as IR 75-750

The Calibration of IndexingTables by Subdivision

Charles P. Reeve

Institute for Basic StandardsNational Bureau of StandardsWashington , D. C. 20234

July 1975

Final

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(JlttAU of

U S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OfST ANDARDS

NBSIR 75-750

THE CALIBRATION OF INDEXINGTABLESBV SUBDIVISION

Charles P. Reeve

Institute for Basic StandardsNational Bureau of StandardsWashington, D. C. 20234

JUly 1975

Final

s. DEPARTMENT OF COMMERCE, Elliot L. Richardson, SecretaryJames A. Baker, III, Under SecretaryDr. Betsy Ancker.Johneon, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Acting Director

Contents

Introduction

The Method of Subdivision

Preparation for Measurement

Complete Closure Design

Partial Closure Design

Sources Error

Applications of Designs

Example

Conclusion

Acknowledgement

Figures

Tables

References

Appendix A .

Appendix B .

Page

. .

5, 7

21, 25, 27, 28

Introduction

The indexing table plays a vital role in the calibration of angle

standards(1 2, 3) 1 . An object which is wrung or clamped to an indexingtable can be rotated through certain angles very precisely. The smal-lest angular increment varies with different indexing taqles but is mostcommonly one degree. The deviation from nominal of any angular intervalin a high quality indexing table is usually no more than 0. 25 second,and the short term repeatability of any setting is usually less than

05 second.

" , . .

In most angular calibrations the indexing table plays one of threeroles

(1) Is used simply to rotate an obj ect through some nominal angIewhose precise value need not be known.

(2) Is calibrated simultaneously with an angular standard in aroutine calibration process. (3) Is used in the calibration of an angular st,andard where someof its angles need to be known precisely beforehand.

The second role is seen in the ca1i.bration of polygons. ' For examplea 300 polygon mag be calibrated b~ comparing each of its twelve anglesto the twelve 30 angles of the iridexing table and then computing ' aleast squares solution for the 24 unknowns. The third role occurs whenit is impossible or impractical to employ a self-calibrating algorithmsuch as in the calibration of a small angle block (less than 150) bydirect comparison to a known interval of an indexing tabl,? In thatcase it is necessary to do a preliminary calibration of the indexingtable angle. One way of accomplishing this is by applying a measure-ment algorithm usually called the "method of subdivision Themathematics of this method are presented in great detail.

., , " ,

2 . The Method , of Subdivision

Angles on the

.'

same indexing table cannot easily be compared witheach other, so it is more convenient to calibrate two indexing tables(denoted by A and B) simultaneously . That way each angle on one tablecan be , compared with several on the other to give a redundant set ofobservations. This idea is incorpo!ated in the method of subdivisionwhich consists ,of two types of measurement designs

, "

complete closureand "partial closure The complete closure design is used to ,subdivide

Figures in ' brackets indicate :t.id~rattire references at the end of thepaper.

the entire 360 degrees of each table intofLo equal segments A

!., . . .,

and B

' ' . .

., B respectively whereO no nE Ai - E Bi - 3600 .i-l i""l

Then by partial closure one seg~nt one~h table, say Al ~~ 81' i$

" .

subdivided iota n1 equal segments A

' . .

., A and 81' . . .. B, n1 n

respectively where

l nt - Al and E B

po Bt-l i-I

.' ,

Similarly Al an4 B1 can be divided int.on

~ equal segments AI' . . . an4 8

1' r . " B respectiv~1y

~~.

2 n2 n

~.. . ",,

.. A1 and ~ 8 '" 8i'F1 i-I'

Tnis pra~e8s can b~ continued ~ntil th. desired level is rea~hed.

As an e~ample, if it were desired to k~ow the va1~ .of the 0.0 -, 10intervaloneacbtable. they could ))e calibrated using the campleteclosure design with nO '" 12 . and two partia~ closure designs with nl ~6and n2 '" 5. Thus each 300 interval wou14 be calibrated. then Qo - 300on ea~h table would be subdivide4 into 5.0 intervals, and the 00 - 5

interval on each table wauldbe subdivided' into 10 intervals. Thissubdivisian is illustrated in figure 1.

It should be noted that .only those increments which are c.ommon toboth tables can be calibrated. For tnfftance , if table A has a smallestincrement pf ope degree and table ~ bas a smallest in~re1Rent of tenminutes. then t;he sroalles't inctement which can be ca1i))ratecl on eachtab~e i8 pne degree.

)-4\-4

5 'I ~

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0 / I

330 0

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MIRROR

rHOO

.. . "' ,," ' ' , ' , ;,. '

INDEXING, TABLES

. . , : ,

PLATE

, :', ,

Figure

Preparation for Mea~urement

In preparation far measurement table A sho\.lld be mounted on table Bas peady concentric as possib~e. It is then helpflJl if this assemblyis mounted concentrically On table ~ whicp is either an indexing tableor a rotary table. This table is not to be calibrated but serves onlyto rotate the other two tables between sets pf measurements as require4by the measurement design. A mirrar if:; then mounted on table A approxi-mately at the center. An autocollimator is mountep so that the face ofthe mirror is c~ntered in its field .of view. Adjustments are made $0that the autocollimator reading is near the center of i~s scale wheneach of the tables is set in i~s ~era posi~ian. The autocol1imatarshoul4 be .adjusted so that it reads hori~antal angle only.

If table C is not available then either an adjustable mirror mustbe used or the autocollimator must be shifted between sets of measure-ments. The whole assembly should be clamped to a surface plate as shownin figur~ 2. The process of clamping the tabl,s should be done with aminimu~ pf distortion, Ahoad shauld be construc~ed over the autocolli-matpr and m;l'rror sa that no outside light caq 1nterfere with the auto...~allimatar reading,

The assembly is then ready far the measurement Process which in-v()J.ves only ttle appropriate rotations .of the 4=ables. Tables A and 8are always ratated in opposite dire~tions through equal nominal angles.The observed change in autac()llimatpr reading is the~ equal to thepifferencebetween the true values of ~he two angles, This is illustra-ted in f~gure 3 far the angles A

l and 81 whtch are namina11y N withdeviations a

l and al respectively.

Com lete Closure Uesi

Let tables A and B each be initially subdivided into n intervals of3QO/n degrees, and let the deviations from nomina~ of the intervals bedenated by the vectars a;:: (Il " , a.

)'

at'\d a = (al , . . I3respectively. If every a. Were camparedwith every a the result would be

meaauJ.'ements on 20 unknowns. In cases of large n this may requiremore measurements than ~ould be Practical, If every a were comparedwith only m of the 13 t s . where m~n. then there would be mn measurementson 2n unknowns. so for m ~ 2 there would be redundancy in the system.A convenient measurement algorithm is to take n blocks .of m differencesof the form

Pos. I

~/I//I;j MIRROR

n-In 0 A

Pas. 2

111- 'II/II

n-In 0 A,

AU' TO'"COLLIMATOR

...

READING

YI =~+EI

READING '

=~ +(N+al )-(N+ 131 ) +E2

:~+a 13, +E2

z ~ Y2 - Y :::: a I - 131 - E 1 + E

Figure 3.

1 - 13

2 - 13

2 - 13

3 - 13

. . .,

n - I3n-11 - I3

a - a tn+ 1 -13111+2 . m-l m-2

(note that 132n-1 = n-1) where with eac;:h ~uc~~eding block th~ as\fbacri

is increased by one and the 13 subscript by two. The i' black of mdifferences is generated by taking ntH obaervations accor4ing to thefollowing scheme:

il :;: 6i + e:

i2

:;: ~

i + ai - 1321-1 + e:

Y i3

:;: ~

i + ai + ai+1 - 132i-l - 132i + &

y i,m+l~. + a

. +

+ ai+m-l - 132i-l

- .

. - 132i+m-2 + &111+1

where ~ i is the initial:t;'eading of the autQcatlimator and the & ' s are

independent error valqes from a distribution whose mean is ~ero andwhose variance is (1 The subscriptsQf 13 are reduced modulo n. Thecomplete closure design will be denoted by C(n,m).

Before proceeding, a word should be aaid about sign convention.This model aS$umes that the two tables are numbered in opposite directions,and that increasing the angle of table A g~ves a positive deflection ofthe autocollimator. If the tables do not conform try this cQnventian,they can be made to conform by reversing the assignment of the anglesor by reversing the sign of the observations or both.th

The new random variables for the i block, ~i - (z il' . . . z

are formed by

il = YiZ - Yi1 = ai - 82i-l + EZ - E

iZ = Yi3 - Yi2 = ai+l - I3Zi + EJ - E

im - Y m+l - Y im = ai+m-l - 132i+m-2 + €m+l - E

or in matrix notation ~i - My i where

~ince Var(y i) - a Im+1' then Var(~

) - a Vm where

2 -1 0 -1 2 -1 0

-MM' - 0 -1 2-f

0 0

. .

. -1

Let ~ - (~1 z

' . . z

formThen the least squares estimati.on takes the

E (z) . X (:)

where

D pm n -D pm n

D pm n -D pm n

D pn-l

m n-D pn-2m n

and P = (I J where I is the m x midentity matrix andm m,is the m x (n-m) zero matrix, and p

l i$ the n x n permutation

matrix which is given by

the full variance-covariance matrix of the observations is given bythe nm x block diagonal matrix

w =

Then

m -1

where

m-1 m-2m-1 2 (m- l) 2 (m-2)m-2 2 (m- 3 (m-2)

m+1

2 (m- m-1m-1

The normal equations (incorporating the restraints

i-1 i-1~. = 0) take the form

(;:W

~. ~

(fJ ~ (X'

where Al .and A2 are Lagrangian multipliers entering in the minimizationprocess and 1 - (1 1

. .

. 1)' (The normal equations ar.e developedfurther in Appendix A in a form suitable for computer programming.

The estimates are given by

X ,1 00 1

l' 0 0 l'

( ~: ;

n r ~'

where C is the variance-covariance matrix of the estimates.

The predicted values are given by

z =

and the deviations by

d - z - z.

The estimate .of (1 is given by

(1 = ' d ' d! (mn-2n+2)

and the standard deviation of theeatiI!lat~s by

- &

and8 = j+n, j+n'

the ' coVariance of i and 8

iso given by

(1 a Ci, n+j

For the cumulative values let

cp - E Qi andi-1 k - ?: 8J.-1

and

The standard deviations are given by

0 O

k - j=l

. .

J.)and

'"

1/i i-l j=l n+i, n+j" "" ..If

There is no error entering from the restraints since theyrepresent an exact relationship.

Partial Closure Design

Let Ai and Bi be two calibrated intervals of the same namina1

angle on tables A and B respectively and let their observed deviationsfrom nominal be given by m

A and mB respectively. Let the corresponding

variance-covariance matrix of the values be given by

The intervals may be subdivided into n segments denoted by

& - (&

1 . . . a

)'

and a - cr\ . . a )' by using the partial closure

design under the restraint that

E&. = mi=l and L: s. - mB .i= 1

Each a can be compared to each 8 so that there are n measurementsof the 2n unknowns. One measurement algorithm which is conVenientforms the 2n-l groups of differences

1 " a I. (~~ = ::"1 r. t . . ., al - 8

2 - 8

Z - 8

3 - 8

, . . ., (~

n - 8 1 J

n-l n-l n - 8n-la -

where the central block has n differences and the adjacent b10ck~'decrease in size by 1 until the end blocks have only 1 difference.Each block of k differences requires k+ 1 measurements. ror e"amp1ethe central block is generated by n+1 measurements according to thefallowing scheme:

nl = An + 8

Y .. n2 = An + a1 - 81 + 8

. .

- A

+ . .

. + a - 8

- .

. - 8 + 8n+1 -

where A is the initial reading of the autocollimator and ~he& ' s

are independent error values from a distribution whose mean is zero

and whose variance is o This partial closure design will be deBated

by P(n). The new random variables z = (z

. .

. z )' are fo~ed byn nn

n1 - Yn2 - Ynl = al - 81 + 82 - 8

n2 - Yn3 - Yn2 - a2 - 82 + 83 - 8

nn - Y n+1 - Ynn = an - 8n + 8n+1 - 8

or in matrix notation z = My where

0 -1

M -

Since Var(Y ) - a In+1' then Var(z

) = a Vn where Vn is defined

as in the previous section.

Let z ' = (zi zi . . . zin - 1) be the complete set of difference

measurement.s. The size of the vector zk is k JC 1 if k .::. n. and (2n-k) x 1if k .~ n, Corresponding to each z

k is a variance-covariance matrix

k whic11 is kx k if k ~n and (2n-k) x (2n-k) if k ~ n. For

k - 1 2, and 3 the .ffiatrices take the form

1 = (2), V2 = (-i -

~),

3 = (-L~-n

The least squares estimation takes the form

E(z) - X (i)

where

X -

-D pn-l1 n

-D pn- 2

2 n

-D . pn n

D pn-1 n n-l

n,..11 n

and Dk = (Ik 8 , n-k where Ik' 8 , n-k' and p~ are defined as .in the

previous section. The variance-covariance matrix of observations isgiven by

W =:

.;,

n.,...l,

whereW is a block diagonal matrix whose blocks vary in size from1 x 1 ~o n x n. Then

..1

,..1

,..1W -

V-In-1

. -1

where v~l is the same as in the previaus section.

V~l == f

(1J, v;1 ~. v;l

The normal equations (incorporating the restraints

i - mA andi-1

~ B

== ~)

take the form

i=l

For eump,le,

. '..

i' 0 0i' 0 ~J OJ ~ lX'

where Al' A2' anA i are defined as before.

(The normal equatians ~re

developed further in Appendix B in a form. suitable for computerprogramming. )

The estimates ~re given by

'"'

x i -1

0 i ' M '

. ,

i' 0 It. - 0 i' O ,

:)

, r~l

,,' ." "

wQere C is the variance-covariance mat,.r1x of the estimates.

The predicted values are given by

~ ~ X

and the devi~tions by

...

d - 2 ... Z.

The es t ima te 0 f 0 is given by

...

' -1 0::: . d' W dl (n -2n+2).

Since the ranqom errar entering through the restraints is non-2ero itmust be tak,n into account when co~puting the tot~l standard deviatipnof the estirrtates. Th~ complete e~pr~ssion for the variances andcoyariancea of the estimates is &iven by

... 2 aVar - 0 C ~ (8 b) S (

)'

Thus

A2 2 A2 2 ~2' 0" - 0 C + a + h . 0 + 2ai II1A lI1 .~ m II1~2 AZ A2 A ' :10" ;: 0 +n ' +n + a + 0 + b + 2a', J n m A II1 m A B '

and the cov~riance of ai and B

j i$~2 ~2 0". - 0" C + a ~ 0" + (a b + a . b ,j i,n+~ ~ n+j m 1. n+j n J :L ,m II1

" .' ,

A2,+ b

~ ,

For the c1,1111u1ative value", let

a' " k i "

k = .E (X i and 'Il '" E Bi-I i-1

Th~ variances are given by

,.

&Z

~ j

t Cij + (it aJ ""'4. + (~b

Y &~ ..

'.

2(i a1(tl b

) " "",!!

' and

k ~ .2

~. j

t Ci+n:- j+n +

(ttHUY ~;A + Ct, tfnY~~ 2 Ct.t+nh~l ifn

) ~

A"JI .

Sources of Error

In the mathematical models just described the only error~;accountedfor were the random errors of measurement and the propagationof tneE;eerrors into the following series. The error of a single measuJ::ement.which was given as cr is the combination of random errors from threesources:

random setting error in indexing table A

random setting error in inde~ing table B

random error of the autocollimator reading

Since SOme of the physical parameters of the system are unknown andcannot be ,modeled, there are systematic errors in ,the measured values.These probably are small , but it has not been determined if they arenegligible. Factors such as the aut;ocollimator reading being 1Jiasedby vertical angle, imperfections in the mirror , effects of varyingtemperature , and foreign particles in the setting mechanisms .of theindexing tables can influence the -accuracy of calibrated values Thereis evidence that over a short period of time , such as a week " that theseeffects are negligible in comparison to the random errors.

There are several reasons why long term changes in the indexingtable angles may not be negligible. If the table is move~ from onelocation to another it may be clamped down in a different manner thusresulting in slight distortions. The grease Which lubricates. thetables may collect small amounts of dus.t over long p, iods of time andas this dust works its way into the teeth of t~e tables i~ causes avariability in the setting of different angles.

Over a period of years the values of two indexing tables at; NBShave been observed to drift slowly, but whenever .a table was dismantledcleaned , and reassembled there was usually a significant change in theangular values. The conclusion to be drawn is that over short periodsof time systematic errors can probably be ignored while over longerperiods of time, such as several months, they become significant and itis better to recalibrate the table ,before using it.

Applications of DesignS

As described in section 2 , a combination of the two types of designsallows the calibration of. intervals of various sizes by subdivision.However, each design can be useful by itself When the .completeclosure design is used alone, it- subdivides the two tables into nequal angles each. The standard deviation of all the angles on indexingtable A are equal , and likewise for indexing table B.

Ta1Jle 1 lists .the standard deviation coefficients , kA and kB' ofthe angles for all values of n up to 36 and for all appropriate values

of m. Note that the standard devtatipn coefficients for the angleson inde~ing tables A and n are not always equal for a given design~ Thestandard devfatidns'of the individua~ angles are given by S

t ='kA '

s' 'arid

a = kB s where s is the estimated standard deviation of a singHr .

measurement.

The standard deviationcoefftcierits for the cumulative angles arealso given. They are the max:i.mum v~lues of the k' s which correspondto all possible sums of consec4tfve'angles. '

Table 1 is given pr!marily as an aid indetetmining w4ich value of;m to use when subdividing two indexing tables into 0, ' equal parts ' each.Assuming that there is a desired upper limit for the standard deviationof eaCli angle (s ) and that an appro~imate value of the observed max standard deviation (s) is knpwo, theIl-any value of m is acceptabl~ - whose corres ondin k value is .lesa t4an the ratio s Is. . For exatn le,'" max

s = 02 and n = 24 , then k

-( .

667;~x values and m ~ 10 for cumulative values.and m;:'3 for individual va+ues and m~) for

given the parameters s .

~ .

03,therefore , m~6 for individualIf s

= .

05 then k -( 1. 667max cumulative values.

The least squares splution to each complete closure design isrestrained by the fact that the sum of the angles on each table isexactly zero, therefore there is no error entering the measurementprocess from that source

Table 2 lists the standard deviation coefficients ~ k = kA ,= k

B' .o~ ,

the angles which are calibrated by the partial closure designs for vc;t,1,uesof n 'UP to ten. In this case the standard deviation coefficients ,are the same for the coresponding angles on indexing tables A and B , that

, kA = ' B for each position. As in the first Case s = sa = ks,

~ '

but now there is an error term entering from the restraint since thevalues of the angles being subdivided are not known exactly. This erroris spread out over the smaller angles and generally will be small com-pared with the randpmmeasurement error within the series.

Thecombtnationa'f . 8 complet~ closure design and one or roo're partialclosure designs gives ,1t flexible system which should be adequate to

' '

economically measure any possible combination of indeX,ing table .angleswhich orfecould imagtne' Tne onlyrequiremertt for using these tools isa modest, atl1ount of imagination. An example is given in th.e next' section.

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STANDARD DFVTATlefN CfHWFTCTF.NTS ~F ANGLES 6N INDEXINGTABU~S A AND B ' CAL I13RATriD wt Ttf CrlMPLETr- Cl.t"ls,J'RF.Dri:~hGN"g!'

. " ,;! , , ".j' ,

r~;~~'

HTDIVIPU AL. . . . . CU~ULATI VF .

...

)01' IC( A)t \f ~(

"#!;

K( A? l td.A X Ie( R)

. :;.

'It

325 . J;?~ 325 326316 316 316 316

~"

30~ 310 309 310

507 424 444 246033 004 785 97~729 696 602 658'622 195 866545 531 886 266

496500 7~6 980465 458 617 75~

1 0 439 437 548 637543417 413 492

1:2 399 398 452' 482

. ,.

4181 'Ft13' 363 .381 43414' 36E)' 39~ 40~369

176356 355 373357 .t, . 358"1:

!'$~;

345 345335 334 342 . 34:f ;w"

. 326 326 . 3. 30 33r:.3 18 317 320 . 320)

20, 310' 310 311: 311

~ (

21)' 303 303 302 1;; 3032915 296 296 296

23' 289 28~ 2 f\C;,fil r" 289283 283 283 283

086665 1 . 588 6353 . 7821 21 093 085'

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STASJ)ARft DEv 1 ATJt1N Cf'E"'" P Ie rENTS I'-'F ANGl,FS~N TNOPXISG T-\BLFS A .~Nf) H CALIH~.A.TEO wI,"!PARTIAL CLM$"VE DESIONS WHPPP R(A )"'g( n..

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443 456 432420 434

400

These designs are applicable to other areas where angle measure-ments are involved. The complete closure method is applicab1~ to thecalibration of a polygon. When a polygon is substituted :!i,or- themirror its exterior angles are calibrated. If the polygon has twelvesides or less it is not too burdensome to take the full number of -

' '.'

measurements. For a twelve sided polygon the C(12 12) design could beused requiring (12 x 13) = 156 measurements, a reasonable number.However, if a 36 sided polygon were calibrated by the C (36,36) seriesit would require (36 x 31) = 1332 measurements which normally wouldnot be economical. Instead , a design such as' G(36, 6) could be usedrequiring a more modest (36 x 7) = 252 measurements. In cases whereis large and .m is small the error in cumulative values greatly exceedsthe error in the individual values. This may be an important factor inpolygon calibrations , so the value of m must be carefully chosenaccording to the required precision.

Rotary tables may be used in place of indexing tables , but theobserved standard deviation will be much larger because rotary tablesetting errors are generally much larger than i~dexing table settingerrors.

Example

The first application of the method of subdivision was in connec-tion with the absolute measurement of a set of angle blocks. Angularintervals on each indexing table which needed to be calibrated were00 - 600 , 00 - 300 , 00 - 150 , 00 - 50 , and 00 - 10 On table A these

angles were denoted by (1 +(12' (1

+0. +0.3' aI' andai respectively. The

corresponding angles on table B were similarly denoted using 13 insteadof (1. The most natural subdivision scheme seemed to be completeclosure on the 300 intervals and two partial closures on 50 and 10intervals. The three designs chosen were C(12 6), P(6) and P(5).

The observed y values from the three series are given in table 3.The computed values for the individual angles and their correspondingstandard deviations are given in table 4 along with the observedstandard deviation of a single measurement , s , and the degrees offreedom , df.

Conclusion

Computer programs have been written for the calibration of indexingtables by the methods described in this paper. The programs have beentested and successfully implemented. As a result several angle cali-brations have been made more efficient by reducing both the timerequired to make the measurements and the time required to reduce thedata.

1'\BtE 3.

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Acknowledgement

I wish to thank Mr. Ralph Veale of the Dimensional TechnologySection for acquainting me with the various ways in which indexingtables are used .in calibrations and for suggesting areas where sub-division schemes might be applicable. I especially thank Mr. Joseph M.Cameron of the Office of Measurement Services for reviewing the paperand for offering se'\i"eral helpful suggestions on improving it.

RefeJ'ences

Cook, A. H., The Calibration of ~ircu1ar Scales a~d Precision.Polygons, British Journal of Applied Physics, Vol. 5, 19$4,pp. 367-371.

2.. Hutne, K. J., Metrolagy with Aptoc:o;Llimators, Hilger and Watts, Ltd.,

'London , 1965 , Ch. 6, 7.

Moore, Wayne R., FoundatioI1$ of Mec:han:f,.caJ. Accuracy, the MooreSpecial Tool Co., Bridgeport, CoQ.n., 1970 , pp. 201-250.

Ze1en, MaJ'vin, Survey of Numerical Analysis , edited by John Todd,McGraw-Hill, 1957, Ch. 17.

Appendix A

Reduction of the Normal Equations of the Complete Closure Model

The normal equations given in section 4 can be expressed in aform which is useful in programming the model for the computer. Insteadof performing the matrix multiplications the normal equations are formeddirectly so the amount of memory needed is substantially reduced. The

product .x ' is given by

l r:

n-1

n-2m m

n m m

' . . _p

n m m

- (-~ ~

z . . :-:: J

where Mi .and ~i and n x m matrices.

Now

n m m

(p: ) ~j .

a T

nm ij nm i+a,

where the subscripts ale reduced modulo n.

the observations , X'W- , 1s then given byThe coefficent matrix of

(~)

ij = (T i+n-k+1 j and

(~)

ij - (T i+n-2k+2 j where i - 1 n; j = I m; k - l

Let Q and Q ben x 1 vectors defined by

~ J ~ (-

:: - : : : -

::J . z

. .

where each zi is an m x 1 vector of observations.

be' expressed byThen Q andQ can

(Q)i - k-I j-l (~) ij (z

) j

L: E (T ) i+n-k+1 j (z

j,

andk-I j-l

(Q) 1 - E I: (~) 1j (z - I: k-1 j-I k-I j-I i+n-2k+2 j (Z

where i - 1,

The product X'W Xis given by

X -

n-I n-k D' V

-I D p

k==O n m m m nn""l n-k

D' v-1 D p

k-O n m m . m n

n-l n-2k1: D' v

-I D

' p

k-O n m m m nn-l n-2k 15' V"'l D pk-O n m m m n

- L:. -~ J wb.re N ,N aOON ar. n " " matr!ces,

-1 Now D p

""

. m m ,n-m pn m m m n ij n ijn-m,m n-m n-m

u pn nm n ij ) i+a,j-b where the subscrip~s are reduced

modulo n..

. .

The matrix of normal equations, X' X, is then given by

n-1(N\j - :E (U i+n-k j-k'k-O

. ,

n-1(N)

"" ~o (U i+n-k,j-2k' and

n-1(N) i - E. (U ) i+n-2k -2k where i - l,

n; j - 1k-O After augmenting with the two restraints the normal equations take

the form

3:'

~'

Appendix B

Reduction of the Normal Equations of the Partial Closure Method

The normal equations in section 5 can be treated in the samemanner as those in section 4. The product X' W-l is given by

D' v D' vn nn-l D' v "" 1

n-l n-l

-D' . vn-l n-l

. . . p~ -

:~ :~:J

~p~

D' Vn D' vn n

r:

. .

, ~n H I . . . H

t-M

. .

. -Mn-1 . .

. -M

where Mk and are n x k matrices. Now

(D' Va a - (~n:~a))ij(T )..na ~J

and

a D'. Va a ij

a T

na ij ) i+a,

where the subscripts are reduced modulo n.

the observations, x, , is given by

The coefficient matrix of

(~) ij = (T ) ijand

(~) ij = (T ) i+k, where i = l n; j - l,

Let Q and Q be n x 1 vectors defined by

l:J ~

: : : _

n-l . . . '\1

n-l . . . -M n+l

2n-l

where the zk are vectors

.of observations of size k x 1 if k and (2n-It) x 1 if k ,. n. Then Q and Q can be expressed by

n-l k(Q) . ~ L:

(~)

ij- J + L: E (~) ij (z2n-kk-l j=l It-I j-l

n-l = E. L (T j + I: i+k, (2:2n-k andk=l j-l k-l

n-l k (in i - L: E (~) ij (z

) j

+ I: E (~) ij (z2n-kk-1 j-1 k-l j-l

n-l k

= ,

L: (T ) i+k j (z

) j

L: L: (T ) ij (z2n-k

) j

1=1 j-l k-1 j-1

where i - 1,

The product X' W X is given by. ,

)!:' \/ )( = ; '

. -N' where

n n-

. .

-1

. " p

n-krD," V-I D p

L..J Dk V k Dk + L..J n . n,-k n-k n-k nk-1 k-l

N -n-l

D' VI D pn-k +

E pn-k D' V-I D

k-1 k k k n k=l n n-k n-k n-k' and

N -n-1

k D' V-I D. p

n-k + L'

' -

k-1 n k k k-l

-1 Now P

n Db Vb Db Pn ij (P: (::~b.::~:~:J P~)

. ,

n Unb P ) ij ) i+a, j-c

where the subscripts are reduced modulo n.equations, X' W- X, is then given by

The matrix of normal

i' 1.

(N) ij ~l (U nk) ij

+ E (Un n-k i+n-k j-kk-l

(in ij

n-lE (U i j-n+k + E (Un n-k i+n-k k-1 ' k-1 and

(N) ij

n-1i+k j-n+k + E (Un n-kk-1

where i . l.nl j - 1,

After augmenting with the two re~traints the normal equationstake the form

-N i -N I 0 iit 0 0

it 0

where mA and mB are restraint va1ues

fJ,".om the previous series.

USCOMM.NBS.oC