

a

20 (160) 30 (150) 40 (140) 50 (130)

4.0721 2.7321 2.0674 1.6732

201.8 91.8 53 m 5 36.0

55 .O 37.6 29.3 24.6

0 \=Area o f $=Area of 95% Circ le 95% El l ipse

Y

60 (120)

;; I;;:; 90

734.71 1 078.5 73.47 216.0 14 . 69 108.4

le4142 1.2328 1m1001 1 .oooo

m7180 .7321 .7525 .7802

26e9 22 -0 19.6 10.8

2 1 e 7 20 a 0 19.1 18.8

~ 3.67

1.83 1.47

I 2.44

e8165 .86 3 2 .9231

1 mO000

~ 1 e24 1 a10 1.02 1 m o o

-20-

ALGORITHMS FOR CONFIDENCE XIRCLES AND ERLIPSES

Ffgure 7 Radius of the 95% Confidence C f r c l e when a1 02 and p12 ' -21-


4.4 Example 4

Repeat Example 3 w i t h cor re la t ion p12 = 0.5. A lgor i thn 1 with n = 3 gives uX =

36.1325 m,ay = 9.3864 m, 8 = 19.5924', and p(30 m) - 0.5666 with E = 1.9*10-8.

E = 5.8*10". The radius o f the 99.9% c i r c l e i s calculated wi th n = 7 t o be

R = 119.2786 m where E = 6.1*10-'.

Algorithm 2 with n = 6 gives f o r . t h e 95% c i rc le , R = 71.4658 m and

The 95% e l l i p s e has parameters kux = 88.4433 m, by = 22.9756 m, and 2

7he area o f the 95% c i r c l e , 16045.2 m i s 151% greater than the area 8 = 19.5924'.

o f the 95% confidence e l l ipse, 6383.8 m . 2

Comparing the resu l ts o f Examples 3 and 4, i t may be observed t h a t the e f f e c t

o f changing the cor re la t ion from zero t o 0.5 i s t o increase by 41% the area o f the

95% c i r c l e while the area of the 95% e l l i p s e i s decreased by 13%. Moreover, the

or ien ta t ion o f the 95% e l l i p s e i s increased from 15.7733' t o 19.5924'.

These examples suggest t ha t confidence e l l ipses are super ior t o confidence

c i r c l e s since they provide the same p r o b a b i l i t y o f loca t ion bu t over a s i g n i f i c a n t l y

smaller area. To be more precise, f o r any leg i t imate values o f ol, 02, a, and p12 ,

the area o f the 95% e l l i p s e i s 11*ln(400)a#~ whi le the area o f the 95% c i r c l e i s

less than the area o f the 2dRMS c i r c le , 4n(ux t uy). 2 2

I n the best o f circumstances, that i s when o1 = 02, a = n/2, and p12 = 0, t he

area o f the 95% c i r c l e i s equal t o the area o f the 95% el l ipse. However, as Example

1 shows, i n less than ideal condi t ions the 95% c i r c l e can be several hundred percent

la rger than the 95% el l ipse. Clearly, i n such si tuat ions, for any p robab i l i t y the

confidence e l l i p s e i s t o be prefer red over the confidence c i r c l e s ince a

substant ia l ly smaller area provides the same probab i l i t y o f location.

- 2 2 -


5.0 EQUIVALENT FORMJLAS FOR THE ERROR ELLIPSE

Def i n i ng 2 2 A o1 + 2Pl2o1o2co~(a) + o2

2 C = ol cot(a) + pl2olo2csc(a)

i t can be shown t h a t the semimajor and semiminor axes of the e r ro r e l l i p s e may

be calculated from

A c s z (a)[ A + [A2 - B2I2] 2 -

O x - 1

a y 'T CSC' (a)[A - [A2 - B2I2 ]

or

oy 'T A*cs?(a) - C*csc(a).

5.1 Special Cases

For the special case Q1 = a2 = o , and p12 = 0, It can be shown that the

parameters o f the e r ro r e l l i p s e s imp l i f y t o the fo l lawing:

-23-


Burt, Kaplan, and Keenly's [4] and Bowditch's [2] f o m l a s f o r t h i s

special case must be used with caution since t h e i r formulas f o r ox and oy

i m p l i c i t l y require that the crossing angle between the two LOPS nust be

acute. Their formulas give incorrect resul ts f o r obtuse crossing angles.

Now i f u l = u p = u , p12 = 0, and a i s rest r ic ted t o values s t r i c t l y

between 0 and n/2, then ox and uy may be fur ther s impl i f ied t o

'L 2

a 2 O * C S C ( O / ~ )

3- u = 2 u*sec(a/2) Y

Final ly, i f u1 = u2 = u, p12 = 0, and a - n/2, then a l l calculat ions can

be greatly s impl i f ied t o the c i r cu la r normal distr ibut ion:

= a

= U

Q X

Y U

e = o

h R(p) = a [ -2* ln( l - p)] .

- 2 4 -

ALGORITHHS FOR CONFIDENCE CIRCLES AND ELLIPSES

6.0 APPLICATION TO LORAN-C

Bregstone [3], Col l ins [SI, Pierce, kKenzie, and Woo&ard [8], and

Worrell [12] state e x p l i c i t l y o r assume i m p l i c i t l y t h a t assunptions (11, (2),

and (3) l i s t e d i n Section 2.2 with p12 = 0 may be applied t o LORAN-C. Swanson

[9] also accepts the three assumptions but suggests a value o f 0.5 fo r the

co r re la t i on o f the time-difference o r TD errors.

Amos and Feldman [l] po in t out t h a t t he TD e r r o r i s a funct ion o f .many

variables. In rea l i t y , because o f t h e current design o f many LORAN-C

receivers, the centra l l i m i t theorem o f p robab i l i t y theory appl ies and i t i s

reasonable t o assume tha t the TD er ro rs are approximately n o m l l y

d is t r ibuted.

The value f o r the cor re la t ion p12 i s often taken as zero; hcmever, it i s

l i k e l y tha t another value such as 0.5 should be used. S ign i f i can t di f ferences

i n the sizes and or ientat ions o f confidence e l l i pses as we l l as t h e s izes o f

confidence c i r c les may be observed i f the cor re la t ion i s taken as 0.5 instead

o f zero.

The U.S. Coast Guard pe r iod i ca l l y publishes revised speci f icat fons o f the

I n t h i s respect, see reference Cll]. The current t ransmit ted LORAN-C signal.

value given f o r the standard dev iat ion of. the TD erors i s 108 nanoseconds.

7 m0 CONCLUSIONS AND RECOMMENDATIONS

Algorithms with new stopping c r i t e r i a have been given which may be used

t o solve two standard problems i n pos i t i on locat ion: (1) Find the p r o b a b i l i t y

p t h a t the t rue pos i t ion T i s w i t h i n a c i r c l e o f radius R centered a t the

observed pos i t ion 0; and, (2) Find the radius R o f the c i r c l e C centered a t 0

such that the p robab i l i t y i s p t h a t T l i e s w i th in C.

-25-


It i s assumed tha t the errors associated w i t h t h e l i n e s of pos i t ion my

be approximated by a nonorthogonal b i va r ia te dependent Gaussian d i s t r i bution

where the e r ro rs are measured orthogonally t o the LOPs. The algorithms

presented f o r t h i s model are readi ly implemented on a microcomputer.

Moreover, they are p rac t i ca l since they avoid t h e use o f p robab i l i t y curves,

tables, charts, nomograms, f i c t i t i o u s functions and angles of cut, special

rat ios, sigma s t a r factors, double Langrangian interpolat ion, and Bessel

funct ions which are required by some methods o f solution.

Numerical resu l t s confirm the h igh accuracy and e f f i c i ency of t h e

algorithms presented herein f o r t he ca lcu lat ion o f t he parameters associated

w i th the e r r o r e l l i p s e and confidence c i rc les.

Confidence c i r c l e s are conceptually easi l y understood and frequently

used; however, w i t h the advent o f mlcroconputers w i t h powerful graphics

capabi l i t ies , confidence e l l i pses should be considered as a superior

a l t e rna t i ve I n appl icat ions where confidence c i r c l e s have t r a d i t i o n a l l y been

used since nuch less computation is required f o r t h e parameters o f a

confidence e l l i p s e than f o r a confidence c i r c le . kreover , t h e area of a

confidence e l l i p s e i s generally substant ia l ly l ess .than the area o f a

confidence c i r c l e having the same associated probabi l i ty ; th is can be

important not only i n rout ine pos i t ion location, but even more so, i n c r i t i ca l ’

search and rescue missions.

F ina l ly , as previously stated, the algorithms are appropriate only when

the error model described i n Section 2.2 i s v a l i d f o r the p a r t i c u l a r pos i t ion

locat ion system under consideration. Also note t h a t the algorithms nust be

modified i n s i t ua t i ons such as the m iss i l e o r t a rge t problem where the errors

are measured p a r a l l e l t o t he axes o f a coordinate system rather than

orthogonally t o the LOPs as i s t h e case i n pos i t i on locat ion calculat ions.

-26-

ALGORITHMS FOR CONFIDENCE CIRCLES AND ELL1 PSES

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 I D 1 1.8 I .9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1

,15149376 .51607270

.63187975 ,68268949 .72866788

,80639903 .83848668 .8668560 .89010142 a91086907 .92813936 .94256688

,964271 16 .97219310 .97855178 .98360193

e57628920

. m w m 6

.9n49974

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. w i m 6

.5115181 ,572595669 .62887213 .68023254 .7266 9 6 7 .76822148 .80505480 .83740(89 ,86551266 .88970083

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.99486123 ,99624767 ,99728531 .9980W17

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.42556056 A9606835 .5644571

,67235867 ,72026823 .7630349

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-99561263 .99517918 .99682936 .99652052

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.94593857 .930682ll ,9305133 .gin7137

,958373ae .9458mu .9~126981 .gsamo39 e97605221 e96791351

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. t s m s g i

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3.6 ,99968118 3.7 .99978440 3.8 .99985530 3.9 .99990381 4.0 .99993666 4.1 .99995868 4.2 .99997331 4.3 .99998292 4.4 ,99998911 4.5 .99999320 4.6 .99999578 4.7 ,99999740 4.8 .99999841

5.0 .99999943 4.9 .99wggm

5.1 .99999966 5.2 .99999980 5.3 .99999988 5.4 ,99999993 5.5 .99999996 5.6 .99999998 5.7 .99999999 5.8 .99999999 5.9 1.oooooO0O 6.0 6.1 6.2

.99968007

.99918324 ,99985453 .9999a)29 .99993632 .99995847 .99997317 .99998283 .99998912 .99999317 ,9999957 5 .9999978 .99999841

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.99999lW .99999727

.99999838 .99999833

.99999902 .99999899 -99999941 -99999940

.999993m .99999286

,99999966 .99999965 ,99999980 .99999980 .99999988 .99999988 .99999993 .99999993 .99999996 .99999996 ,99999998 .99999998 .99999999 .99999999 .99999999 .99999999

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-99999964 ,99999979 .99999988 .99999993 .99999996 .99999998 .99999999 ,99999999

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1.0000004) 1.00000000

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1.00owo000 .99999999 .99999999 .99999999 .99999997 1.00000Ooo 1.00000000 1.ooO0Oooo .99999998

.999(39999 1.-

p(K.c) = probablllty that a polnt l l e s wlthln a c l r c l r I h s e center i s at the origin and rhose radlus I s I KO,. Mere c 0 /a where a, i s the larger standard dcvlatlon. Ihc table glrw r o l u n of the standard orthogonal b l rar la te lndcpmdent Gausslan distrlbutlon.

Y ‘

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REFERENCES

1. Amos, 0. and Feldman, 0. * A Systematic Method o f LORAN-C Accuracy Contour Estimation." Washington, D.C.: U.S. Coast Guard, unpublished report, pp 1-18.

2. Bowditch. Nathanlel Ameiican Pract ica l Navigator. Washington, D.C.: Defense Mapping Agency Hydrographic Center, Vol. 1, (1977) pp 1204-1237.

3.

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11 .

Bregstone, Edward Washington, D.C.: U.S. Coast Guard, telephone comn ica t i on , (1981) .

Burt, W. A., Kaplan, 0. J . , and Keenly, R,* R., e t a1 "Mathematical Considerations Per ta in ing t o the Accuracy of Pos i t ion l oca t i on and Navigation Systems.' k n l o Park: Stanford Research Ins t i t u te , Naval Warfare Research Center Research Memorpdum NWRC-RM 34, (1965) pp 1-21.

Col 1 ins, James "Formulas f o r Posi t ion ing a t Sea by Circular, Hyperbolic, and Astronomic Methods." Washington, D.C.: National Oceanic and Atmospheric Administration,NOAA Technical Report NOS81, (1980) pp 1-26.

Fet t i s , Henry E. "Numerical Calculat ion o f Cer ta in D e f i n i t e In tegra ls by Poisson's Summation Formula.'' Math. Tables and Other Aids t o Computation, 9 (1955) pp 85-92.

Frame, 3. Sutherland ' "Numerical In tegrat ion and the Euler-Maclaurin Summation Formula." East Lansing: Michigan State University, unpublished report, pp 1-9.

Pierce, 3. A., McKenzie, A. A., and Woodward, R. H. - LORAN. M I T Radiation laboratory Series Vol. 4, New York: McGraw- H i l l Book CO., (1948) pp 425-431.

Swanson, E. R. "Estimating the Accuracy o f Navigation Systems. A S t a t i s t i c a l Method o f Determining Ci rcu lar Errors Probable." San Diego: U.S. Navy E l e c t r i c a l Labs., Research Report 1188, (1963) pp 1-5.

"An E f f i c i e n t Modi f icat ion o f Euler-Maclaurin's Formul a." Trans. Japan Society f o r C i v i l Engineers, 24 (1955) pp 1-5.

U.S. Coast Guard "Speci f icat ion o f the Transmitted LORAN-C Signal." Washington, D.C.: Department o f Transportation, U.S. Coast Guard COMDTINST M16562.4, July 14, 1981.

Tanimoto, B.

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12, Worrell, Clarence L. "Production of LORAN-C Reliability Diagrams a t the Defense Mapping Agency." Washington, D.C.: Defense Mappfng Agency Hydrographic Center, unpubli shed report, pp 1-9.

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