DMA TR 8400.1

7.2 Propagation Through Known Functions

Another method for the propagation of errors [1] is the
Propagation of known errors through known functions. This method assumes

the errors are independent and known.

are derived and shown in Appendix C.
measured quantities u and v, where x

The equations for this propagation
A quantity, x, is computed from two
= f(u,v) denotes a function of u and

v. The error o, of x is affected by the errors in both u and v. When

% and o, are randomly distributed,
computed by the general equation:

x
where: % =
Oy % =

2x ax

du? Sy 7 Partial deri

the propagated error, o, » can be

2
+ (yo? (7-6)

the standard error of x

the standard errors of u and v

vatives of x, with

respect to u and v.

Application of the general equation
following rules for the function f(u,v

Rule 1. “Addition and Subtraction

to specific conditions produces the
):

xX = au tby
9, = aé o,2+ b 25 2 (7-7)
Rule 2. Multiplication and Division
x = tauv
2 2
~. “us 4 (7-8)
u v



OMA TR 8400.1

4.3 Precision Indexes

A precision index [7] reveals how errors are dispersed or scattered
about zero and reflects the limiting magnitude of error for various
probabilities. For example, 50% of all errors in a serfes of measurements do
not exceed +20 feet; 90% do not exceed +49 feet. Although different errors
are given, each expresses the same precision of, the measuring process (Figure
4-3, page 22). The standard deviation ( a ) and average error (n) are two
indexes with theoretical derivations, Common usage has included three
additional probability levels which are, in effect, precision indexes: (1)
probable error (PE), (2) map accuracy standard (MAS), and (3) and three sigma
error (30) .

The standard deviation is the most important of the indexes and has
the probability of:

to,
P(x) = f Ax) dx = 0.6827 (4-9)
0,

Or, 68.27% of all errors will occur within the limits of + Oy +

The average error is defined as the mean of the sum of the absolute
values of all errors:

n
Y 1Qy-%|

net Lisl (4-10) r

The probability represented by the average error is 0.5751, or 57.51%. The r
average error is easily computed from the standard deviation: ~

n= 0.7979 oy (4-11)

20



fre

cee ee Oe eee ee ee ee

REPORT DOCUMENTATION PAGE one ae ono o18e

Tehor a SMTEH at nigrmatn w erumanee io average nour poy reponse monaing me TRE WON ieeracny sonny ened ata aa
ans the data needed, and completing sod reviewing tne colecion'olintermnionTM Send comments regarding es burden emote other spect ts
stions for sper) this burden, fo Washington Headquarters Services, Directorate for information Operations a1 para a
way, Suite 1204, Taringeaas UX 72202-4903, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington,

. AGENCY USE ONLY (Leave blank) | 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

‘AND SUBTITLE 5. FUNDING NUMBERS
Error Theory as Applied to Mapping, Charting,

and Geodesy
—

z ING ORGANIZATION NAME(S) AND ADORESSIES) . BERF
Defense Mapping Agency REPORT MUMIER
8613 Lee Highway . , so,
Fairfax, VA 22031-2137 DMA TR 8400.1
SORING/ MONITORING AGENCY NAME(S) AND ADDRESS(ES, FONSORING / m
N/A : N/A : }
ft. SRREERTRT WoTes 7 4
Report based on previous DMA Technical Reports
[Tie OSTRIBUTION /AVAILABLLITY STATEMENT 12b. DISTRIBUTION CODE
Distribution authorized to U.S. Government

Agencies and their Contractors (Administrative
or Operational Use). Requests for this document
should be referred to the Defense Technical
Information Center as stated in the Foreword. 4
STRACT (Maximum 200 words)

Error theory provides the precision index which identifies the error
distribution and specifies the probability that the true error does
not exceed a certain value. These indexes are applicable to the
evaluation of cartographic and geodetic information and make possible
accuracy statements having a uniform interpretation. Standardized
procedures and supporting theory for computing linear, circular, and
spherical precision indexes are presented. The recommended procedure
for computing the circular or spherical standard error from the linear
standard errors in their components is to average the linear standard
errors. Other precision indexes in the same error distribution are
easily computed from the linear, circular, and spherical standard
errors.

(Ta. SUalECT TERMS
Probability Error Propagation
Circular Errors Precision Indexes

Map Accurac’
. 7. Y CLAS ni
GF REPORT OF THIS PAGE ‘ABSTRACT
UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED
. ‘Standard Form 296

bic) a Brescribed by ANS! Std. 233-1 18
7 - OC 7 298-1

erical Errors

iON



DMA TR 8400.1

mean that the. error in position is exactly the value shown, rather it
expresses the probability that the true error in a given position will not
be larger than the error given.

_ Positional error should be expressed by precision indexes which
immediately identify the form and probability represented by a given
error. The precision index chosen should be’ that which has been
conventionally used for the type of data represented. For instance, the
90 percent probability used at DMA for map errors. Another example is the
error in position associated with artillery or missile testing, or weapons
delivery. In these cases the errors have conventionally been expressed as
circular error probable (CEP). Using error theory and the factors given
in this report, the different accuracies and precision indexes can be
related. For example, if an accuracy is given by a weapons analyst as 100
feet CEP, we know the form of the error is circular and the probability is
50 percent. This implies a 50-50 chance that the geodetic position in
question does not vary more than 100 feet from the true geodetic
position, If we are interested in the equivalant error at the 90 percent
probability level, multiply 100 ft. by 1.8227 (from Table 5-5) to yield a
90 percent probability that the positional error will not exceed 182 ft.

Errors in different forms are more easily understood when
precision indexes comion to Jinear, circular, and spherical error
distributions are used. Precision indexes suitable for expressing
positional error include (1) the linear, circular, and spherical standard
errors representing 68.27%, 39.35%, and 19.9% probabilities, respectively,
(2) the linear probable error, circular error probable, and spherical
error probable representing 50% probability in each distribution, (3) the
map accuracy standard, circular map accuracy standard, and spherical
accuracy standard representing a 90% probability level, and (4) a
probability level approaching near-certainty for each distribution which
the positional error is theoretically unlikely to exceed; (a) three sigma
(linear, 99.73%), (b) three-five sigma (circular, 99.78%), and (c) four

66

hy e_a any ayaa ss ss ST SLU SKA 3S SC

—


DMA TR 8400.1

Appendix C
‘PROPAGATION OF ERRORS THROUGH KNOWN FUNCTIONS

Suppose that we wish to determine a quantity, x, which is a function
of at least two other variables, u and v, which are actually observed
(measured), We will determine the errors in x from those for u and v and
from the functional dependence [1] [2] [4].

x= fluyv, 2. .) {C-1)
The errors in x can be determined by considering the dispersion in

the values of x resulting from combining the individual measurements Uy oy
+ «+ into individual results Xj.

X= f (Uy, yy...)
x2 = f (Ug, va, «)

Xy = FlUns Vas se) (C-2)

It will be assumed that the most probable value of xX, x fs:

x = f (U0, Ve...) : (C-3)
where U,V are averages of the measured variables.

It ts shown [1] that the variance of x, a? will be given by:

C-1

a a



DMA TR 8400.1

An example is the probability of efther a 3 or 4 occurring on the single roll
of die:

P(3 or 4) = 1/6 + 1/6 = 1/3

Rule 4. The probability of two {ndependent events occurring
simultaneously is equal to the product of their individual
probabilities.

P(A and B) = P(A) * P(B).

If A is the value of the first die and B is the value of the second die, then
the probability that A = 3 and B = 4 in a single roll of both dice,

P(A = 3 and B = 4) = 1/6 * 1/6 = 1/36,

The probabilities of occurrence of the numbers summed from each of 36
possible combinations resulting from the single rot] of two dice are presented
in Figure 3-1. The probability of rolling the number 7, for example, ts 6/36
or 1/6 since there are six combinations which have a sum of seven. A histo-
gram shown in Figure 3-1 is a discrete representation of the normal
probability curve for the roll of two dice.

3.2 The Normal Distribution of a Continuous Random Variable [6] [7]

The area under the normal probability density curve (Figure 3-2a,
page 14) represents the total probability of the occurrence of the continuous
random variable x and is equal to one, or 100%, The mathematical expression
of the curve is the normal probability density function, p(x):

aw
p(x) = = ae (3-1)
a 7

where: x = the random variable
10



DMA TR 8400.1

TABLE A-1 (cont'd.)
% 9, o o,
03 0.0376 0.2468 0.4951
04 0.0502 0.2857 0.5479
05 0.0627 0.3203 0.5932
06 0.0753 0.3518 0.6334
07 0.0878 0.3810, 0.6699
08 0.1004 0.4084 0. 7035
09 0.1130 0.4343 0.7349
10 0.1257 0.4590 0.7644
11 0.1383 0.4828 0.7924
12 0.1510 0.5056 0.8192
13 0.1637 0.5278 0.8447:
14 0.1764 0.5492 0.8694
15 0.1891 0.5701 0.8932
16 0.2019 0.5905 0.9162
17 0.2147 0.6105 0.9386
18 0.2275 0.6300 0.9605
19 0.2404 0.6492 0.9818
19.9 : 1.0000 .
20 0.2533 0.6680 1.0026
21 0.2663 0.6866 1.0230
22 0.2793 0.7049 1.0430
23 0.2924 0.7230 1.0627
24 0.3055 0.7409 1.0821
25 0.3186 0.7585 1.1012
26 0.3319 0.7760 1.1200
27 0.3451 0.7934 1.1386
28 0.3585 0.8106 1.1570
29 0.3719 0.8276 © 121751
30 0.3853 0.8446 1.1932
31 0.3989 0.8615 1.2110
32 0.4125 0.8783 1.2288
33 0.4261 0.8950 1.2464
34 0.4399 0.9116 1.2638
35 0.4538 0.9282 1.2812
36 0.4677 0.9448 1.2985
37 0.4817 0.9613 1.3158
38 0.4959 0.9778 1.3330
39 0.5101 0.9943 1.3501
39.35 1.0000
cu) 0.5244 - ° 153672
41 0.5388 1.0273 1.3842
42 0.5534 1.0438 1.4033
43 0.5681 1.0603 1.4183
44 0.5828 1.0769 1.4354
45 0.5978 1.0935 1.4524

A-2

Pee eee



DMA TR 8400.1

R2
aa
P(R) = P= 1-e (5-8)

where:

~
.

c the circular probability distribution function, a spectal
case of P(R)

the radius of the probabilfty circle

z
i

a
"

c the circular standard error, a spectal case of o,when

Or = NR Tye

When 6, and oy are not equal, the P(R) function, (5-7), is
modified by letting “a" equal the ratio o,/ dy where oy is the smaller
standard error of the two. Then from Appendix D, part 3:

p(R) = —22 i eY 1 (vk) 4
roy ° v (5-9)
lta 0
where:
2 2
xs R' tt +a J
40, a
y
2 2
ve +; bay
4oy a
k= 1- a@

Equation (5-9) can be solved for different probabilities or values of P(R)
representing precision indexes of the error distribution.

c
oF r
i]
c

28



DMA TR 8400.1

11, Kalafus, R.M. and G.Y. Chin; “Measures of Accuracy in the NAVSTAR/GPS:
2DRM vs. CEP", Proceedings of the National Technical Meeting, The
Institute of Navigation, Long Beach, California, January 21-23, 1986.

12. Defense Mapping Agency: "DMA Product Maintenance System Manual"; DMAM
8570.1, Washington, D.C., January 1988.

13. Rice, S.0.; "Properties of Sine Wave Plus Noise"; Bell System
Technical Journal, Vol. 27, No. 1, January 1948.

14. Grad, A. and H. Solomon; "Distribution of Quadratic Forms and Some
Applications", The Annuals of Mathematical Statistics, Vol. 26, 1955.
ot

15. Hart, R.G., "A Formula for the Approximation of Definite Integrals
of the Normal Distribution Function", Mathematical Tables and Other
Aides to Computation, Vol. XI, No. 60, October 1957.

16, Shultz, M.E.3 Circular Error Probability of a Quantity Affected by a
Bias; ACIC Study No. 6; United States Air Force Aeronautical Chart

and Information Center (ACIC, now Defense Mapping Agency Aerospace
Center); St. Louis, Missouri; June 1963.

17. Stroud, AsH. Approximate Calculation of Multiple Integrals; Prentice-
Hall; Englewood Cliffs, New Jersey; 1971.

18. Greenwalt, C.R. and ME, Shultz: Principles of Error Theory and

Cartographic Applications; ACIC TR No. 96; United States Air Force
Aeronautical Chart and Information Center (ACIC, now Defense

Mapping Agency Aerospace Center); St. Louis, Missouri, February 1962.

19. Greenwalt, C.R.; User's Guile to Understanding Chart and Geodetic
Accuracies; ACIC RP No. 28; United States Air Force Aeronautical

Chart and Information Center (ACIC, now Defense Mapping Agency
Aerospace Center); St. Louis, Missourt; September 1971.

CO OC" TD mas ee. eU[UL sSlUmestlUme le


OMA TR 8400.1

Figure 5-1. Normal Circular Distribution

35



DMA TR 8400.1

Assuming the horizontal and vertical components are independent,

2
Fag Pagar °
Qp= le 0 (7-16)
pp Aghr 7 Ar
2
0 0 © ale

From here the horizontal circular relative error and vertical linear
relative error are determined with the same formulas used for the absolute
accuracies.

62

r
r
:
;
|



OMA TR 8400.1

An approximate circular standard error is determined from
equation (5-9) by letting P(R) = 39.35% and R = % + Values of o/on,,
for ratios of o,;,/o,,, from 0.0 to 1.0 are contained in Table 5-2 and
plotted in Figure 5-2 (page 39). For the Guin’ Snax ratio between 1.0 and

0.6, the curve is a straight line with the equation:

o0~ (0.5222 Gnin + 044778 quay) « (5-11)

A rapid approximation gives a slightly targer % value for the same
Onin! Snax ratio:

9g, ~ 0.5000 (o, + oy) . (5-12)

As nin’ Snax approaches zero, the 39.35% probability curve follows a
transition from circular, through elliptical, to the linear distribution
form. The curve does not effectively represent a circular standard error
for nin’ Snax ratios less than 0.6 because it is not compatible with
other circular precision indexes. For example, the factor 1.774 converts
a circular error at 39% probability to a circular error at 50% probability
when Onin! Snax = 1.0, but when Snin = 0, the factor converting a linear
error at 39% probabiltiy to a linear error at 50% probability is 1.3094.
The circular standard error computed from equation (5-12), however, can be
converted to other circular precision indexes by constant circular
conversion factors for Omi n/ Snax ratios between 1.0 and 0.2 and is,
therefore, the preferred method for approximating the circular standard
error.

The circular error probable is the circular error which 50% of
all errors in a circular distribution will not exceed, or the value of R
in equation (5-5) which makes P, = 0.5, The CEP in a truly circular
distribution (i.e. Oy = oy = 9 ) ts computed by:
R2
~ F628

c

0.5= l-e

31



DMA TR 8400.1

Vartous types of points require different parameters to establish
Precise positions. These have been discussed as one, two, or three
dimensional coordinates, For example, a vertical position (elevation)
requires only a one-dimensional Coordinate - the height of the Point above
a reference datum; a geodetic Position is expressed by two-dimensional
Coordinates - latitude and longitude referenced to a specific datum; and
Spatial positions require three-dimensional coordinates such as the x, y,
Z coordinates in a rectangular system. The errors accumulated in the
Process of determining the various Positions must be evaluated in the same
dimensions required to express the position. Errors for vertical
Positioning can be assumed to follow a normal tinear distribution; those
for a geodetic position -~ a Circular distribution; and the errors for a
Spatial point can be assumed to follow a normal spherical distribution.

8.2 The Accuracy Statement

Three major groups of data fall within Department of Defense

] Positioning requirements: (1) maps, charts and other graphics; (2) data
bases of digttal data such as Digital Terrain Elevation Data (DTED); and
(3) specific points. By the use of error theory, a horizontal accuracy
evaluation of the DMA Positioning product as a whole can be obtained. In
the case of maps or graphics, a specified, probability (90 percent) that
the true errors in well-defined planimetry will not exceed, is used as an
accuracy statement for the map. Map accuracy can also be interpreted as a
Percentage - the Percentage of well-defined Points which contain errors
not exceeding the given Magnitude, Similarly, vertical accuracy is stated
as a 90 percent probability that the linear errors in vertical position
will not exceed a specific value. DMA also expresses the accuracy of its
digital data bases in terms of 90 percent probability. For some data
bases, for instance Digital Terrain Elevation Data (DTED), the header
record contains both absolute and point-to-point (relative) error
values. These accuracies are given as 90 percent horizontal (circular)
and 90 percent vertical (linear) errors. The accuracy statement does not

65

eee ae



20

21.

22.

23.

24,

25.

i
DMA TR 8400.1

Bjerhammar, E.A.; Theory of Errors and Generalized Matrix Inverses;
Elsevier, New York, New York, 1973.

Defense Mapping Agency; "Accuracy Evaluation Methods"; DMA INST
8421.1, Washington, D.C., 2 dune 1987.

Ostle, B. and R.W. Mensing; Statistics in Research; 3rd Edition, The
Towa State University Press, Anes, Iowa, 1975.

Clifford, A.A.; Multivariate Error Analysis; Applied Science
Publishers, London, 1973,

Gelb, A., Editor; Applied Optimal Estimation; The M.I.T. Press,
Cambridge, Massachusetts, 1974,

Worthington, A.G.; Treatment of Experimental Data; John Wiley and
Sons, Inc., New York, New York, 1959,

H-3



DMA TR 8400.1

oO

@aing AWIGEGOId 3dSW“b-G BNI
atte“)
=
L 60 80 £0 90 so v0
q

€0o



OMA TR 8400.1
Step B
2 2 f
To get the quantity t+ ct, - 1)] in the form of (vk):
fo, o, [
2
2 0
ve Pye dy) r
4o ° . =
y K
~
L

02 «2
Gz rast
o, 2
Cy - 1)
TR

a7

ac

6 te
1+

a+ -

a t

Let a= a where oy is the smaller of the two: -

y c

he

ces

) &-y eet

|

ata

k= . « . bn 6 \
9: (1 +5) Lee lta (6)

Satz YY eS) -
x x a t
Step C 7
=
Getting oy and sy in terms of a: r

2 2

20, 0, a2 a2

aa: 6, "2 Ohe of +g, tigte er a C
(+32) ae y
wv

D-6



DNA TR 8400.1

1. INTRODUCTION

Cartography, photogrammetry, and geodesy, as practiced at the Defense
Mapping Agency (DMA), involve the measurement of physical quantities ‘and
the utilization of such measurements. Regardless of the precision of the
instrument, no measurement device or method gives the true value of the
quantity measured. Mechanical imperfections in instruments and the limi-
tations introduced by human factors are such that repeated measurements of
the same quantity result in different values. Variations among successive
values are caused by errors in the observations. The true error of each

observation is the difference between the true value of a quantity and the
measured value, -

While the theory of errors does not yield a true value nor improve the
quality of observations, it does Provide a way of estimating the most
Probable value for the quantity and of determining the certainty attribu-
table to the estimate. This measure of the certainty will be called the
Precision index and will be the value that is attached to the DMA product
to express the product's reliability. On such DMA graphics as the Air
Target Chart this evaluation is expressed in the form of a reliability
diagram, The reliability diagram will express the horizontal accuracy and
vertical accuracy of the chart. Although some charts lack an accuracy
statement, the basic concepts of horizontal and vertical accuracy apply to
all series of charts and other products such as positioning data bases.

Using error theory, the user of DMA products can develop for himself
accuracy estimates for products that do not furnish reliability infor-
mation, This report will describe how to combine error measures to obtain
the reliability estimates desired by the user. Use of error propagation

techniques allow the user to relate DMA furnished accuracies to the
variables of his interest.



DMA TR 8400.1 Ly
2 May 1991
Lal

DEFENSE MAPPING AGENCY B®
(0013 LEE WQHwaY f
FAIRFAX, VIRGWWA 22031-2137 i
e
“See?

DEFENSE MAPPING AGENCY TECHNICAL REPORT 8400.1

ERROR THEORY AS APPLIED TO MAPPING, CHARTING AND GEODESY

FOREWORD

1. This publication is issued to disseminate timely and useful
technical information to organizations engaged in cartography,
geodesy, and related subjects.

2. DMA TR 8400.1 is based on ACIC Technical Report No. 96,

. February
1962; ACIC Reference Publication No. 28, User's Guide to
» September 1971; and
ACIC Study No. 6,

, June 1963. This report draws on material in these three

reports.

3. This report has been approved for release to United States
Government Agencies and their Contractors. Copies may be obtained

by forwarding a request to the Defense Technical Information Center,
Cameron Station, Alexandria, VA 22304-6145.

CHARLES M. ROSE
Colonel, USAF
Acting Chief of Staff


OMA TR 8400.1

5.6 Circular Conversion Factors

The relationships of the circular standard error to other
circular precision indexes [10] are summarized in Table 5-4. Conversion
factors’ (Table 5-5) computed from these relationships convert a circular
error at a given probability to a circular error .at another probability.
When a circular error distribution is substituted for an elliptical
distribution, the circular conversion factors are retained,

Table 5-4
Summary of Circular Precision Indexes

Symbol Probability Derivation

% 23935 1.0000 o,

CEP -5000 1.1774 %

MSPE 6321 1.4142 %

CMAS ] -9000 2.1460 %

3.5 9978 3.5000 8

Table 5-5
Circular Error Conversion Factors
To

From 39.35% 50.00% 63.21% | 90.00% 99.78%
39.35% 1.0000 1.1774 1.4142 | 2.1460 3.5000
50.00% 0.8493 1.0000 1.2011 | 1.8227 2.9726
63.21% 0.7071 0.8325 ‘1.0000 | 1.5174 2.4749
90.00% 0.4660 0.5486 0.6590 | 1.0000 1.6309
99.78% 0.2857 0.3364 0.4040 | 0.6131 1.0000

38



OMA TR 8400.1

Appendix A

PERCENTAGE PROBABILITY FOR
STANDARD ERROR INCREMENTS

The “following table presents the increments of linear ( oy ), circular
( o ), and spherical ( o, ) standard errors for intervals of one percent

probability. Percentage levels corresponding to precision indexes are
underlined.

Factors for converting the error at one Percentage probability to
another within the same “distribution are derived by dividing the standard
error increment of the new percentage probability by the standard error
increment of the given percentage probability. An example is the

conversion from the circular map accuracy standard (90%) to the circular
probable error (50%):

CEP = 1.1774 %

CMAS = 2.1460 o,
1774

> 25140 OMS
«*. CEP = 0.5486 CMAS

TABLE A-1

Conversion Factors for Converting Standard Errors
to Various Different Percentage Probabilities

% oy % o,

00 0.0000 0.0000 0.0000
01 0.0125 0.1418 0.3389
02 0.0251 0.2010 0.4299



DMA TR 8400.1

Appendix D

DERIVATION AND SOLUTION OF THE TWO-DIMENSIONAL
PROBABILITY DISTRIBUTION FUNCTION

1. Derivation. The probability density functions [9] of the independent
errors "x" and "y" are :

2 2
x
p(x) = ~tee 27 and py) = ——ee Ze
x y

Using Rule 4, Section 2.1.:

2 2
wl ( + )
Pou) = wage 7 "4,2 az
x
2 2
: 1 x
-o boat)
P (x,y) wee IJ e 2 “9,2 az dx dy (1)

Using polar coordinates:

2 = re cos 8

y =r? sine 8
where r is,the radial error and r = x2 + y?

P(r)=P (re dxtey? cr) = P( xy R) (2)

where R is the radius of the probability circle.

D-1



Mi

on



DMA TR 8400.1

b DMA MISSION STATEMENT

The Defense Mapping Agency shall provide support to the Office

of the Secretary of Defense (OSD); the Military Departments; the
Chairman, Joint Chiefs of Staff and Joint Staff; The Unified and
Specified Commands; and the Defense Agencies (hereafter referred
to collectively as "DoD Components") and other Federal Government
Departments and Agencies on matters concerning mapping, charting

and geodesy (MC&G).

iii



OMA TR 6400.1

Then
2 2

x2 = A; + ad (5-3)
% %

For values of p(x,y) from 0 to = , a family of equal probability densit,
ellipses are formed with axes K 9, and K % .

When o, = oy >» equation (5-2) becomes : (5-4)

-202

fin [plx.y) 0,2 24] = x? + y?

For a given value of p{x,y), the left side of equation (5-4) is a constant
which is the square of the radius of an equal probability density circle.

The probability density function integrated over a certain region
becomes the probability distribution function which yields the probability
that x and y will occur simultaneously within that regton, or:

Puy) = f J p(x.y) dx dy (5-5)

However, since both positive and negative values of efther "x" or "y" will
occur with equal frequency, the errors may be considered as radial errors,
designated by “*r", where r = xvoeyh.

§.2 Elliptical Errors

The probability of an ellipse [10] is given by the distribution

function:
2

K'
P(xy) = L-e Zs (5-6)

The solution of equation (5-6) with values of K for different
Probabilities yields the results shown in Table 5-1. For a 39%

26



DMA TR 8400.1

4. Solution. of Modifted Function. To compute the CEP [13] [14] (CEP = R
when P(R) = 0.5) for values of a, * 1.6 and a #7 4, two methods are

available:

Method 1:

To determine the value for x by Rice's table’of I, (vk) dv, enter the
table with values of k and the, required probability. 1 2
<a

P(R) = 50% probability; a= —%= 0.8165; a¢ = 0.6667; k « = 0.2
(R) p 3 ey H Tea?

P(R) werd i €Y 1, (vk) dv

2

x
50 (1 +a . “v
80 (Lea) fre 1, (vk)dv
0
x
0.5103 = fe 1, (vk) av
0

Enter the table with k = 0.2 and interpolate for 0.5103 to get the value
of x.

«6 = 4517
5103
8 = 5516
ote X = 0.71732
2

x pit 2, = 0.71732
Toyz a °

0-8

Spo om om me oe ge

a



OMA TR 8400.1

Then:
1 oR ped ot Fd A cos 4
P(R) = ye a 6, for x a 4 f é % % 3 dr
r=0 0
Rearranging terms:
r2 5 J Tv rd 6, 2
R - [1 + ] - L - 1] cos @
P(R) = 4 fore 40? a th se 40” a dé] dr
x ’ y
r=0 470
Let: - .
e (Se ] 2
17 - - 1] cos 6 re ©,
a e fo, oy d¢ J=1, vee oe ea
y x

where I, is a Bessel Function, zero order, modified first kind.

Therefore: 2 2
GC.
R eat [1 + ] 2 2
PIR) = SA fore 407 we 1g 2-1 dr (3)
x 'Y p29 Sy oy

2. Special Case of Two-Dimensional Probability Distribution Function.

When o, = % = OG, from equation (3):



OMA TR 8400.1

Precision is affected only by the random errors in the measuring
process while accuracy is affected by the precision as well as the
existence of .unknown or systematic errors. The difference between
precision and accuracy is illustrated in Figure 2-1 where the plots of
errors in a circular distribution are shown, In Figure 2-la, the points —
are grouped closely together and the measurement is said to be "precise".

It is also accurate because the center of the group coincides
with the center of the circle. In Figure 2-1b, the grouping is still
precise but inaccurate because it is not centered on the center of the
circle. Instead, the mean of the points is offset by a systematic error
or bias, The measurements shown in Figure 2-1b are “inaccurate because
of the bias even though they are "precise" (grouped closely). In
Figure 2-lc, the points exhibit neither clase grouping nor nearness to the
center, They are, therefore, not precise and not accurate. Measurements
can be precise and inaccurate at the same time, but they can never be
accurate unless they are precise also.

The basic definition of an error distribution assumes that
systematic errors and blunders have been removed and only random errors
are left. However, systematic errors cannot be removed from positional
information unless some means exist for their detection such as comparing
this information against given control. Consequently, if systematic
errors are not removed, they will have an effect (for example) on geodetic
and photogrammetric measurements and the resulting positional information.
It is not always possible to remove systematic errors from positioning
information. To give the user some knowledge of the accuracy of the
product he is using, methods have been devised to state the uncertainty of
the products.

In the discussion of horizontal and vertical accuracy, the terms
relative and absolute accuracy are often used. Absolute accuracy has the
Same meaning as accuracy, that is, how wel] does the measured quantity



SPOWRaW UOReUUIKOSddy 40113 puepURS jeoaYds Jo UOSEdWIOg —-T-g aunBig

DNA TR 8400.1

(0 “op p)

(€‘oL‘or)—_(v‘or ‘ous (S ‘on ‘1 (9-01 ‘ou (2 ‘Ot OL) (8 ‘01 ‘OW) (601 ‘OL) (OL ‘OL ‘on .
F T T r T t T OL

(70+ 0+") eeeeo

€

=
{0+ 20+70



DMA TR 8400.1

Appendix F
SUBSTITUTION OF THE CIRCULAR FORM FOR ELLIPTICAL ERROR DISTRIBUTIONS

SE

——E=e”
6 . Goun
Tam = O9 Tmon 8

No
XS

SZ ee
Tan. Srnie.
a =0.7 rae =0.6
.
Fel



DMA TR 8400.1

TABLE A-1 (cont'd)
% ; oy o, 9,
46 0.6128 1.1101 1.4695
47 0.6280 1.1268 1.4866
48 0.6433 1.1436, 1.5037
49 0.6588 1.1605 1.5209
50 0.6745 1.1774 1.5382
+5 . 1. 194% .
52 0.7063 1.2116 1.5729
53 0.7225 1.2288 1.5904
54 0.7388 1.2462 1.6080
55 0.7554 1.2637 1.6257
56 0.7722 1.2814 1.6436
57 0.7892 1.2992 1.6616 |
57.51 “0.7979 ;
3 0.8064 1.3172 1.6797 4
59 0.8239 1.3354 1.6980 |
60 0.8416 1.3537 1.7164 |
60.82 1,7321
a 0.8596 1.3723 :
62 0.8779 1.3911 1.7540
63 0.8965 1.4101 1.7730
63.21 1.4142 ii
tr 0.9154 Tat 1.7924 |
65 0.9346 1.4490 1.8119
66 0.9542 1.4689 1.8318
67 0.9741 1.4891 1.8519
68 0.9945 1.5096 1.8724 il
68.27 | - 1.0000
ism T0152 1.5305 1.8932 |
70 1.0364 1.5518 1.9144
71 1.0581 1.5735 1.9360
72 1.0803 1.5956 1.9580
73 1.1031 1.6182 1.9804 a
74 1.1264 1.6414 2.0034
75 1.1503 1.6651 2.0269 il
76 1.1750 1.6894 2.0510
77 1.2004 1.7145 2.0757
78 1.2265 1.7402 2.1012
79 1.2536 1.7667 2.1274
80 1.2816 1.7941 2.1544
81 1.3106 1.8225 2.1825
82 1.3408 1.8519 2.2114
83 1.3722 1.8825 2.2416
84 1.4051 1.9145 2.2730
85 1.4395 1.9479 2.3059



DMA TR 8400.1

1-05 = e c

oak

20,

In 0.5
R2 = 0.69315 (29,7)

Ro = 1.1774 %
CEP = 1.1774 % (5-13)

When 9, and o are not equal, an approximate CEP is determined from
equation (5-9) by letting P(R) = 50.00% and R = CEP. Values of CEP/ aaa y

for ratios of Omin/ max from 1.0 to 0.0 are tabulated in Table 5-3, The
50% probability curve plotted in Figure 5-3 is approximated by a sertes of
Strafght lines for different ratios of nin’ Snax with the equations:

CEP ~ (0.6142 Onint 0.5632 Ona x)
when g.1,/o,,, 18 between 1.0 and 0.3

CEP ~ (0.4263 min + 026196 Cmax)

when quin/oma 1S between 0.3 and 0.2

A rapid approximation of the CEP plots as a straight line which intersects
the 50% probability curve at the point where Fain! Snax = 0.2 and has the
equation:

CEP ~ 0.5887 (o, + sy)
when o4,/omqx 1s between 1.0 and 0.2

The CEP computed by this equation is compatible with the circular standard
error computed by equation (5-12) and is, therefore, the preferred method

32



DMA TR 8400.1

6. THREE DIMENSIONAL (ELLIPSOIDAL, SPHERICAL) ERRORS

6.1 Introduction

_ A three-dimensional error is the error, in a quantity defined by
three random variables. Expanding on the example in Section 5.1., a point
is referred to X, Y, and Z axes which establish the Spatial position of
the point. When random and independent, the errors x, y, and z each have
a linear probability distribution. The three-dimensional probability
density function is expressed by:

2 2 2
1; x z
. “2a apt ay)
Pix.y.z) = ——pt—— ° (6-1)
(2m)? % % 9

Similar to Section 5.1, the probability density function can be written:

2 22

2.
x
War tar t gr e)
where:

3
we = -2 In (plx.y,z) % % °, (2n)e }

For values of the constant W2 from 0 +, the density function defines a
family of ellipsoids of equal probability density.

6.2 Ellipsoidal Errors

“The probability of an error ellipsoid is given [10] by the
probability distribution function:

. W 1 42
P(s) = Es fe-~2* at (6-3)

45


DMA TR 8400.1

The spherical accuracy standard is defined as the magnitude of
the. spherical radius S when the function P(S).= 0.9 or 90%. Expressed in
the form S = C os the spherical accuracy standard fs computed by:

SAS = 2.5000, (6-13)

The four sigma error, representing a spherical probability of
99.89%, approaches near-certainty in a spherical. distribution and has a
magnitude four times that of the spherical standard error.

Table 6-2
Solution of P(S) Function for P(S) = 50.00%

au nin SEP ~ Ssog SEP ~ 0,512/ (9, ty + o,)
ax Snax Letting o.3, 2}
0.866 0.866 1.4016 o,, 1.4007
1.0 0.707 1.3892 o,., 1.3879
0.775 0.632 | 1.2341 a,” 1.2341
0,577 0.577 1.1016 ¢,,, 1.1044
0.894 0.447 1.2104 ¢,., 1.2002
0.707 0.408 1.0894 a, : : 1.0844
0.535 0.378 0.9791 oy : 0.9808
1
0.354 0.354 0.8689 o,,, 0.8757

50



OMA TR 8400.1

where: =

the radial error;

s« Vea yeke

Ss
te
ont |
Oy, TM Standard error of the radial error *s"

The solution of equation (6-3) for W yields the values given in Table 6-1.

Table 6-1
Values for the Constant W

aaa s3 53 3 5 8 ee oe

“Probability W
19.9% 1,000
50 1.538
60.8 1,732
90 2.500
99 3.368
99.89 4.000
fea
46



DMA TR 8400,1

From equation (2);
t n n (4)
= - % = 0

A A

This shows that the sum of the differences about the mean is zero,
which.was expected, but if equation (3) is squared and then summed:

xy?» 42-2 K+ T? (5)

Ko? = xg? = 2Xo ¥+ x2
vee eeeaseeececcoreeeseees

2. y2 2
x, x 7 XN + ¥

2 mye % 2 (6)
Px =f xp > yxy +
i=l

cy
The most probable value will be found when J x4? = 0, or the
i=l
most probable value of X will be that which makes

n
+ x? = a minimum, In order to find this minimum, differentiate
i=l

equation (6) with respect to K and equate to 0:

a) 2 T x,+20k © 0
a xo = -2 +2nk «
ax i=l 1 iy!
n x;
ow. X « oh TT (7)

Equation (7) proves that the mean value X is the most probable
value of a set of independent observations. Therefore, in the
determination of the residual value it is correct to use the mean value as
an approximation of the true value.

B-2

= Se ee ee ee ee



DMA TR 8400.1

1 o G, % 9 °%
oo Ey ms won
20 areeel * 2 eater] = 2 be 1° (7)
i sees ee
Step D
2 2 1 o,2 +o 6,2 +?
¢, g. o o oe
1+ eth ee oe a o,2 y-£1 (8)
x x o2 x x a
yor ge
Combining Steps A, B, C, D and equation (3):
2 2
R' lt+a 2 2
Te i] wf pity 2 2 2
2a y a To 2 r lta lea
P(R) -—> e y’ a If = (5) ] dv. (9)
lta J ° a2 an Oa )
Rewriting equation (9):
2a *% wy
p(R) = —“A5 f &Y 1, (vk) dv (10)
lta 0
where:
2 2 2 2 2
R lta r lta l-a
= ——] ; Vegegl—pd] i ke eS
* % c a + a G ta )
D-7



es

WSE'GE = (Y)d YOYM UOHDUN, (y)d @Y) JO aAIND Z-g andy

DMA TR 8400.1

ma
s
od

+ SAIN ANgeqoig %GE'6E

si . dor
ao" (o +*0).0005'0 9p -———o Timo 2
_—— TMo
-_ and
7 08'0
+
4060



|ENERAL INSTR!

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Standard Form 298 Back (



DMA TR 8400.1

requirement for high absolute accuracy for point positioning, the
recommended product is the Point Positioning Data Base. This product
allows positioning directly from a data base that avofds the errors caused
by map construction and the representation of the positioning data on a
Paper copy.

‘The statement [12] for a map of descriptor 1 is that. 90 percent

of all well-defined planimetric features are located within 0.5mm (0.02.

inches) of their geographic position with reference to a prescribed
datum, Table 8-1 gives the ground distance in feet of 0.02 inches on a
map for various scales of DMA maps. This number forms a limit for the 90
percent accuracy of a map at the given scale. These limiting case
accuracies for descriptor 1 products should not be confused with general
product accuracy which will usually be less accurate.

Among the positioning errors in photogrammetric adjustments or on
maps, there are often those which are not measureable or able to be
modeled and which must be estimated by empirical methods. When this is
necessary, an additional assumption must be made to the effect that such
data is compatible with the computed data and that the empirically derived
error data will also follow the theoretical normal error distribution.

Table 8-1
Ground Distance Equivalent to
0.02 of an Inch at Chart Scale

Chart Scale Distance (feet)
ONC 1:1,000,000 1667
TPC/PC 1:500,000 833
JOG 1:250,000 417
ATC 1:200,000 333
Topo 1:100,000 167
Topo 1: 50,000 83
64

[are os ome ne OO DO ee es Po Wl ces Vs ee ee Oe WU, Oe,

|



a aen kt ek maiikiaiz

DMA TR 8400.1

The following alternate method of approximating o, is not
recommended because of limited applicability:

" Z
a~ {/ ~—— (6-8)
when nin’ nax is between 1.0 and 0.9

The spherical error probable is defined as the magnitude of the
spherical radius S when the function P(S) = 0.5 or 50%. Expressed in the
form S = Co, » the spherical probable error is computed by:

SEP = 1.5382 os (6-9)

The P(R) function for two-dimensional errors is solved by the use of Grad
and Solomon's tables. Expanding this method into the spherical
distribution, the radius S for a 50% probability sphere (Ssog) was
computed in terms of Trax for ratios of Grin’ Snax and Gnid/ Snax and
tabulated in Table 6-2. Utilizing these values, an approximation of the
spherical probable error can be computed:

SEP ~ 0.5127 (9, +o + o,) (6-11)

Y
when Smin/nax is between 1.0 and 0.35

The mean radial spherical error is the radius of the error sphere
equal to 1.7320,, or 3 Oy» in a truly spherical distribution. When
o, #0, # a, , the MRSE is computed by:

x y Zz
/ 2 2 2
MRSE ~ Yo," + oy +o, (6-12)

when “nin/ Snax is between 1.0 and 0.9

The probabilities represented by the MRSE are computed by equation (6-6).
Because of the variation in probability, the MRSE is not recommended for
use as a precision index.



OMA TR 8400.1

In Equation (G-8), S denotes a sphere of radius o, -

cc] oo oe

for the linear case, Equation (G-4), the numerical integration may be
performed using Simpson's Rule. For the two and three-dimensional cases,
Equations (G-6) and (6-8), numerical integration may use the 15th degree

Gauss Product Formulas [17]. .

tv} ry

r

6-4



DMA TR 8400.1

The. probable error is that error which 50% of all errors ina.
linear distribution will not exceed. Specifically, the true error is equally
likely to be larger or smaller than the probable error. Expressed
mathematically:

b
PE = f p(x) dx = 0.50 . (4-12)
a

The probable error is computed from the standard error:

Tx
PE = 0.6745 Tt? 0.6745 oy (4-13)

The U.S. National Map Accuracy Standards specify that no more than
10% of map elevations (a one-dimensional error) shall be in error by more than
a given limit. The standards are commonly interpreted as limiting the size of
error of which 90% of the elevations will not exceed. Therefore, the map.
accuracy standard is represented by:
bt
MAS = f ox) dx = 0.90 (4-14)

or, computed from the standard deviation:
MAS = 1.6449 o, (4-15)

The three sigma error, as the name implies, fs an error three times
the magnitude of the standard deviation, The 3o0error is used because it
approaches near certainty -- 0.9973 or 99.73% probability. Since the
Probability of a linear random error falling outside these limits is
sufficient cause to consider such errors as "blunders".

The meaning of the standard deviatton with respect to the normal
distribution function is illustrated tn Figure 4-3. The vertical axis, p(x),



DMA TR 8400.1

PX)
4

Xj=Hy — €j* -o My +0 Xj, =Pyt €)

Population

p(X)

Xp = Ky 0x x tax Xj =X xj

Figure 4-1. Normal Probability Density Curve of Observed Values



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WOwaA AVANIT NOUUA UVANTT ‘Bowes BVANTT WOUNE AVANTT
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wOwss TOMAS: WOUda TVOATHAS FTCTE MIA CLG YOUN TVOREGHAS
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coool | sers'o | ecee'o | s¥zz'o | £2°66 oooor | sars0 | eeeeo | BrezO | £166 000! | eerse | eect’ | spzz"o | e266 Oooo" | SerS'O | ERED | BPZz‘O | C166

creel joooe'r [esos j1o1ro | 06 6eze'1 |oo00r |osos'o |torve | 06 eezet |oooot |oses'o |101ro | 06 6ez8'T |oo00T Jogoso | 1orro | 06

ooo's |err9't | 00001 | stz9'.0 | 27°89 0000'E }6rr9'T | 0000! | s29°0 | 27°89 oooo's |erre't |ooo0't | srz90 | uze9 oo00's |erra'l 0000"! | srz9'0 | az89

sever [Zev | 9zBFT | 0000'T | 40S surry | ieee |sceri | oo00T | %0s SLvYD | L8Er% | 9c8Y1 | OOOO! | OS SLPF'D | 298h'2 | 9zer1 | 0000 | %OS

wares | 206 |uezeo | sos | > weres | oe feezes | wos | ‘wares | soe fuzzes | vor [ wez'es | %06 fulzes | wes | o

sowda HVANIT ‘aOuda AVANTT ‘MOUNT YVANTT MOUNT AVENIT



DMA TR 8400.1

Table 5-2
Solution of P(R) Function for P(R) = 39.35%

comin —<
- Trex wax
1.0000 1.0000
0.8165 9.9063
0.6547 0.8197
0.5000 0.7323
0.3333 0.6327
0.2294 0.5727
0.1005 0.5274
0.0 0.5151

Note: When P(R) = 39.35%, R ~ Oe

: Table 5-3
Solution of P(R) Function for P(R) = 50.00%

Pain _teP

max Fina x
1.000 1.1774
0.8165 1.0683
0.6547 0.9690
0.5000 0.8707
0.3333 0.7696
0.2294 0.7174
0.1005 0.6835
0.0 006745

Note: When P(R) = 50.00%, R ~ CEP

30



OMA TR 8400.1

or written in matrix form:
y =c+ x (7-3)

where ¥ = [yy yglt

% = [xy xg]t .
c = [ay by It

a, a
C | |

It can be shown £2] that the covarfance matrices are related by:

s t -
Ey * SExy © (7-4)
b
Yy Wye a) a2 A ° xX a, P)
or y. =
b
VYe fy, oe a er on ay D2

The matrix C is called the Jacobian of y with respect to x,

denoted by dyxe

J, (7-5)

yx *

ET)

It should be noted that even if there fs no covariance in x,
that is

there may still be a covariance in y .

56

e

r
r
r.
r
r
r
r
r
I
r
n
Hi



DMA TR 8400.1

TABLE A-1 (cont'd.)
% 9 % Og
86 1.4758 1.9830 2.3404
87 1.5141 2.0200 2.3767
88 1.5548 2.0593 2.4153
89 1.5982 2.1011 24063
90 1.6449 51460" 2,500.
92 1.7507 2.2475 2.5998
93 1.8119 2.3062 2.6571
94 1.8808 2.3721 2.7216
95 1.9600 2.4477 2.7955
96 2.0537 2.5373 2.8829
97 2.1701 2.6482 2.9912
98 2.3263 2.7971 3.1365
99 2.5758 3.0349 3.3683
99.73 3.0000
99.78 3.5000
99.89 4.0000
99.9 3.2905 3.7169 .
99.99 3.8905 4.2919 4.6094

_

Loose bias Uitkiens Wiens Mien Wace Uithann iene Wibase Wiese Wien Wibnas Wits Wiis: Wi.

=



DMA TR 8400.1

Factors for converting from one probability level to another are
shown in Table 4-1. For example, an error of #20 feet in elevation at 90%
probability can be converted to #12 feet at 68.27% by multiplying 20 feet by
the factor 0.6080.

Table 4-1

Linear Error Conversion Factors

To
From 50.00% 68.27% 90. 00% 99.73%
50.002 1.0000 1.4826 2.4387 4.4475
68.27% 0.6745 1.0000 1.6449 3.0000
90.00% 0.4101 0.6080 1.0000 1.8239
99.73% 0.2248 0.3333 0.5483 1.0000

4.4 Examples of Linear Errors

The foregoing discussion demonstrates the use of the normal distribu-
tion in the analysis of random errors. There are numerous opportunities for
the occurrence of random variables in cartographic and geodetic work. For
example, the base lines and measured angles, observed lengths of lines, eleva-
tions, etc., resulting from geodetic triangulation, traverse, and leveling all
contain error. Celestial and gravimetric observations as well as distances
measured by trilateration are also examples of measures where linear errors
occur. The principles of error theory can be used advantageously to analyze
the results in terms of the specifications established for the survey.

At DMA, the normal linear error distribution has important applica-
tions with respect to evaluating the accuracy of positional information. In
addition to the one-dimensional errors which exist in such positional data as
elevations above mean sea level, the linear error components of two-dimen-
sional positions can be analyzed by applying principles of the normal linear
error distribution. The following sections contain discussions of the utility
of the linear standard error for analyzing two and three-dimensional
distributions.

23



DMA TR 8400.1

s2
ft Ss 5? 2a,"
2 P(S) =¥5 f ye ds
0 9,
Integrating by parts:
- sz
rs
s s 20,
Let us, dy = —ye dS
s 6
s
2¢,
du = &, v= -e 5
s
a a
Jz 78 s e 255
p(s) = VETS) (2 )+ j = ds] (5)
s s

In order to use approximation formula [14], P(S) must be transformed to
the integral of .-t7/2g _

Letting C a >» 4S = o, dC, where o, = constant :
os s , s

2 oS
ns) /2 [-ce7* oa Ly °s dc] (6)

From above reference when x > 0:



rn
A
a
L
1
1
1
1
1
4
i
L
+

4
:
=

DMA TR 8400.1

2. ERRORS
2.1 Classes of Errors

Errors fall into three general classes which may be categorized
by origin as (1) blunders, (2) systematic, and (3) random [1] [2] [3].

Blunders are mistakes or gross errors. Blunders may be caused by
the wrong reading of instruments, transposing numbers or equipment
failures. Blunders are usually large and easily detected. Blunders
cannot be considered part of the sample from a statistical point of
view. For this reason great care should be taken to avoid blunders when
making observations. Observation and data collection methods should be
planned to include redundancy and reasonableness checks. Since blunders
are considered to be outside the population, they should be investigated
and explained before being eliminated from the population. If there is no
evidence of disturbance in the observations or data causing the lack of
homogeneity in the population, data should not be eliminated solely on the
basis of the magnitude of the error. Often data or observations are
edited on the basis of a three sigma test. In this test the observations
are averaged, and any that are more than three standard deviations from
the mean are rejected. When an accuracy evaluation is being based on a
data set there can be no automatic elimination of data as blunders.

Systematic errors affect the observations in the same way, hence
they are hard to detect by repeated observation. Systematic errors may
have the same sign and value, the so-called constant error or bias. In
cartographic applications, systematic errors may occur due to instrumental
factors or due to human limitations. Systematic errors may follow some
pattern such as refraction or distortion due to the curvature of the
earth. The method of compensating for systematic errors is to model them
mathematically. The theorical model of the observations should attempt to
mathematically compensate for a1] known systematic errors. If the effect



| DMA TR 8400.1

Appendix B
THE MOST PROBABLE VALUE

| . Since the true value of a measured quantity is never known, the most

probable value of the quantity must be determined from the observed
| values. The following proof [4] will show that the arithmetic mean of the
. observed values is the most probable value of the quantity:

Symbols:

| X = an unknown quantity

X; = the observed values of the unknown quantity;
Xe = Nye Kos Xz nee Ky (1)

X = the arithmetic mean of the observed values;

n x;
yosf woe ornk =
f=1 i

ina

x (2)
1 i
X; = the error in an observation; (3)

xp eX - X

Proof:



DMA TR 8400.1

The two-dimensional probability distribution functions is:

v2 ain? C) cost C)
R ray “3 C +
1 e aye % r dr d0

vr do dr (small increment resulting
from dx and dy)

Using identities: sinte = 4a - cos 20)

costo = $a + cos 26)

oe

2
R on af paces ze , L+cos 20
Tr 2
PR) = pot ff e ay o, r dr do
XY x0  o=0
Rearranging terms: 2 . 2
1 1 Zr 1 1
R -Tl—yt—y) - DL —y—-—s] cosz0
PR) = aegng re vee ‘feta? oF dedr
Y p=0 6-0
let ¢ = 2,
dg = 240
0-2



DMA TR 8400.1

TABLE OF CONTENTS (Cont'd)

PAGE
E. Derivation of the Spherical Probability Distribution
FUNCEION. soseeeseseeececescesscsccscesessnestavesescasees E-1

F. Substitution of the Circular Form for Elliptical Error
Distributions.............

G. Error Probability of a Quantity Affected by a Blas....... G-1

REFERENCES. 40. 000ceesscesseccessecssceancseesscessserssensecs H-1

xi



or more symmetrically,

Rule 3. Powers
If x is obtained by factors raised to various Powers
x = yayd

Then (C-5) becomes:

o* J 4228-2204 2 222 y2b-2,, 2

or symmetrically (obtained by dividing by x = uayd)

C-3

DMA TR 8400.1

(C-8)

(C-9)



DMA TR 8400.1

Appendix E
DERIVATION OF THE SPHERICAL PROBABILITY DISTRIBUTION FUNCTION

The combined probability density distribution function of the
independent errors x, y and z are;

x2 2 22
1 20, 1 26 , 1 20”
p(x,y,z) = te er a a e q)
a, a 3,

In the spherical case where = = = OF t

5 2,2 2
= x* + y* +z
Pixsysz)éx dy dz = 1 dog dx dy dz (2)
a? (2%)?
Converting to 3-dimenstonal coordinates;
£=2 cof ¥ cost a
f = coe ¥ sine a
Za? sity
£ +¥ +2«? cof ¥ cost r+ 5? cos? ¥ sin a+ s? sin? y
=¢ cok (cos? A+ sin? + s? stn? y

= 52



OMA TR 8400.1

Rule 3. Powers

x = udyd
6, o 2 o 2
a eo) (7-9)

Indexes other than the standard error can be used to propagate
errors. For example, using Rule 1:

7 (pe)? + (PE)?

(PE), =

% . TM * ny

% . o + oy

(was), = % (mas)? + (Mas),?

(30), = 4 (30),2 + (30),

However, note that the index must be consistent throughout the formula.
Similarly precision indexes in two and three dimensions may be
propagated. For example:

7.3 Application of Error Propagation Methods

The principle source of covariance information to be used fn the
propagation of errors is least squares adjustment. In the case of
positional information for DMA products these adjustments may be
analytical photogrammetric triangulation. In strict terms, the inverse
matrix (wrt) is not a covariance matrix, but a cofactor matrix usually
denoted by Q.

qg= wl

58

c
J
r
r
r
FE
r
i
F
I

z


OMA TR 8400.1

6.5 Spherical Conversion Factors

The relationships of the Spherical standard error to other
Spherical precision indexes [10] are summarized in Table 6-3. Conversion
factors (Table 6-4) computed from these relationships convert a spherical
error at a given probability to a spherical error at another Probability.

Table 6-3
Summary of Spherical Precision Indexes

x

Symbo? Probability Derivation
9, 2199 1.000 9,
SEP 50 1,538 9,
MRSE -608 1.732 9,
SAS 90 2.500 9,
4 9, 29989 4.000 9,

Table 6-4

Spherical Error Conversion Factors

To \ |

From 19.9% 50% 60.8% 90% 99.89% a

19.9% 1.000 1,538 1.732 2.500 4.000 i
50% 0.650 1.000 1.126 1.625 2.600
60.8% 0.577 0.888 1.000 1.443 2.309
90% 0.400 0.615 0.693 1.000 1.600
99.89% 0.250 0.385 0.433 0.625 1,000

i Te ee ea



OMA TR 8400.1

x
P(x) = f p(x) dx

2
x
x -&-y
Pix) = f Le 2a ax (3-4)
= of Br

The value of P(x) ranges between 0 and 1, illustrated in Figure 3-2b,
approaches its upper limit, P(x) approaches 1;
limit, P(x) approaches zero.
than its defined limits.

As x
as X approaches its lower
This is true since x cannot exceed nor be less



DMA TR 8400.1

Aunqeqoid %SE'GE ©} %00'0S 40) $10}984 UOISIaAUOD Jo Ydely

80. “£0 90 EX*) v0 €0

‘9-G aunBiy

43


DMA TR 8400.1

and (4) the spherical near-certainty error, four sigma ( 4a, ). The mean
radial spherical error (MRSE), an index which has been used at DMA, is not
recommended © because the probability represented varies when
yr Oye and a, are not equal.

The probability of an error sphere of radius equal to the
spherical standard error is computed by equation (6-6) for the condition

c = 1 as follows:

Ss
fz = 0.7978846

e 7 B= 0.60653

e “0-4 . 0.67032

0.8e “4 = 0.53626
P(S) ~ 0.79788 (1.253 - 06065 ~ 0.3948)

ot. P(S) ~ 0,20 or 20% . (6-7)
For a truly spherical distribution, the linear standard error are

equal and identical to the spherical standard error (9, =o =o = o) +

When = os Sys and o,are not equal, the spherical standard error is
approximated by:

5, ~ 0.333 (o, + vy +,)

when Swin/nax is between"1.0 ahd 0,35

The substitution of a spherical form for an ellipsoidal distribution is
not recommended when the o,;,/o,,, ratio is less than 0.35.

48

t

I



OMA TR $400.1

2 2
2 . aye @&) + a? @) + 20, ) ) (C-4)

If u and v are independent ( wy * 0) then the standard deviation of
x ts:

2 2
o,* / a Ht) + a? (#) . (+5)

Special Formulas for Error Propagation

Rule 1. Addition and Subtraction

The parameter's a and b are defined as positive constants. If x is
the weighted sum of u and v,

x = au + bv
Then (C-5) becomes:

o = eo? + ro? (C-6)

x

or in the special case where a = b = 1
a = F, + a? (c-7)

Rule 2. Multiplication and Division

If x is the weighted product of u and v
x = + auv
Then (C-5) becomes:

oa, avea +arwe,

ee Fe ee eee oe oe oe ee a Um lee

a. nce


ee et ed dS

DMA TR 8400.1

compare to the true value. DMA defines absolute horizontal accuracy as
the statistical evaluation of all random and systematic errors encountered
in determining the horizontal position of a single data point with respect
to a specified geodetic reference datum. It is expressed as a circular
error at the 90 percent probability level. Absolute vertical accuracy is
defined as the statistical evaluation of all random and systematic errors
encountered in determining the elevation of a single data point with
respect to mean sea level. It is expressed as a linear error at the 90
Percent probability level. Absolute accuracy ts determined by comparing
data points on a map, chart or related product to points with known
horizontal positions and elevation on the specified geodetic datum and
mean sea level. .The absolute error will include all uncertainties and
biases associated with relating the product to the specified datum. DMA
defines relative (point-to-point) horizontal accuracy as the statistical
evaluation of all random errors encountered in determining the horizontal
position of one data point with respect to another. It is expressed as a
circular error over a specified distance at the 90 percent probability
level. It defines relative (point-to point) vertical accuracy as the
statistical evaluation of all random errors encountered in determining the
elevation of one data point with respect to another. It fs expressed as a
linear error over a specified distance at the 90 percent probability
level, The - main thing that should be pointed out about relative
accuracies is that they do not translate into absolute accuracies. For
instance, the relative accuracy involved in resolving a point on an image
is not the absolute accuracy of positioning that point on the ground with
respect to a geodetic datum.

Obviously due to their nature, relative errors will be smaller
than absolute errors. If an absolute measure of the accuracy of a data
set or map is required, one way to obtain it is to compare the data to an
independent set of data that is known to be accurate in the geodetic frame
of reference. An example of this would be compartng map points to points
positioned by field survey which is on the datum for which the absolute
accuracy is desired.



OO ee ee ee ee eee ee

aang AWgeqold ISUW 2-9 eunBiy

(0 “49 0)
. (6‘2‘ou)
(9 ‘e ‘ou (4‘v‘08) = (8 ‘S OL) (89 ‘ou ftor ‘OL OW)
t T t t T v

52

7 %S9

DMA TR 8400.1
g


DMA TR 8400.1

PREFACE

Optimum utilization of Defense Mapping Agency (DMA) production
requires that the accuracy of the source material, interim and final
products be considered. This accuracy is expressed by an error statement
which indicates whether the product is reliable and acceptable or should
be used with discretion. Therefore, the error statement must be
representative of the product and have a sound statistical basis. The
purpose of this report is to present and explain the theory and procedures
for providing a valid and meaningful error statement.

The normal distribution of linear errors is explained in detail
because two and three-dimensional error distributions are more easily
analyzed statistically by individual treatment of the linear components.
The principles of the linear error distribution apply only to independent
random errors, assuming that systematic errors have been eliminated or
reduced sufficiently to permit treatment as random errors.

This report is based on three reports produced at the Aeronautical
Chart and Information Center (now the Defense Mapping Agency Aerospace
Center) . These reports are Principles of Error Theory and Cartographic
Application, ACIC Technical Report No. 96, February 1962; Users Guide to
Understanding Chart and Geodetic Accuracies, ACIC Reference Publication
No. 28, September 1971; and Circular Error Probability of a Quantity
Affected by a Bias, Studey No. 6, June 1963. This report borrows heavily
from the material in these three reports.

L vii


DMA TR 8400.1

Spherical Error Formulas

Percentage

Precision Index Symbol Probability Formula
Spherical o 19.9% a, = 0.3333(g + % + 9,)
Standard Error when nin’ Snax 2 9-35
Spherical SEP 50% SEP = 1.5382 o.
Error Probable

SEP = 0.5127 (o, + % + o,)

when nin’ Snax > 0.35

Spherical SAS 90% SAS = 2.5003 9,
Accuracy Standard
Spherical Near- 4a, 99.89%

Certainty Error
(Four sigma)

4.0000 o,

Spherical Error Conversion Factors

To
From 19.9% 50% 612% 90% 99.89%
19.9% 1.000 1.538 1.732 2.500 4.000
50 0.650 1.000 1.126 1.625 2.600
61 0.577 0.888 1.000 1.443 2.309
90 0.400 0.615 0.693 1.000 1.600
99.89 0.250 0.385 0.433 0.625 1,000

71



DMA TR 8400.1

represents the mean value for the measured quantity. The normal error
distribution function about the mean is expressed in one sigma units centered
on the mean. . The property of the distribution curve fs twofold:

- The total area under the distribution curve is equal to unity.

- The area under the curve between any two values’ of x; and x» fs
equal to the probability of an error occurring between these limits.

- The area under the curve between the limits Xp" o and Xt - 0
is 68.27% of the total area under the curve. Under the assumption that the
errors are normally distributed, this means there is a 68.27% probability that
errors in any further measurements under the same conditions will not exceed
the standard deviation. The standard deviation does not indicate the
probability that an error of a certain size will occur, it only indicates that
approximately 68% of the errors will fall within the specified limits of plus
or minus one sigma.

90% (MAS)
99.7% (30)

Figure 4-3: Normal Linear, Distribution

22



DMA TR 8400.1

Summary of Formulas and Conversion Factors

Linear Error Formulas
Percentage
Precision Index Symbol Probability Formula

Standard Error o 68.27% .
. /3 (X; - Xe . Y) x2
% n-1 n-1

where: X; = measured value
of the quantity X; Xy, Xp

see Xn

X= the most probable value
(arithmetic mean) of xX

Ts
7 rey
x= the error; x =X; - X

n = number of measurements

Probable Error PE 50% PE = 0.6745 oy
Map Accuracy MAS 90% MAS = 1.6449 ay
Standard

Near-Certainty 36 99.73% 3.0000 o

x

Error (Three sigma)

Linear Error Conversion Factors

To
From 50% 68.27% 90% 99.73%
50% 1.0000 1.4826 2.4387 4.4475
68.27 0.6745 1,0000 1.6449 3.0000
90 0.4101 0.6080 1.0000 1.8239
99.73 0.2248 0.3333 0.5483 1.0000

69



- DMA TR 8400.1
by 215o values

Fist (0, + a,”) + ae, + 7, + (0,77

and (7-14) -
2.1 2,. fl, 2, 22 2
wy 210, +0,” - Flo toy + (oy,)
Now
% “3 (a, + 9)
and 2,
90% CE = 2.146 %
for 2K y/o <1.

To compute the relative accuracy using the 6 x 6 covariance
matrix Q, change the varfables such that:

A6 = 6) = 64
| : Ad = dg = Ay
| the hy = hy.

/

The point-to-point covariance matrix will be:

= ggut -
pp JQ (7-15)
where
“1004100
J=/ 0-1 0010
00-1001

61



DMA TR 8400.1

4. ONE-DIMENSIONAL (LINEAR) ERRORS

4.1 Linear Errors [4] [8]

_ An error in a measurement is the difference between the "true" value
of a quantity and the measured or derived value. The "true" value can never
be determined because of instrument limitations and human fallibility, but can
be estimated by taking a sufficient number of measurements. In determining
the value of a quantity, only one measurement may be necessary when an approx-
imate value is sufficient. If, on the other hand, the quantity is important
enough to require a more precise value, repeated measurements are made.
Variations will exist between the values obtained from several measurements.
Applying the theory of the normal distribution to these measurements, the
"best" value for the quantity is the Jean or average of all the observed
values. The differences between the mean, usually denoted by the Greek letter
“mu" (u}, and the observed values are the apparent errors or residuals which
are used to estimate a statement of preciston for the measuring process. When
the residual errors (the measurement minus the mean) here denoted by (x), are
randomly distributed about the mean, the precision of the measurements is
expressed by a single term, the standard deviation. The standard deviation is
designated by the Greek letter "sigma" (o)and sometimes referred to as the one
sigma (lo) error. The square of the standard deviation (0%) is called the
variance, For a linear distribution, the standard deviation is computed by
squaring all the residual errors, adding the squared values, dividing by the
number of errors (less one), and taking the square root:

4.2 Application of the Probability Density Function to Random Errors [7]

The normal probability density curve of an infinite number of mea-
Surements of the unknown quantity X is expressed by Parameters analogous to
those of equation (3-1). The true value, us is the mean of the distribution
of the observed values Xy> Xp, X35 aeons The curve, illustrated in
Figure 4-1 (page 17), has the mathematical form:

15



DMA TR 8400.1

for approximating the circular probable error within the specified limits.

Although a circular error concept jis not recommended for

min! Snax ratios less than 0.2, a near-linear 50% probabilty error may be

computed to represent a CEP for lower ratios when a comparison of circular
errors derived from different sources is required:

CEP ~ (0.2141 Grin * +6621 o,,,)
when Snin/%max 1S between 0.2 and 0.1

CEP ~ (0.0900 Guin + 026745 Cmax)

‘when Omin/max 1S between 0.1 and 0.0

CEP ~ 0.6745 Cmax

when Gin = 9

The following alternate methods of computing an approximate CEP are not
recommended because of limited applicabflity:

2 2

0

o
cep ~ 1.1774 / 4t
and CEP ~ 0.8325 Va? +02

When Oni n/Omax 1S between 1.0 and 0.8

The mean square positional error is defined as the radius of the
error circle equal to 1.41420, and has little significance in a truly
circular error distribution. However, when o, and
equal, the MSPE defines the error in a
computed:

%y are approximately
geographic position and is

msPE = fo? + 2 (5-14)

when Onin’ Imax 18 between 1.0 and 0.8

33


DMA TR 8400.1

7. PROPAGATION OF ERRORS

7.1 Variance and Covariance Propagation

_ The propagation of errors [1] [2] is sometimes called the
propagation of variances and covariances. Variances, the measure of
dispersion of a random variable, has already been defined. The covariance
is a measure of the mutual variation of two random variables. Covariance
describes the correlation between two variables. Covariance is computed
from the equation:

n
Sy Tg, Oo) Oy > wy) (7-1)

where x,y; are the observations of the variables x, y

u, mean of the set of observations

Ws 'y

n the number of observations

The covarfance matrix is a square symmetric matrix of the
variance and covariances of the random variables we are using. The matrix
may be any dimension depending on the number of random variables.

2
o o o
ast “1 %2 0 *1 %3
; 2
lo=lo o o (7-2)
x x) Xo Xo 2 X3

o o 0
x %3 Xo x3 x3

If the random varfables y;, and y, are linear functions of the random
variables, x), and xp, they may be written:

Yi = Ap + ayX] + aQxX2
Yo = byt byxy + boxo

55



DMA TR 8400.1

i ,
t
’
'
'

—€ ad ) +6 +€
(Xj Say) (Xj = py) (X, > px)
Population
,

t

!
i
|
H
“x ~ 6, 0: +6, +X
(X,<X) (xj =®) (j>%)
Sample

Figure 4-2. Normal Probability Density Curve of. Errors



OMA TR 8400.1

RL
°y

The - radius of the 50% Probability ciréle (CEP) resulting from
Oy» vy is R = 1.0713 oy

= 1.0713

Method 2:

Using tables computed by Arthur Grad and Herbert Solomon [14]:
From equation (2):

PR) = P (¥ xe sy? ¢ RY = PGeaye

Since x and y have unit standard errors,

< ®)
they can be written as:
x= % Xand y = x» Ye

Therefore:

P(R). = P (0, ¥ +02 x? ¢ Re)

= Py ae < “a? (11)

From Grad and Solomon Tables:

2 2
Play tar yy <t) ata =

a (12)

D-9



“owiehe? Lo Da ba Lo Lo Lo Lob Lo“

OMA TR 8400.1

%00'0S = (Y)d UayM UORDUN (u)d aip Jo eAIND E-G aunty

m9
ip
oL 60 80 Lo 90 so v0 £0 zo ix) 00,
vr T T T t T T T Uy t
; 3
q ;-~AYAN
> 0 ON
on:
- .

OL
(40+"0) 4889°0 =4
CTM 9 Sb490+ “0 00600) =3

CTMo 17990+ ““o Lyizo)=a4

OTMo 96190+ ““o Egzy'0) = 9

pueda, ~ ot

40



OMA TR 8400.1

R om 2 a?
rin) s Sy lgme tr tokla Ca n
r r t

2

Ro
p(R) * yf " Boye 1 (0) dr .
ee 0

rLaIGa aa saIis Ose Se cas! es 6 le

hs

27. PCR) ® Lee!

t



DMA TR 8400.1

Q should include the effects of residuals from the adjustment,
the covariance of the control used as dfagnostics, a covariance of the
measurement errors, and the covariance of the comparisons. The points
( ops Ape by )¢ do» Age hy ) should be chosen in pairs separated by the
distance over which the point-to-point relative accuracy is desired.

To compute the absolute accuracy, consider the 6 x 6 Q matrix to
be partitioned such that:

%1 G2
Q= . (7-11)

M12 Ye

The absolute accuracy ‘may be determined from Qi; or Qoo. The units of the
matrix should all be meters. Assume that the horizontal and vertical
components of Qj are independent but the latitude and longitude may be
correlated, That is:

*, o, 0 (7-12)

The 90% LE or MAS value for the vertical accuracy is computed from:
90% LE = 1.6449 | Yor, |. (7-13)

The factor 1.6449 is from Table 4.1 relating the one sigma error to the
90% error.

To compute the circular accuracy in this case, we will consider
the latitude and longitude as correlated. The equations to relate these
variances and covariances to two independent variables, u,v, in the
horizontal plane are:

60

peat

T= Mion Wheaiie: Han ten When Wes Wize Wen fae ian LL



DMA TR 8400.1

3. Modified Form of the Two-Dimensional Probability Distribution
Function. To solve equation (3) by the use of tables, the equation must
be modified. From S.0. Rice's “Properties of Sine Wave Plus Noise" [13]:

I, (kx) = I eY 1, (vk) dv (5)

Modifying equation (3):

2
2 o,
r y
- [1+ ]
Lp YO ee (ay «
re ° I. [-— -1 r
0, 0 2
% Y neo 4oy 9,
Step A
Letting:
2
ve a+ 2)
fo, ox
2 “y
dv=—", (1+-%) ar
te Orth

D-5



DMA TR 8400.1

probability, the axes of the ellipse are 1.000 9, and 1.0000 oy 3 for a

50% probability, the axes are 1.1774 a, and 1.1774 %y

.

Table 5-1

Values of the Constant K

Probability K
39.35% 1,0000
50.00% 1.1774
63.21% 1.4142
90.00% 2.1460
99.00% 3.0349
99.78% 3.5000

The use of the error ellipse is complicated by the problem of
axes orientation and propagation of elliptical errors. Therefore, the
ellipse is commonly replaced by a circular approximation which is easier
to use and understand.

5.3 Circular Errors

The probability distribution function [9] of the radial error
expressing the probability that “r" wil? be equal to or less than radius
R, or the probability that the point (x,y) will be contained within a
circle of radius R, is derived in Appendix D and stated as:

2 9,
.
R "7 2 {1 +5] 2 6, 2
P(RY = ay Jre 4% Ita Cy = ar (5-7)
0 y x

A special case of the P(R) function (5-7) is formed when r=R, and

o % FOF O% + From Appendix D, part 2:



FIGURE
2-1
3-1
3-2a
3-2b

4-2
4-3
5-1
5-2
5-3
5-4
5-5

5-6
6-1

6-2
G-1

DMA TR 8400.1
LIST OF FIGURES
TITLE PAGE

Precision and ACCUPACYsessessesseetscasecsececoeeece 5
Probabilities of Numbers from the Roll of Two Dice.. 11
Normal Probability Density Curve..sscccccscssceccece 14
Normal Probabtlity Distribution CUPVE. cc ccevcesceece 14
Normal Probability Density Curve of Observed Values. 17
Normal Probability Density Curve of Errors...ssecece 18
Normal Linear DIStribUtTON......ceseceecscccccsccce 22
Normal Circulal Distribution... ....ccccccccsccccsees 35
Curve of the P{R) Function when P(R) = 39.35%......4 39
Curve of the P(R) Function when P(R) =-50.00%....... 40
MSPE Probability CUPVE...ceeeecccccccccccccecsccsces 41
Graph of Conversion Factors for 39.35% to 50%
PrODabElity..cccccccecccccccscccscccvsscnsnsscescecs 42
Graph of Conversion Factors for 50.00% to 39.35%
Probability. ccccccccecccccsccsrcessccesccscrccsscens 43
Comparision of Spherical Standard Error Approxi-

mation Methods.....sccccccccccsscsccescsscceccseece §1
MRSE Probability CUPVE. .esceeccccccccsesccsscccccecs 52
Circular Error with a BIAS. .cccescccncvcesccnsscecs G-1

xv



“hel ee es Ue

: AWIAEROLd 9405 01 9656°6E 40) s10;D8 UOISJeAUOD jo YdeID

S-g ound

oo.
wp
oF LX) UX) £0 LX) 0 vo £0 co LX*) oo |
t T T T T T T T T T one
StL
—-— —e—
ort
4sz1
OE:
“AGet

UMA TR 8400.1

42


a ecauideeEeeeae a

TABLE

4-1
5-1
5-2
5-3
5-4
5-5
6-1
6-2
6-3
6-4
8-1

A-1

DMA TR 8400.1

LIST OF TABLES

UME PAGE
Linear Conversion FACtOrS. ce ccccccccccasaccscesccces 23
Values of the Constant K....ssccccsecscesccsecssence 27
Sotution of P{R) Function for P(R) = 39.35%......00. 30
Solution of P(R) Function for P(R) = 50.00%.......4. 30
Summary of Circular Precision Indexes. ...ccccccseees 38
Circular Error Conversion Factors. .cccecccccccccccs 38
Values for the Constant Weecsecccccecsccscccccecccce 46
Solution of P(S) Function for P(S) = 50.00%......006 50
Summary of Spherical Precision Indexes. .scecccccccce 53
Spherical Error Conversion Factors. ...csccceccccece 53
Ground Distance Equivalent to 0.02 of an Inch at
Chart SCATC. cc ceseccccccccccccccccecarcesecsevescece 64
Conversion Factors for Converting Standard Errors to
Various Different Percentage Probabilities.........5 A-1

xiii



DMA TR 8400.1

The circular _map accuracy standard jis based on the percentage
level in use by the U.S. National Map Accuracy Standards which specify
that no more than 10% of the well-defined points in a map will exceed a
given error, The standards are commonly interpreted as limiting the size
of error which 90% of the well-defined points will not exceed. Therefore,
the circular map accuracy standard is represented by the value of R in
equation (5-8) when P, = 0.90, and is computed:

CMAS = 2.1460 (5-17)

or CMAS = 1.8227 CEP (5-18)

The three-five sigma error, representing a circular probability
of 99.78% approaches near-certainty in a circular distribution and has a

magnitude 3.5 times that of the circular standard error.

5.5 Discussion of Circular Errors

The normal circular error distribution is derived from the
bivariate or two-dimensional distribution of errors. For typical
applications, the variables x and y are random errors defined as Eastings
and Northings, downrange and crossrange components, or latitude and
longitude converted to some metric unit. It is rare that these errors are
uncorrelated and their standard errors equal. As such, they are referred
to as elliptical distributions. To simplify probability calculations, the
elliptical distribution is converted to an equivalent § circular
distribution. For the following discussion, it will be assumed such a
conversion has taken place according to the following formula which jis
adequate for cases where the larger standard error does not exceed five
times the smaller:

9, * 0.5 (0, + sy (5-19)

36

embers gL: . 7 og a



DMA TR 8400.1

iP ix)

7

; Figure 3-2a. Normal Probability Density Curve

Figure 3-2b, — Normal Probability Distribution Curve

14



DMA TR 8400.1

3. REVIEW OF THE BASIC CONCEPTS OF PROBABILITY

3.1 Probability

Probability [5] [6] is defined as the frequency of occurrence in
Proportion to the number of possible occurrences, or simply, the ratio of the
number of successes to the number of trials. Let A and B symbolize two com-
pletely independent events. P(A) is called the probability set function of
the event "A". The value of P(A) is called the probability of event "A" and
the value of P(B) is called the probability of event "B", The probability of
any event must be between 0 and 1. That is, zero Probability means the parti-
cular event will never take place, and a probability of one means that the
Particular event will occur with each trial. For example, the probability of
rolling the number 7 with a single die is 0.0 (an impossible event), but the
Probability of rolling a number from and including 1 through 6 is 1.0.

Rule 1. The probability of event A is equal to or greater than 0 but
equal_to or less than 1.

O< P(A) <1
Rule 2, The probability of a failure, or the probability of an event
not occurring, is 1 minus the Probability that it will
occur.
1 - P(A) = failure of event A
Rule 3. The probability of two mutually exclusive events A or B
occurring is equal to the sum of their individual

probabilities.

P(A or B) = P(A) + P(B)


DMA TR 8400.1

TABLE OF CONTENTS

‘

DMA MISSION STATEMENT. .00sescescsccsecscccccscescescssoeceseass

FOREWORD. .....00ccsccecceccccesccsceese sence eeeceeccsceseecene

PREFACE. sc ecccccccccccccece

SOO m cere recccevecesvecsccecesesee

TABLE OF CONTENTS. +. .sceesssesseceescersscccssccescetceceeesees

LIST
LIST
1.
2.

4,

OF TABLES. +s sseesseeccsscesccsssevecesssensenssescesceucs

OF FIGURES... cscsssscececcscsceccccsscsceascese se eeseees

ERRORS». ++ ssesseecsccssesccsssccscoscncesscecessrseesees

2.1 Classes of EVPOVS.ccscsecsercvccssesccsscsensscevece
2.2 Precision and ACCUTECY sa cececesscccccccccccssccesece
REVIEW OF THE BASIC CONCEPTS OF PROBABILITY. ...ecsccececs
3.1 Probabtlity.sssssesssessesessessseesesoreecececs...,

3.2 The Normal Distribution of a Continuous Random

Vartablesscscecssscecececsscscscessecasserscscesrece
ONE-DIMENSIONAL (LINEAR) ERRORS. cc cccccccccccccccvcceeses
4.1 Ltnear EFLOTS..cccscccsccccccccecccscsccscccccceecce

4.2 Application of the Probability Density Function to
Random EVTOrS..cccccccccsccccnces

4.3 Precision INd@X€S....cesccecccscccecscccceccecccecee
4.4 Examples of Linear EVTOrS.cceescsccseescccscccsccses
TWO DIMENSIONAL (ELLIPTICAL, CIRCULAR) ERRORS. ...eccceses

§.1 Introduction... .cssscscecsccsscscseacscscscecessuees

5.2 Elliptical EVPOrS.cccccessccccccccccccesuccsecceecce

PAGE
tii
v
vii
ix
xiii

xv

o © &F ww w

10
15
15

15
20
23
25
25
26
27



DMA TR 8400.1

sigma (spherical, 99.89%). Since error values are easily converted from
one precision index to another in the same distribution, the use of any
index is largely a matter of choice. However, in presenting positional
information, the position error is best expressed by the precision index
that fs conventional for the type of positional error being described,

67



oe ae oe ae a oe oe ho he 6 ae oe

DMA TR 8400.1

The true error is approximated by the residuat "x", hereafter designated the

error and defined as the difference between the observed value and the most
probable value:

xk =X . (4-4)

The standard deviation computed from a sample (9) is identified by a
subscript and computed: from:

(4-5)

This term is sometimes referred to as the standard error. The normal
probability density function of errors now becomes:

x2
°
p(x)=—L eg (4-6)
oy 7

The parameters X and o, may assume different values as various sam-
ples are selected from the same population and are, therefore, random vari-
ables with dispersion expressed by similar parameters. The standard deviation
of mean, Oy» and the standard deviation of the standard deviation, oy» indi-
cate the reliability of the estimate and help "round off" the computed values:

(4-7)

(4-8)



DMA TR 8400.1

of the error is not included in the theoretical model, the error should he
removed from the observations.

Random errors result from accidental and unknown combination of
causes beyond ‘the control of the observer. They are characterized by:
positive and negative errors occuring with equal frequency, small errors
occuring more frequently than large errors, and extremely large errors
occurring rarely, Random errors are the errors that remain after the
blunders and systematic errors have been removed. Due to their
unpredictably random errors cannot be eliminated from observations.
Because of these random errors it is impossible to measure (observe) the
‘true’ value.

Though it is impossible to predict random errors, they have
characteristics that may be expressed mathematically. The random errors
of repeated observations usually display a normal frequency distribution.
Because of the nature of these characteristics, the frequency distribution
of random errors can be expressed mathematically hy the normal distribu-
tion function. Assuming all errors are independent and random (conforming
to the normal distribution function) the analysis of these errors allows
us to derive accuracy information on the observation. The probability
that a random error will not exceed a certain Magnitude may be inferred
from an analysis of the normal frequency distribution of the random
errors,

2.2 Precision and Accuracy

Although the terms precision and accuracy [1] [2] [4] are often
used interchangeably, there is an important difference between them. By
definition, precision is the closeness with which repeated measurements
made under similar conditions are grouped together; and accuracy ts the
Closeness of the best estimated value obtained by the measurements to the
“true” value of the quantity measured.



DMA TR 8400.1

5. TWO-DIMENSIONAL (ELLIPTICAL, CIRCULAR) ERRORS

5.1 Introduction

_ A two-dimensional error is the error in a quantity defined by two
random variables. For example, consider the true geographic position of a
point referred to the X and Y axes. Each observation of the X and Y
coordinates will contain the errors "x" and “y". When assumed random and
independent, each error has a probability density distribution of:

- 2
0,
p(x) = wlie *
0, Van
a
~ 202
and: ply) = 1 e Y
oy ven

Applying Rule 4 of Section 3.1., the two-dimensfonal probability density
function becomes:

2 2
. i ( x + )
p(xsy) = —b—e 2% °y (5-1)
an % vy
B Rearranging terms: 2 2
f 1; x
fo . -3(-y+*))
not P(xsy) oo, a= e yy
Therefore:
(3 2 2
x
-21nf play) oo tr] =p tty (5-2)
dann . x y
For given values of p(x,y), the left side of equation (5-2) is a constant
K2
25
pr es



DMA TR 8400.1

Appendix G
Error Probability of a Quantity Affected by a Bias

In error analysis it is assumed that all systematic errors have been
removed from the observational data., This is seldom true for it jis
virtually impossible to eliminate all Systematic errors. In the analysis
it is hoped that the systematic errors of consequente have been eliminated -
leaving only numerous small systematic errors whose combinations cannot be
distinguished from random errors. When a systematic error of consequence
has not been removed, the value obtained fs safd to be biased. That is,
the value deviated from its true or accepted value by some known or
undetermined amount.

This appendix is concerned with the effect these undetected, biased
quantities have on the probability interpretation applied in error
analysis [16]. In the two-dimensional case, this would be the radius of
circle which will include a certain portion of an error distribution
affected by a biased quantity. As an example, consider a missile with a
circular standard error 6, =r , aimed at a point T. The missile is
biased (or the target misidentified) by an amount d; therefore, the
distribution of missile impacts is around the point A (a distance of d
from T) not T. (see Figure G-1)

&

Figure G-1. Circular Error with a Bias
G-1



OMA TR 8400.1

Correlation between equations (11) and (12) will permit use of the tabled

values.
S& fa, RL Sk
o, a .

y % 2
Enter the tables with values of ay, ag and the regutred probability. Then

interpolate for values of
Raft,

% a 2

a
Since 3" v4, then a, = 4) a) = 66

t = .6 = 4589
§000
7 5080

ad

tt , 0.60864 | Re
ay As q ¥

R 0, 68464
‘0, ——* 1,068
R = 1,068 sy

D-10

maomUugau I 4437" "3 353 8 &

Ae


DMA TR 8400.1

aq
al
7
aq
ai

wd

J

an
soft . . ‘

bh

ho, ,

Figure 2-1. Precision and Accuracy



DMA TR 8400.1

Onin, _ aun _
Gre = 08 Tran 24

oS
>

o min.
Teen =01

Cen ee ees GO ee SO amen as WO id es OO nae OE nes as Os


1.

2.

3.

4.

6.

10.

Sciences; McGraw-Hill Book Company, New York, New York, 1969.

and Statistics; John Wiley and Sons, Inc., New York, New York, 1950.

DMA TR 8400.1

REFERENCES

Bevington, P.R.; Data Reduction and Error Analysis for the Physical

Mikhail, E.M.; Observations and Least Squares; Reprint, University
Press of America; Washington, D.C., 1976.

Rainsford, H.F.; Survey Adjustments and Least Squares; Frederick
Ungar Publishing Co.; New York, New York, 1958,

Beers, Y.; Introduction to the Theory of Errors; 2nd Edition, Addison-
Wesley, Reading Massachusetts, 1958.

Papoulis, A.; Probability, Random Variables, and Stochastic Processes;
McGraw-Hill Book Company, New York, New York, 1965.

Hogg, R.V. and A.T. Craig; Introduction to Mathematical Statistics;
4th Edition, Macmillan Publishing Company, New York, New York; 1978.

Arley, N. and K.R. Buch; Introduction to the Theory of Probability

Barford, N.C.; Experimental Measurements: Precision, Error and Truth;
2nd Edition, John Wiley and Sons, New York, New York, 1985.

Space Technology Laboratories, Inc.; Ballistic Missfles; Vol. IV,
Revision 1, Los Angeles, California, 1960.

Merrill, G. Editor; Principles of Guided Missile Design; D. Van
Nostrand Co., Princeton, New Jersey, 1956.

Bk

Sa

H-1



a ee =

DMA TR 8400.1

8. APPLICATION OF ERROR THEORY TO POSITIONAL INFORMATION

8.1 Positional Errors

_ By the use of error theory in the evaluation of positional
information, it is possible to establish a meaningful accuracy subject to
uniform interpretation. To provide a logical and acceptable basis for
computation and comparison, positional errors are assumed to follow a
normal distribution. This assumption is valid because positional error
components generally follow a normal distribution pattern when sufficient
data is available.

The statistical treatment of errors is applied to measureable
quantities found in the sources of positional information. The
differences between diagnostic surveyed values of ground control (not used
in the photogrammetric adjustment) and the coordinates of that control
from a photogrammetric adjustment are considered to be the errors in the
Photogrammetric adjustment. Analysis of the linear components, latitude
and longitude expressed in meters, provide a two-dimensional expression
for the accuracy of the adjustment. When all the linear standard errors
occurring in the adjustment are combined and converted to a circular
distribution, as described in the Chapter 7 on error propagation, it will
be part of the accuracy statement for the product. In the case of maps,
the final product accuracy will be based on this adjustment error and the
errors associated with map construction. In the case of digital data, the
accuracy statement contained in the header record is for any single data
point in the cell. Although the accuracy is based on comparing diagnostic
points to data interpolated from the digital data, the accuracy does not
include any interpolation caused errors of the user.

Because of the different sources contributing to the accuracy of
various DMA products, it is important for the user to carefully choose the
right product to fulfill his positioning requirements. If there is a



DMA TR 8400.1

Circular Error Formulas

=
Percentage
Precision Index Symbol Probability Formula
Circular % 39.35% % * 0.5000 (a, + 4)
Standard Error , when Grin’ Snax 2 0.2
Circular CEP 50% CEP = 1.1774 9,
Error Probable CEP = 0.5887 (0, + ay)
when nin’ Snax > 0.2
CEP ~ (0.2141 Gin * 0.6621 Snax)
when 0.1 < qiin/oiay £ 062
CEP ~ (0.0900 Onin t 0.6745 Snax)
when 0.0 < qii,/a,,, < O01
Circular Map CBAS 90% HAS = 2.1460 %
Accuracy Standard
CMAS = 1.0730 (o, + oy)
when Sntn/ Tmax 2 0.2
Circular Near-.... | 3.5 Be 99.78% 3.5000 o
Certainty Error
(Three-five sigma)

Circular Error Converston Factors:

To
‘From 39.35% 50% 63% 90% 99.78%
39.35% 1.0000 1.1774 1.4142 2.1460 3.5000
50 0.8493 1.0000 1.2011 1.8227 2.9726
63 0.7071 0.8325 1,0000 1.5174 2.4749
90 0.4660 0.5486 | 0.6590 1.0000 1.6309
99.78 0.2857 0.3364 | 0.4040 0.6131 1.0000

70



eae ee ee Bee ERE EE SE EC

© ONAN wD

Number

ade ae?
one o

Probability

0

1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36

(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)

i)
ae

t

1

(2,1)
(3,1)

(2,2)

OMA TR 8400.1

Combinations

(4,1) (2,3) (3,2)

(2,4) (4,2) £3,3)
(2,5) (5,2) (4,3) (3,4)
(6,2) (3,5) (5,3) (4,4)
(4,5) (5,4)

(5,1)
(6,1)

(6,3)
(6,4)
(6,5)

rey
eto
aie

L

(5,5)

7 8 9 10 11 12

Figure 3-1. ; Probabilities of Numbers from the Roll of Two Dice

11



DMA TR 8400.1

The probability error space containing the true value when the error
source has both a random and a bias component can be expressed
mathematically. The probability density function for an error, x, in one
dimension, f(x) is: :

ly ai
1°? C(x - HY) (6-1)

f(x) =
Von oy . {

where x = random variable

XK = the bias in x,
The probability function for two dimensions is:

-4 L bag we J
flay) = wee e (8-2)
where x,y = random variables

X = bias in x
yY = bias iny

In three-dimensfonal, the probability density function is:

xy? <2 <2

- (x), yn, @ az)" y

f(x,ysz) -—yl— z % vy s, (G-3)
(29) o Clo

xFy%e

where Xsyoz = random variables

fe |

PS

ied ves ees TO ee Bee ee es aces |

il |



DMA TR 8400.1

1 200
p(X) = a e (4-1)
oven

. 2
/ 1 Mare
where: o# a

The normal probability density curve of errors has a mean of zero and is
identical in form to that of the observed values. Illustrated in Figure 4-2
(page 18), the curve is described by the function:

- Pa
ee

e (4-2)

a

P(e)

where: e = the true error;
er Xy- uy

o = the standard deviation of the errors

Since the true value of a quantity cannot be measured and an infinite
number of measurements is impractical, estimated values obtained from a finite
number or sample of measurements must be. substituted for the true value and
the parameters of the density function. As the number of measurements in the
samplé becomes larger, the reliability of the estimate tncreases, Often, 30
values provide an adequate estimate. The most probable value ( X )

approximates the true value and is determined from the arithmetic mean of
observed values:

1
x
i
421
y-tt_

(4-3)



DMA TR 8400.1 I

6.3 Spherical Probability Distribution Function

When oo, = oF = g Crs = Gg» equation (6-1) becomes

2
the spherical probability density function [10]:

* We

p(s) = —l—e (6-4)
2") o3
s
where: o, = spherical standard error

Integrating p(s) from s = 0 tos = S, equation (6-4) becomes the spherical
probability distrtbution function:

s? s2
“32. om
2 Ss 2
P(s) = [Ft <3 +] "5 as] (6-5)

where: S = radius of the probability sphere
Equation (6-5) can be solved by an approximation formula:

ln
iN)

2
c? -&
P(S) ~ fe (1.253:- ce 7 . are ) (6-6)
C+0.8 e “*" .
where: C= &
| %s

| 6.4 Spherical Precision Indexes

| A spherical error distribution is represented by indexes [10]
similar to those in Sections 4.4. and 5.4, Preferred spherical precision
indexes include: (1) the spherical standard error Co, ), (2) the
Spherical error probable (SEP), (3) the spherical accuracy standard (SAS),

47



DMA TR 8400.1
+? . x?
- -& ed
fe dt ~ ———— (7)
% x+0,8e°
2 4
i “2 gc = 1.253
x td . J
peg
| 2 2
(8) AE P-ce % + 1.283- - a) (8)
8 @

E-4



OMA TR 8400.1

5.4 Circular Precision Indexes

The precision indexes [10] itlustrated in Figure 5.1 (page 35)
are measures of the dispersion of errors in a distribution and represent
the error which is unlikely to be exceeded for a given probability. ‘The
preferred circular precision indexes, consistent with indexes used in the
linear distribution, are: (1) the circular standard error ( % ), (2) the
circular error probable (CEP), (3) the circular map accuracy standard
(CMAS), and (4) the circular near-certainty error, three point five sigma
(3.5 % ). The mean square positional error (MSPE), an additional index
which has been used at DMA, is not recommended because the probability
represented varies when o and vy are not equal.

The probability of the circular standard error is found by
solving equation (5-8) for P, when eR, thus:

o 2
° G
Poz l-e *
“3
Po= l-e
Po = 1 = 0.60653
ote Po = 0.3935 (5-10)

That is, 39.35% of all errors in a circular distribution are not expected
to exceed the circular error.

For a truly circular distribution, the linear standard errors are
equal and identical to the circular standard error (o, = gy = o,) « When
% and %y are not equal, a normal circular error distribution may be
substituted for the elliptical distribution. The substitution is
satisfactory for error analysis within specified Onin! max ratios.
Because of distortion in the error distribution for low ratios, however,
the circular concept should be used with discretion.

29



DMA TR 8400.1

w= a parameter representing the mean value
of x

o = a parameter representing the standard deviation,
a measure of the dispersion of the random
variable from the mean, wu. (The square
of the standard deviation is called the vartance.)
Wim = 2,5066.66

e =the base of natural logarithms, 2,71828...

The parameters are computed from an infinite number of random variables:

y ede (3-2)

i ,

(3-3) rc

where: * c
n= the number of random variables ,

The normal probability distribution function, P(x), determines the

probability that the random variable wilT assume a value within a certain

interval and is derived from the notmal probability density function by

integrating between limits of the desired interval. Letting the limits range
from - = to x:

12


DMA TR 8400.1

A cofactor matrix is related to the covariance matrix by a scalar
multiplier.

Q= ky
1
where k= 37>
0
and aoe is called the reference variance,

It should be pointed out that only random errors will be
Propagated by:

: T
yy Jyx8 o yx °

Hence this will be a measure of the precision of the adjustment.

Another method is called error propagation from sample

statistics. This method evaluates the Product by comparison to diagnostic

Control points. This method will take into account biases in the Product,
random errors in the Product, random errors in the diagnostic control and
random error in measuring the control on the product. In this method a 6
x 6 cross-covariance matrix is computed. Let this matrix be denoted by Q.

To 7
o o o oO oa o
#1 My Ay hy O42 Opry Moyne
2

“ory Ay AM, Ady *AyAg PAyhy
2
oO G
my Thy ty Thy Ay Thyhy

(7-10)
2
“it, "te yy toh

20
Bry %p%y dy gh,

oO o. oO
hg “typ Aghy 9 hy

59



DHA TR 8400.1

7.

8.

TABLE OF CONTENTS (Cont'd)

5.4 Circular Precision IndexeS..sscccocsccscnsaccsccvscce
5.5 Discussion of Circular ErrorS.sseccccseccccvcsccvecs
5.6 Circular Conversion Factors.cscccsccscsccccccccesces
THREE DIMENSIONAL (ELLIPSOIDAL, SPHERICAL) ERRORS. .e0eee
6.1 Introduction. ..cccccccccccccccsecussscuccssssccsence
6.2 Ellipsoidal ErrorS.ccccccccccuccscscevacesesessccece
6.3 Spherical Probability Distributfon Function......se0«
6.4 Spherical Prectston Indexes....ssescccccsceccecscess
6.5 Spherical Conversion Factors.csccccccccsccccccscccece
PROPAGATION OF ERRORS. .ccecccccccccccccessecssescccscccus

7.1 Variance and Covariance Propagation. ..cccccccccccees

7.2 Propagation Through Known Functions..sessee
7.3 Application of Error Propagatton Methods. .s.cieseeee
APPLICATION OF ERROR THEORY TO POSITIONAL INFORMATION...
8.1 Positional Errors... ccccccccccccccccscsescsessvsoecs

8.2 The Accuracy Statement. ccccccccescscnggeccpscoscsece

Summary of Formulas and Conversion FactorsS..csccscsespecccveses

APPENDIXES .
A. Percentage Probability for Standard Error Intrements.....
B. The Most Probable Value.....ccccccccccccscuccsesccvsccace
C. Propagation of Errors Through Known FUNCEIONS...ccccceeee
D. Derivation and Solution of the Two-Dimehs tonal

Probability Distribution Function. ..cccscsccsecnsovosceus

¥ i. wig a Z rl ie



DMA TR 8400.1

Let $= radfus of sphere, replacing radial error s.

Then: dS Sdv$ cos ¥ da = S@ cos ¥ d¥ da dS

- s?
wpe} 7 ; —t—e 2s 2 cos vay da dS (3)
SO Ow - 3S (2a) °?
. Pa
(s) “4 (2) (2x) ; Se lg
(2x)* - so "8

£-2

~ =



DMA TR 8400.1

The probability represented by the MSPE can be found by solving equation
(5-8) for P, , when R = MSPE and o, is approximated by equation (5-12),

thus: 2
R
=
Po= l-e %
c
. (2 + 6, 2)
Pos Lee 20, (5-15)
When % = oy t
. -1.0
Po = l-e
P, = 1 - 0.3679
Po = 63.21% , (5-16)

When o, # oy, the solution of (5-15) yields values, of P, ranging from 64%
when Onin’ max = 228 to 77% when Gain! Smax = 0.3, Because of the
variation in probability, the MSPE is not recommended for use as a

precision index.

The 2drms accuracy standard has been suggested [11] for use in
the navigation community, The 2drms is defined by:

2drns = Yo, + 0,"
Hence, it will have a radius twice the NSPE. The percentage level for the
2drms is 95.4%. Like the MSPE, it will vary in probability and may be
skewed by large errors.

C
P

Ly

2drm = 2.99 o,
for o = a.

34



DMA TR 8400.1

x = bias in x
y = bias iny
z = bias inz

The one standard deviation probability level may be found from the
integrals of the probability density functions, Equations (6-1), (6-2),
and (G-3), using numerical integration. In one dimension, using equation
(G-1), the linear standard deviation probability is:

oh,

Pr (- oh <x < ot,) = f F(x)dx (6-4)
~ on,
x

Pr (-oh, < x< oh) = .6827 (G-5)

where ot, ts the linear one-sigma error in x.

The probability density function is two dimensions, Equation (G-2),

may be numerically integrated to find the circular one standard deviation
probability level:

2,2 2
Pr E(x +y") <a J =f i f(x,y) dxdy (6-6)
Pr E(x +9") ¢ 0,2] = .3935 (6-7)

In Equation (G-6), R denotes a circular disc of radius %

For the three-dimensional case, Equation (G-3) may be numerically
integrated to find the spherical one standard deviation probability level:
2 2 2 2
Pr [(x* + y® + 2°) < o,°] = J f £ Flxy2) dxdydz (6-8)

Pr E(x? + y+ 22) « 92] = 1990 (6-9)

G-3



DMA TR 8400.1

The quantity (a) is defined as the circular standard error and is the
basic statistical parameter used in probability estimates based upon the
normal circular distribution.

The probability associated with circular errors is used in the
Same way as for Tinear errors, However, instead of the area under the
normal distribution curve, the Probability in a circular distribution js a
function of the radius of a circle centered on the mean of the error
distribution (Figure 5-1, page 35). A circle with a radius equal to the
circular standard error represents 39.35% probability in a normal circular
distribution, Other probability levels can be defined by circles of
larger radii. ‘For example, a circle with a radius of 1.1774 times the
circular standard error represents 50% probability and is known as the
circular error probable (CEP) which is common to missile, bombing, and
artillery error terminology.

The radius can be increased further to describe a circle
representing 90% probability. The latter, which is approximately twice
the circular standard error (or 2.14600, ) sis known as the circular map
accuracy standard (CMAS). Circular errors (CE) at the 90% level are
Sometimes expressed as CE 90%. The expression "90% assurance" is another
term used to express confidence in an estimate. However, probability
rather than assurance is the preferred term because it has a statistical
meaning.

Circular distribution probabilities can be converted from one
level to another by the factors in Table 5-4. For example, a geographic
position fs assigned a CMAS of 500 feet. The CEP (50% probability) of the’
position is estimated by multiplying 500 feet by the factor 0.5486 from
Table 5-5 to obtain 274 feet. This does not imply that there is a 50%
probability that an error of 274 feet will occur, rather it means that
there is a 50% probability that the error will not be larger than 274
feet.

37