NBSIR 80-1967

The Calibration of Angle Blocks

by Intercomparison

Charles P. Reeve

Statistical Engineering Division

Center for Applied Mathematics

National Bureau of Standards

S. Department of Commerce

Washington, DC 20234

April 1980

Final

~f."'10" Co

 

:tv

..  . ......

. z

!. .. . , 

(I.rAU of

S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
NBSIR 80- 1967

THE CALIBRATION OF ANGLE BLOCKS

BYINTERCOMPARISON

Charles P. Reeve

Statistical Engineering Division

Center for Applied Mathematics

National Bureau of Standards

S. Department of Commerce

Washington, DC 20234

April 1980

Final

S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Productivity, Technology, and Innovation

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Dir8ctor
TABLE CONTENTS

Page

Introduction 

Measurement Flatness 

Measurement Squareness 

Measurement Angle between Faces

Setup Procedures

Intercomparison Scheme 

Measurement Equations

Method Solution 

Sources Error

Statistical Tests

Exam~le 

Conclusion 

Tables

References

Appendix 

Figures
Introduct ion

The calibration of angle blocks is a routine service provided by

the National Bureau of Standards. Test blocks normally come in sets of

16 nominal sizes: 

, 3, 5, 20, 30 seconds,

1, 3, 5, 20 , 30 minutes, and

, 3, 5, 15, 30 , 45 degrees.

Periodically all tests sets which ha~e been received are measured

against two ets of NBS master angle blocks by intercomparing all blocks

of the same nominal angle. The two sets of master blocks are called

reference blocks and check blocks in accordance with the role they play

in the measurement process The reference blocks have been calibrated

by the "absolute" method and serve as "ground zero" for establishing the values of the other blocks. The check blocks also have known values but

are treated as unknowns during calibration. Their calibrated valnes are

compared to their historical values in order to maintain control over

the measurement process.

As part of the routine calibration each face of the test blocks is

measured for flatness and for squareness to its top and bottom surfaces.

(These measurements may be omitted on test blocks which have previously

been calibrated at NBS.

The purpose of this paper is to describe each phase of the c.alibration proces's and give a detailed description of the mathematical model

for the intercomparisonscheme. Three statistical tests for process

control are described and an example is given.

There is not an abundance of literature on angle block calibration

available. However , general (.and often brief) discussions of angle

blocks may be found in 12, 81.

2. Measurement of Flatness

The two polished faces of each test block may be denoted by b

(base) and (hypotenuse) as shown in Figure 2. 1. The flatness each

face is measured in a light box using a Fizeau-type interferometer. An

optical flat is placed over the base and adjusted until a series of

interference fringes .is ob~~rved as sl1Qwn j,.g F:i,gure 2.2. A moveable

hairline is adjusted to touch the ends of one fringe and the ratio a/b

is estimated where.!. is the maximum deviation of the fringe .from the

Customary U. S. units are used in this report rather than the recogniz.ed metric (SI) units. The well established ongoing calibration

procedures described here employ customary units exclusively. The

conversion to SI units will be made at a future date.

Numbers in brackets indicate references listed at the end of this

paper.
hairline and b is the fringe gi ven by separation.

b =-

The out-of- flatness is then

(2-1 )

where A is the wavelength of the monochromatic light in the light box.

The value fh is similarly derived for the hypotenuse, and the value

max = max( fb , (2-

is reported as the maximum out""of-flatness of the angle block faces.

Currently this value is rounded off to the nearest microinch.

The measurement of flatness is not intended to be of the highest

accuracy. The maximum out-of-flatness figure is an indicator of the

quality of the angle block and, depending on its magnitude, may indicate

a source of long-term variability in the measured values of the ~ngle

between the faces.

Measurement of Squareness

The squareness of the base and hypotenuse to the top and bottom is

measured for each test block. Let the four interior angles between these faces be given by -1:900 + Bit i=1 ) as shown in Figure 3. 1. 

autocollimator which is set to read vertical angle is mounted near the

anvil upon which the angle block rests. The block is wrung to the anvil

so that one of the four surfaces of interest faces the autocollimator

(as does the base in Figure 3. 1). The autocollimator is adjusted

vertically to read near the center of its scale and the reading sl is

recorded. The block is then rotated 900 and re'Wrung to the anvil so that the adjacent surface faces the autocollimator. The reading s2 is

is recorded. The rotation pattern is continued in the same direction

unt;i.l readings s3 and s4 have been recorded. The readings take the form

(ignoring possible error terms)

si = 

!!. 

Bi , (3-

for i = 1 4 where !!. is some initial reading of the autocollimator.

A~suming J~at the sl1.I"f~ces ~..!e .!,rue ~lanes the~ B1 + B2 + B3 + 84 = o. Incorporating this equation 'Wi thEqua.tion 3-1 and solving for the unknown

parameters gives

(sl + s2 + s3 + s4)/4 and

(3-2 ) 8i = 

for i = 1 The value

max = max( 1811, I B21, I B31, 1841) (3-3 )

Throughout this paper the abbreviated notation i = p, q indicates that i

takes on the consecut i ve integer values p, p+ 1

.. , q, q.
bairlille

o,tical flat

DESIGNATION OF FACES

hypoten use (h)

20 DEG

base (b)

Figure 2.

MEASUREMENT. OF FlATIESS

Figure ;~. 2

angle block
a.tocomlat.,

MEASURfMENT Of SQUARfNESS

: fEND VIEW OF ANG1E BLOCK)

. . .

900 + 132 -- 900 + 133

_ase 

tGO + I3t

anvil

Figure 3.

hypoten use
is then reported as the maximum out-of-squareness of the angle block surfaces. Currently this value is rounded off to the nearest arc

second.

The measurement of squareness is not intended to be of the highest

accuracy. As in the case of flatness measurement, the maximum Qut-ofsquareness figure indicates the quality of the angle block and a

possible sou~ce of long~term variability.

Measurement of Angle between Faces

Setup Proced~res

Two autocollimators and a serrated anvil are mounted on a surface

plate as shown in Figure 1. The anvil has three stops which enable

the angle blocks to be inserted at the same position repeat.edly. Since

it is usually most convenient to measure the smallest blocks first, a 1"

block is inserted on the anvil and the two autocollimators are aligned

so that the block is centered in their respective fields of view and

each reads near the ' center of its scale. After this point autocollimator A does not have to be realigned , but autocollimator B must be

realigned for e~ch different nominal angle. The autocoliimators are

connected to a digital voltmeter 'W'hich displays the difference between

their readings. The voltmeter ls set so that a larger block size gives

a larger reading.

In order to keep track of the blocks during measurement , small

pieces of masking tape can be stuck to the side of the surface plate in

front of the anvil and numbered from 1 to n where n is the total number

of; blo~~s ~o be intercompar~d.. 

__

(Normally there is one reference block

one check block, andn-2 test blocks although an extra check block is

sometimes included. The corresponding identification numbers which are

engraved on each block are also written on the pieces of tape, and the

blocks are then !3et in the proper positions. The blocks should be

checked to see that they are free of lint and smudges. The anvil should

also be cleared of lint and dust particles so that the blocks will sit flat during measurement. 

Intercomparison Scheme

Two series of measurements are made on each set of blocks of the

same no~in - ~i?!~o~ J~L.:t~ f.:i"r~t. se~;j,_~~- the blocks are in the

t~p-up" post tion, and in the .second series they are in the "bottom-up

position (Figure 4.2). In the "top-up" position the inscribed block

size reads from + to - while in the "bottom " position it reads from -

to +.

The intercomparison scheme is illustrated in Figure 4. 3 for n :: 7.

The check block and each of the test bloCks is compared to the reference

block and to the two blocks ahead of it in a counterclockwise direction.

The reference block must always appear in position 1 (center). The

check block may appear anywhere on the perimeter, but for the sake of
CALIBRA nON SETa.tocollillter A

CDJ digital yoltmeter

surface plate

aRgle black

Figure 4.

TOp.UP" POSITION IonOM.UP" POSITION

.....

c:a

c::t

....

C&I

.....

, &::II

c:a

Figure 4.
uniformity it is normally put1n position 2. Angular differences

between the blocks are obtained by making n-l groups of seven

measurements each. From each group three differences are derived in the

manner described in the following section.

Measurement Equations

For each block size the first group of seven measurements is obtained by inserting the blocks on the anvil in the order 2-3-

~et t~~correspon4i.M readin~ of the d:i,gita,l voltm~ter be g:i ve;n ~y the

vector Yl =~ll Y12 Y13 Y14 Y15 Y16 Y17)' where the symbol

' 1nd1cates

vector (or matrix) transposition. Let the deviations from nominal angle

of the n blocks be given by a = (a

l a2... a ) I , The first group of seven

measurement equation9 is then given by 

Yll = b. + a2 + E:ll

. 112 = b. + a3 + d + E:12

Y13 = b. + a2 +2d + E:13

Y14 = b. + al + 3d + E:14

Y15 = b. + ae + 4d + E:15

(4-

~16 = b. + a4 + 5d + E:16

Y17 = b. + ae + 6d + E:17

where A is some inittal reading of the digital voltmeter , d is a linear

drift factor, and the E:ij ' s are independent error values from a

distribution with mean zero and variance , (see section 4. 5 for a

discussion of 0;). Let three new computed observations for the first

group be given by the vector zl = (Zll z12 Z13)' where

zll (Yll 2Y12 + Y13) /2

z12 = (Y13 2y14 + Y15)/2 ( 4-2 )

z13 (Y15 2y16 + Y17)/2

Then in matrix notation Zl = MYl where

1 -2

:J .

(4-3 )

Since Var(Yl) = o I then Var( zl) = where-
IITERCOMPARISON SCHEME FOR 7 ANGLE BLOCKS

. .

T=TEST BLOCK

C=CHECK ILOCK

R=REFEREM CE BLOCK

Figure 4.
:

::: MM'

:::

(4-

(Note that in forming zl the A and d terms drop out.

The remaining n-2 groups of measurements are taken ' and transformed

in a similar manner to give the complete ,'3-( n-l) vector of new observations z::: (zl ~2 ... ' Z

) , .

Method of Solution

Let E( ~) and, Var( ~) denote the statistical expectation and variance

of the random vector ~ just described. Then the least squares estimation (1,101 of the angular values takes the form

E(~) ::: XCI (4-5 )

where X is the (3n-3) x n matrix

1 -

0, 0

1 0-

:::       ... 

( 4-6 )

    

and Var(z) 

::: 

where W is a (3n-3) x (3n-3) block diagonal matrix

given by

W::: (4-

The normal equations (incorporating the restraint al ::: m) take the form
J'W

:~X :) 

(:J - (X'

z J

( 4-8 )

where a ' = (1 0 ... 0) and A is a Lagrangian multiplier entering in the

minimization process. The least squares estimates are given by

tJ 

X'W . X w -1 '1 C i.

1 Fxo

-! z

m J 

.  

oj L m J ( 4-9 )

where I' = (1 1 ... 1) and C is the variance-covariance matrix of the

estimate ex. The predicted values of the observations are given by

z = Xex (4-10)

and the deviations by

(4-11 )

The estimate of 

OW is given by

w = 

' W d/ (2n- ( 4-12 )

and the within-series standard deviation of the estimates by

w,ex. = O' = k

( 4-13 )

for i = 2

" t Let the estimates ex from equation 4-9 be called ex if the blocks

....

were in the "top-up" position and ex if they were in the "bottom-up

position. Then the final angular values assigned to the blocks are

given by

ex =

"t + a (4-14 )

...

where ex = ex 1, ~

...

, n 

*By the symmetry of the intercomparison

... = Cnn. Thus the value of Cu (i 

numbeJ:" oJ a~~_ j)19Cks in the design-. 2 = C.. are tabulated in Table 

design, Cll = 0 and C22 = C

33 =

n) depends only ~ n the

The values of kn = Cii and
Sources of Error

There are three known types of error in the reported test block

... ...

values ~ . . ., a The first type, called long-term systematic error

. , 

is due primarily to the uncertainty in the accepted value of the

reference block. This uncertainty, denoted by E , affects the "top-up

and "bottom-up" values equally and thus must be attached to the reported

values given in equat ion 4-14. For the current set of reference blocks

E is believed not to exceed 0.20 second for each block size. Errors of

this type from other sources are assumed to be negligible.

.

The second type of error, called within-series random error is due

to the "noise" in the measurement system which is caused primarily by

fluctuations in atmosphereic conditions and instabilities in the

autocollimators. The true within-series standard deviation (or , more

precisely, the standard deviation of the within-series error values) 

is estimated from each individual series by sw as described in the

previous section. The sw values from all series done over the first

several years of operation of the measurement system were pooled to give

the best available estimate of 

w. (In the pooling process the Sw

values were weighted according to their degrees of freedom. The value

obtained was

= 0.040 second (4-15 )

which can reasonably be considered the tru~ value since it is based on

N .~ 2000 degrees of freedom.

The within-series standard deviation of the reported value ai is

then eas ily computed from Equations 4-13 and 4-14 to be

,. 

., -_u i ~ ( 4-16 )

(see Table 1 for values of k 

The third type of error, called between-series random error (or

short-term systematic error), is due to changes in the system when the

blocks are flipped from "top-up" to "bottom-up" positions. (The

presence of this type of error WaS first detected atNBS when we began

measuring angle blocks in the two opposite orientations. In many cases

the differences between "top-up*' and "bottom-up values "were much larger

than could be explained by the wi thin-series random error. Some

probable causes for these changes are

(1) imperfections in the master and test block geometries such

out-of-flatness and out-of-squareness of the faces

( 2) misalignment of the autocollimators (i. e. , reading some

vertical angle), and
(3) the presence of burrs on the anvil which tilt the blocks

differently when they are in different orientations.

A discussion of the estimation and propagation of this type of error is

given in the Appendix. ' One result is that the between-series standard

...

deviation of the r~ported value ex. is a = ab (E~uation A- Appendix).~ Ii in the

,

The uncertainty of the reported value ai is taken to be the sum of

random and systematic components of error. The random component is

obtained by combining the within-series and between-series standard

deviations in ~uadratureand then taking the three standard deviation

limit.

...

Thus. the tqtal uncertainty of a. is

U~i = 3 k~a; + a; + E (4-17)

Statistical Tests

Two statistical tests are applied to each series of measurements in

order to maintain control over the measurement process.

The (two-sided) z test compares the computed value of the ~heck

...

block ex , to it s historical value , a .

forming the statistic

The test is implemented by

...

z =

+ 2a

n w

(4-18 )

and comparing it to the appropriate critical value. 

Note that the

denominator in the above expression is the total standard deviation of

...

ex as given by E~uations . 13 and A-10 (Appendix). The number of

degrees of freedom associated with and a

b (N and M respectively) are

very large, so the denominator can be reasonably considered a "true standard deviation. Thus, under the hypothesis that the true value .

the check standard is ac, z has the standard normal distribution (mean zero and variance one). The critical value at the . 01 level of significance is z 995 = 2.58. If Izi ~ 2. 58 the usual statistical procedure is to rej ect the above hypothesis. However , a large z-value may be caused by something other thana. change in the true value of the

check block. In fact , it may be caused by a change in the true value of

the reference block or by the malfunctioning of a component in the

measurement system. Whichever the case , a large z value indicates that
something extraordinary has happened, and the usual procedure is to

repeat the series of measurements.

The (one-sided) F test compares the computed within series standard

aeviation, sw' to the true within-series standard deviation, w. The

test is implemented by forming the statistic

1 = 

(4-19 )

and comparing it to the appropriate criti.cal value. Under the

hypothesis that s; = a;, 

l has the Snedecor F distribution with 2nand N degrees of freedom. The critical value at the . 01 level of significance isF. 99 (2n- N) which is available in tables (see Table 2

column A). If F1 ~ F .99(2n- N) then the above hypothesis is rejected

because the observed system variability is extraordinarily large. This

indicates the malfunctioning of a component of the measurement system or

a blunder of some sort by the operator. As before, the usual procedure

is to repeat the series of measurements.

If a given series passes both the z and F tests , then that series

is said to be "in statistical control"

A third statistical test is performed on each pair of "top-up" and bottom-up" series. in order to control the between-series variability.

The test is implemented by forming the statistic

(where s; is computed as in the Appendix) and comparing it to the

appropriate critical value. Under the hypothesis that s; = 0;, F 2 has

the F distribution with n-1 and M degrees of freedom (see Table 2,

column B). If F2 ~ F . 99(n- M) then the above hypothesis is rejected.

A large value of F2 may be caused either by a malfunctioning of the

measurement system or by one or more blocks being of inferior quality.

If the operator decides the latter case is true then he may accept the

results and note the problem in the appropriate Report(s) of Calibration.

Otherwise both series should be repeated.

(4-20 )

Example

In August 1974 an intercomparison was made between one set of

reference blocks, one set of check blocks , and five sets of test blocks.

The data sheet for the 10 blocks is shown in Figure 

as an example.
.ANGLE BLOCK CALIBRATION IN'IERCOMP ARISON BLOCKS

Nominal Angle Chec:k Std. Lot L'13Z5 Comments:

cD3 1.5 20 30 45 Observer

see min Date ..1J IS/.::It-

!!2. . pause during any group measurements. pause between groups.

Block Identification

85-

7~ 

'" 85- 7

2. GJ2.

~Ac.-l"

2;,0. U r..c - 31,

HAc.-l4- 2..

!.)? 

2.3

top -2S bottom

S/2-

Z. 04-

II. 2, 'lq II.

0S' 2..

'2.. I. s+ /. 33 2. 

4-1 2., 2. 'S'

ii- 4-3 :2. z,(,

22. 84- z. 4-.

4-1 ~. z.b

III. VI. '1" VI. :::S' III.

I. t~ 4-1

2.. '2 I e,~

/4-

2, fo5 Q4- 2. "I

/4- 4-"2-

2. (.,5 '22.

IV. IV. 4,,0

I. 't2. 1,11-1

..1J2..l

O$"

34-

14-

lJSCo.'NIS-o(: 3/75

Figure 5.
Calibration of 10 Angle Blocks - August 1974

Block

NBSNBS- 7

BACUCCHACNBS512-

(reference)

(check)

Uncertainty (eq. 4-17 J

Accepted value of

check standard 

Computed value .

check standard

z statistic (eq. 4-18)

z .

99.

Accepted within-series

standard deviation (ow

Computed within-series

standard deviation 

Fl statistic (eq. 4-19)

F . 99 ( 12 , co )

Accepted between-series

standard deviation (Ob

Computed between-series

standard deviation (sb

F2 statistic (eq. 4-20)

F . 650)

Series 1

15"

36"

14"

15"

35"

83"

39"

39"

36"

040"

018 "

Deviation from Nominal Angle

063

078

1.55

Series 2

15"

32"

 .

07"

 .

0'2'

11 "

79"

38"

39"

32"

040"

026"

Figure 5.

Avg Diff (1-

34 '

10"

08"

23 "

 .

81"

38"

04"

 .

07"

13"

24"

.

04"

01"

0 . 40"

top-up

bottom-up
The estimated angular values and associated statistics computed from

these observations are shown in Figure 5. 2. The three statistical tests

show both series of measurements to be "in statistical control" The

listed uncertainty value, as computed from Equation 4-17, applies to

each of the average values enclosed in the box.

Conclusion

The lumping of test sets into a single large intercomparison scheme

has proven to be much more economical than calibrating one set at a time. The data reduction process has been fully computerized for

several years. The output includes a Report of Calibration for each set

of test blocks and for the check blocks.

The la.rgerangle blocks may alternately be calibrated by the

absolute" method whereby each test block is compared to several indexing table angles that sum to exactly 3600 There is no reference block

involved, hence the test block is measured more accurately. The drawback is that the number of measurements required becomes cumbersome for

blocks of 50 or less. Currently this method is used only to calibrate

the NBS reference blocks or single large test blocks.

In closing it seems fitting to speculate on how angle blocks might

be measured more accurately in the future by the intercomparison method.

An examination of the three components of error in the uncertainty

statement (Equation 4-17) is revealing. For the case n=7 (as in the

example in Section 5 ) the numerical value of the uncertainty is

u;. = 3 4815)(0.040)

2 + (0. 063) 2 + 0.

= 3 0003 5 + 0. 0039

= 0.20 + 0.

+ 0. (6-1 )

= 0.

The within-series random error component is by far the smallest of the

t~ree , and it is doubtful that any further reduction could be made

there. On the other hand, the between-series component seems to be an

area where real improvements are possible. There has not yet been a

full-scale investigation at NBS into the cause of differences between

top-up" and "bottom-up" values. One might expect large differences to

be positively correlated with the maximum out-of-flatness and out-ofsquareness of the blocks. However , visual examination of the data has

never given a hint of such a correlation. Further investigation into

the problem is needed. If the differences could be explained and

removed then the between-series component of error could be drastically

reduced.
The systematic component consists solely of the uncertainty in the

reference blocks. This component could be reduced to some extent by

painstakingly recalibrating the reference blocks "absolutely However,

in order to obtain realistic uncertainty values the reference blocks

would have to be calibrated in both the "top-up" and "bottom-up!'

positions. It is likely that di~~erences between values obtained in the

two positions wo1.1;~.a. b~ larger than expected , thus the uncertainty values

of the reference blocks would be inflated. A reduction in this

component of error then seems to depend on the reduction in the betweenseries component.

It now seems clear that under the present measurement system the

main obstacle blocking the path toward more accurate angle block

calibrations is the between-series beast which rears its ugly head in

two places. The removal of this obstacle, if it is indeed removable,

would be a significant accomplishment.

\ 
Table 1

Values of knand kn for selected values of n

n = ...)Cii = C..

7949

7465

6318

5572

, .

7111 5057

6939 4815

6824 4657

Table 2

Critical values of two F distributions at the

01 level of significance for selected values of n

No. of Blocks

in series

Column A Column B

, 2.

F . (2n- 00 ) 99(n-l,650 

Since N :;:. 2000 it is reasonable to take N = 00 in tabulating the

critical values.

The value of M is 650 as of the writing of this paper.
10.

References

Cameron, J . , The USe of the Method of Least Squares in

Calibration, NBSIR 74-587, September , 1974.

Farago , Francis T. , Handbook of Dimensional Measurement , Industrial

Press , Inc. , 1968.

Handbook of Industrial Metrology, Society of Manufacturing

Engineers , Prentice-Ifall, Inc., 1967. ,.

Knoyle, C. , Production of Combination Angle Gages, The Engineer

Sept. 1 , 1950 , p. 237-8, Machy (London).

' Miller , L. , Engineering Dimensional Metrology, Edward Arnold Ltd.

London, 1962.

Moore , Wayne R. , Foundations of Mechanical Accuracy, The Moore

Special Tool Company, 1970.

7 . Parsons, AIOJ., Metrology and Ga.uging, MacDonald and Evans Ltd.,

London, 1970.

Thomas, Geoffrey G., Engineering Metrology, Butterworth and

Company, Ltd. , London, 1974.

Veale, R. C., and Reeve, C. , A Survey of the Temporal Stability of

Angle Blocks, NBSIR 74-601 , November , t974.

Zelen, Marvin , Linear Estimation and Related Topics , Chapter 17 of

Survey of Numerical Analysis , edited by John Todd , McGraw-Hill

1957.
App~ndix

Estimation and Propagation of Between-Series Error

Estimation of Between Series Standard Deviation

The true between-series standard deviation crb is estimated f~om

each pair of series in' which the same set of blocks are measured in

opposite orientations. The least squares est-~mates of the angular

values, as given in Equation 4-9 may be expressed as

~~ = (~. + o~ - o ) + a~ and

(A-

(

~ + o~ - o~) + a~

.. 

for the "top-up" and "bottom-up" positions respectively where i

The ~i' s (without superscript) are the "true" angular values of

blocks, and the a' s are error values rep~esenting within-series ability. The a' s have mean zero, and

= 2

the

vari-

...,

Val' Val' Ccr (A-2 )

where the (n-l) x (n-l) matrix C is the variance-covariance matrix of

the estimates ~

2' ~, ..., ~ which is obtained by deleting the first

row and column from the matrix C given in Equation 4-9.

The 0' s are independent error values from a population with mean

zero and variance cr~ repre.senting between-series variability. The subscript indicates the block to which the error is attached (with the

reference block being 1). The 0 values are assumed' to remain constant

throughout their respective series and thus do not contribute to the

within-series error. The standard deviation of the 0 values, crb' is estimated by considering the n-l vector of differences f~om a pair of series
"t 

2 - ex

"t '" b

ex - 

h :: (A-3 )

At 

ex - 

, .

Now from Equations A-

:: o~ - o~ - o~ + o~ + e~ - e~ (A-4 )

for i :: 2 n, thus

Var(a~ - a~) :: ~ b + 2C. 

(A-5 )

and

Cov(a~ - a~, at, &

:: 

+ 2C'

j j 

(A-6 )

for i :: 2 n; j = 2 n; and i * j . Let I denote the (n-l) x (nidentity matrix and J denote the (n-l) x (n-l) matrix of one s. Then

the variance-covariance matrix of the vector h is given by

2(I + J)~ 2Co2 ~ :: 2(I + J + Cr)o

b :: Ho

(A-7 )

where r - 

series is then

a known constant. The estimate of o

b from a pair of

gi yen by

l h

sb :: . (A-8 )

where n-l is the associated degrees of freedom.

Note that during the ongoing measurement process the ratio r is

known because Ow and ob are assumed known. However , before ob was determined the value of r was unknown. To initially determine ob the

estimates sb from each series done on the measurement system during the
first several years of operation were computed according to Equations

A-7 and A-8 with I' = o. Thes~ estimates were then pooled (using weights

l) to give an estimate of ab. The value of r was recomputed using

this estimate of Ob, and the process was repeated in an iterative

fashion until the estimates ofab converged. The value obtained was

ab = 0. 063 second (A-9 )

which can reasonably be considered the true between-series standard

deviation since it is based on M = 650 degr~es of freedom.

Propagation of Between-Series Error

- A

The between-series standard deviation of ai, the estimated angular

value of the ith block from a single series , is easily' computed from

Equations A-I to be

a~ = a

12C1 (A-IO)

This term appears in the statistic given in Equation 4-18.

The between-series standard deviation. of ~i' the estimated mean

angular value o~ . the ith block from a pair of series , is computed from Equations 4-14 and A-I to be

ab,a. = ab

This term appears in the expression for the total uncertainty of the

reported angular values given in Equation 4-17.