Calibration of the clock-phase biases of GNSS networks: the closure-ambiguity approach

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André Lannes, Jean-Louis Prieur

To cite this version:

André Lannes, Jean-Louis Prieur. Calibration of the clock-phase biases of GNSS networks: the closure-

ambiguity approach. Journal of Geodesy, Springer Verlag, 2013, pp.1-23. �10.1007/s00190-013-0641-4�.

�hal-00830353�

ALannes

1

and JLPrieur

2

1

CNRS/Supele /UnivParis-Sud,Fran e

Address: Supéle ,2rueE.Belin,57070Metz,Fran e

e-mail: andre.lannes erfa s.fr

Fax: 33387764700

2

UniversitédeToulouseUPS-OMPIRAP,Toulouse,Fran e

Address: 14avenueEdouardBelin,31400Toulouse,Fran e

Abstra t.Inglobalnavigationsatellitesystems(GNSS),

the problem of retrieving lo k-phase biases from net-

work data hasabasi rank defe t. We analysethe dif-

ferent ways of removing this rank defe t, and dene a

parti ularstrategyforobtainingthesephasebiasesin a

standardform. Theminimum- onstrainedproblemtobe

solvedin theleast-squares (LS) sense dependson some

integerve torwhi h an be xed in an arbitraryman-

ner. We propose to solve the problem viaan undier-

en edapproa h basedon thenotionof losure ambigu-

ity. Wepresentatheoreti aljusti ationofthis losure-

ambiguityapproa h(CAA),andthemainelementsfora

pra ti alimplementation. Thelinkswithothermethods

arealso established. Weanalyse allthosemethods in a

unied interpretative framework, and derive fun tional

relations between the orresponding solutions and our

CAAsolution. This ouldbeinterestingformanyGNSS

appli ations like real-time kinemati pre ise pointposi-

tioningforinstan e. To omparethemethodsproviding

LSestimatesof lo k-phasebiases,wedeneaparti ular

solutionplayingthe role of referen esolution. Forthis

solution,whenaphasebiasisestimatedforthersttime,

itsfra tionalpartis onnedtothe one- y lewidth in-

terval entredonzero;theinteger-ambiguitysetismod-

ied a ordingly. Our theoreti al study is illustrated

with some simple and generi examples; it ould have

appli ationsindatapro essingofmostGNSS networks,

and parti ularlyglobal networksusing GPS, Glonass,

Galileo,orBeiDou/Compasssatellites.

Keywords. Global and regional networks

·

^{Clo k}

biases

·

Un alibrated phase delays (UPD)

·

^{Fra tional}

lo k biases (FCB)

·

^{Network real-time kinemati s}

(NRTK)

·

^{Real-timekinemati pre ise point}positioning (RTK-PPP)

·

^{Closure dieren e (CD)}

·

^{Nearest latti e}

point(NLP)

·

^{Integerleastsquares(ILS)}

1 Introdu tion

Inglobal navigation satellite systems (GNSS), the al-

ibration of the lo k-phase biases of global networks is

a hallenging problem. In parti ular, the knowledge of

thesatellite lo k-phasebiasesisneededforpre isepoint

positioning (PPP); see, e.g., Zumbergeet al. 1997; Ge

et al. 2008; Bertigeretal. 2010;Genget al. 2010;Liet

al. 2013. Inthegeneral ontextdenedbelow,theequa-

tions governing this GNSS alibration problem have a

basi rankdefe t. Inthispaper,weanalysethedierent

waysofremovingthisrankdefe t,anddeneaparti ular

strategy for obtainingthe lo k-phase biasesin a stan-

dardform. Thelinkwithotherrelatedapproa hes,su h

as those proposed by Blewitt (1989), de Jonge (1998),

Collins et al. (2010),and Loyeret al. (2012),is estab-

lished inthat framework.

When modelling the multi-frequen y ( ode and phase)

observations of GNSS networks, the systemto be on-

sideredin lude phasestru turesoftheform











[β rκ (i) − β sκ (j)] + N (i, j) = b κ (i, j)

for

κ = 1, . . . , k

(1)

Here,

κ

^{istheepo hindex;}

k

^{istheindexofthe urrent}

epo h;

β rκ (i)

^and

β sκ (j)

^are lo k-phasebiases. These termsarealso alled`un alibratedphasedelays'(UPD).

They are expressed in y les, and depend on the fre-

quen yofthetransmitted arrierwave;subs riptsrand

s stand for re eiver and satellite, 1

respe tively;

i

^{is the}

indexofthere eiver,and

j

^{thatofthesatellite;}

N (i, j)

^is

theintegerambiguityofthe orresponding arrier-phase

measurement. Theterms

b κ (i, j)

^{in ludethe} orrespond- ing phase data and all the other ontributions of su h

equations; see, e.g., Eqs. (1) and (10) of Lannes and

Teunissen2011,andEqs.(1)and(4)ofLoyeretal. 2012.

Thesetofre eiver-satellitepairs

(i, j)

^{involvedinEq.(1)}

forms theobservationalgraph

H κ

^{oftheGNSS s enario}

1

Inthispaper,satelliteshouldbeunderstoodassatellitetrans-

ofepo h

κ

^{. This graphis assumedtobe onne ted;see}

AppendixA. Note that the wide-lane equation of the

ionosphere-freemodeis typi allyofform (1);

N

^isthen

awide-laneintegerambiguity;see,e.g.,Eq.(4)ofLoyer

etal. 2012.

Asexpli itly lariedfurther on,wheneverphasestru -

turessu has(1)appearinGNSS-networkproblems,are-

latedrankdefe tistoberemoved. Inthispaper,were-

stri tourselvesto therankdefe ts indu edthose phase

stru tures. Thisdoesnotmeanof oursethatthosebasi

rankdefe tsaretheonlyonestobehandledinpra ti e;

see, in parti ular, Teunissen and Odijk(2003). A stan-

dardapproa hforta klingtherankdefe tsis knownas

theS-systemapproa hof Baarda1973, Teunissen1984,

deJonge1998. Examplesofsu hS-systemsolutionsare

to be found in de Jonge 1998; Teunissen et al. 2010;

Zhanget al.2011;Odijket al. 2012.

Inthegeodeti and GNSSliterature,there existseveral

waysofremovingthisbasi rankdefe t. Themostgen-

eral approa h is based on the S-system theory already

mentioned. Other strategiesderivefrom thepioneering

ontributionofBlewitt(1989): therelationshipbetween

theundieren ed (UD)ambiguities andthe double dif-

feren ed(DD) ambiguitiesis ompleted sothat the op-

erator

D

^{thus dened is}invertible. Letus alsomention theapproa hof Collinset al. (2010)whi h is basedon

the on eptof`ambiguitydatumxing.' Theimportant

developmentsof thoseapproa hes,bothata on eptual

andte hni al level,were often ondu ted withdierent

physi al obje tives. They have thus progressivelyand

insidiously masked the fundamental links between the

relatedmethods.

Briey, the Blewitt pro edure an be divided in three

steps. Intherststep,withregardtoEq.(1)forexam-

ple,theUD data arepro essedby onsidering theterm

ontheleft-handsideofthatequationasa` onstantfun -

tionalvariable;'aoatestimateofthis`biased-ambiguity

variable'isthusobtained. Inthese ondstep,the orre-

sponding DDambiguities are omputed,and thenxed

at integervalues. Inthe third step, the lo k-phasebi-

ases

β κ

^{are estimated by using as data the UD ambi-}

guities provided by the a tion of

D ⁻¹

^{on the olumn}

matrixformedbythose xedambiguities. Thetheoret-

i al analysisdevelopedin thepresentpaperprovidesin

parti ular ananswerto the following question: what is

thelinkbetweentheUDambiguitiesthusxedand the

xed` losure-delay'or` losure-dieren e'(CD)ambigui-

tiesoftheUDapproa hofLannesandTeunissen(2011)?

AsimilarquestionarisesfortheUD approa hofCollins

etal. (2010);anansweris alsoprovided.

Inthis generalGNSS ontext,themainobje tiveofthe

paperistopresentauniedinterpretativeframeworkin

whi h the various ontributions in the related elds of

resear h anbe understood and ompared moreeasily.

This an lead to improvements of some related meth-

ods. Forexample,weshow that removingtherankde-

fe tviathe

D

^{-matrixofBlewitt(1989) anbeanalysed}

inatheoreti alframeworktightlylinkedtotheS-system

approa hofTeunissen(1984). Wethusshowthatthein-

termediatedieren ingstageoftheBlewittapproa h an

be avoided, without any ounterpart, via the approa h

of Teunissen (1984) as it is formulated for example in

Lannes and Teunissen (2011): the ` losure ambiguities'

tobexedthenappear,fromtheoutset,intheveryfor-

mulationoftheUDproblemto besolved; omparewith

what isdonein Se t.4ofGe etal. (2005)forinstan e.

Thetheoreti alguidelinesofthispaperarepresentedin

Se t. 2. We rst identify the rank defe t in question.

The minimum- onstrained problem to be solved in the

least-squares(LS) sensedepends onsomeintegerve tor

whi h anbexedin anarbitrarymanner. To ompare

the methods providing LS estimates of the lo k-phase

biases, we then introdu e aparti ular solution playing

the roleof referen esolution. Forthis solution,when a

lo k-phasebias isestimated fortherst time,its fra -

tional part is onned to the one- y le width interval

entred on zero; the integer-ambiguity set is modied

a ordingly. Se tion3isdevotedtothealgebrai frame-

work of our analysis. This framework mainly derives

from the original ontributions of Lannes and Gratton

(2009), andLannes and Teunissen (2011). As asimilar

problemarisesinphase- losureimaginginastronomy,we

also took protoftheanalysis presentedin Lannesand

Prieur (2011). A natural way for nding the referen e

solutionisto adoptanapproa hbasedonthenotionof

losure ambiguity. The prin iple of the orresponding

` losure-ambiguity approa h' (CAA) is dened in that

framework (Se t. 4). The bulk of our ontribution fol-

lowsthemaintheoreti alguidelinespresentedinSe t.2.

In a related option whi h is presented in Se t. 5, the

CAAprin ipleisdire tlyintrodu edviatheS-systemap-

proa hofBaarda(1973),Teunissen(1984)anddeJonge

(1998). The orrespondingdevelopmentisperformedin

the S-system framework dened in Appendix B. The

study developedin Se ts. 3.3, 3.4 and 3.15of de Jonge

(1998) is thus extended to the ases where the union

of the graphs

H _κ

^{is taken into a ount} progressively. Se tion 6 is devoted to the QR implementation of the

CAA prin iple; related information is to be found in

AppendixC. Inmanymethods,therankdefe tinques-

tion isremovedin animpli it mannerorintuitively. In

Se t. 7, on the grounds of some results established in

Se ts.3.5and3.6,weidentifytherelated onstraintsex-

pli itly,andthusestablishthelinkbetweenthesolutions

providedbythosemethodsandtheCAA-(S-System)so-

lutions; seeFig.6inparti ular.

Ouranalysisisillustratedwithsomesimpleandgeneri

examples. It ould haveappli ations in datapro essing

ofmostGNSSnetworks,andparti ularlyglobalnetworks

usingGPS,Glonass,Galileo,orBeiDou/Compasssatel-

lites. Themain resultsprovidedbythis studyare om-

mented upon in Se t. 8; some on lusions are alsopre-

sented with possible appli ations to software pa kages

used forpro essingGNSS networks.

2 Theoreti al guidelines

TheproblemisformulatedinSe t.2.1; therelatedrank

defe tisidentiedin Se t.2.2. This rankdefe t anbe

removedbyimposingsome onstraintswithoutae ting

the GNSS resultssu h as the estimates of the station-

position parameters,forexample. Theparti ularLSso-

lutions thus obtainedare dened in Se t.2.3. Wethen

denethefamilyofthosesolutions(Se t. 2.4). To om-

paretheparti ularsolutionsgivenbythevariousGNSS

methods providing LS estimates of lo k-phase biases,

wethen introdu eaparti ular solutionplayingtherole

ofreferen esolution(Se t.2.5).

2.1 Formulationof the problem

Inourformulationoftheproblem,the omponentsofthe

ambiguity ve tor

N

^{are the integer} ambiguities

N (i, j)

involved in the phase measurements until the urrent

epo h. Wetherebyassumethatthetime-invariantprop-

erty of the ambiguities holds. Regardedas a fun tion,

N

^{thereforetakesitsvaluesontheedgesof}

G k

def =

k

[

κ=1

H κ

⁽²⁾

where

H κ

^istheobservationalgraphofepo h

κ

^{. Inwhat}

follows,

H k

^{denotes the} ` hara teristi fun tion' of

H k

withregardto

G k

^:

forall

(i, j) ∈ G k

^,

H k (i, j) ^def =











1

^if

(i, j) ∈ H k

^;

0

^otherwise.

(3)

The number of edges

(r i , s j )

^of

H _κ

^{is denoted by}

n eκ

^;

n eκ

^{islessthanorequaltothenumberofedgesof}

G _k

^.

Toillustrate ouranalysis,we onsider a`simulatednet-

work'in ludingfourre eiversandvetoeightsatellites;

see Fig. 1. The s enariosof the rst three epo hs are

dened by the hara teristi fun tions

H 1

^,

H 2

^and

H 3

displayed in that gure. While looking simple at rst

sight, this example is rather elaborate. Indeed, it in-

ludesthe aseoftheappearan eofnewsatellitesinthe

eldof viewof thenetwork (

s 6

^and

s 7

^{atepo h}

2

^,

s 8

^at

epo h 3), and alsothe aseof thedisappearan e ofone

satellite(

s 3

^{atepo h}

3

^).

Remark2.1. Whenasatellite omesba kintheeldof

viewof the network, it isdealt withas anew satellite.

Inthe aseofglobalnetworks,ifneedbe,thesu essive

passesarethus dealtwithinasimplemanner

·

At epo h

κ ≤ k

^{, only some omponents of}

N

^{may be}

a tive; seeFig.1. Toformalizethispoint, weintrodu e

theoperator

R ^e _κ

^{thatrestri ts}

N

^{(whi hisdenedonthe}

edgesof

G k

^{)totheedgesof}

H κ

^:

forall

(i, j) ∈ H κ

^,

(R ^e _κ N )(i, j) ^def = N (i, j)

⁽⁴⁾

1 1

·

¹

·

·

^{1 1}

· ·

1

· ^{1 1}

¹

· ·

¹

¹ ·

r 1

r 2

r 3

r 4

s 1 s 2 s 3 s 4 s 5

1 1

·

¹

· ·

¹

·

^{1 0}

· ·

¹

¹

1

1 0 1

1

1 ·

· ·

¹

^{1 1} · ·

r 1

r 2

r 3

r 4

s 1 s 2 s 3 s 4 s 5 s 6 s 7

1 1

·

¹

· ·

¹

·

·

^{1 0}

· ·

¹

¹

¹

0

1 0 1

1

1 · ·

· ·

⁰

^{1 1} · ^{1 1}

r 1

r 2

r 3

r 4

s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8

Figure 1: Chara teristi fun tionsof

H k

^{with regard to}

G k

for

k = 1, 2, 3

^{(example). From top to bottom,}

H 1

^,

H 2

^,

and

H 3

^{. Thedotsdenetheedges}

(r i , s j )

^{forwhi hnodata}

havebeenobtaineduntilepo h

k

^{in luded. Here,}

n e1 = 11

^,

n e2 = 15

^{, and}

n e3 = 16

^{. By denition,}

G k

^{is the unionof}

theobservational graphsuntilepo h

k

^{in luded. Thenum-}

ber of the edges of

G k

^is

11

^{at epo h 1,}

17

^{at epo h 2,}

and

20

^{at epo h3. Six edges appear at epo h 2:}

(r 1 , s 7 )

^,

(r 2 , s 6 )

^,

(r 2 , s 7 )

^,

(r 3 , s 2 )

^,

(r 3 , s 6 )

^and

(r 4 , s 5 )

^{; two edges dis-}

appear:

(r 2 , s 3 )

^and

(r 3 , s 3 )

^{. Notethat satellites}

s 6

^and

s 7

are then dete tedby the network. Three edges appear at

epo h 3:

(r 2 , s 8 )

^,

(r 4 , s 7 )

^and

(r 4 , s 8 )

^{; twoedges disappear:}

(r 3 , s 1 )

^and

(r 4 , s 3 )

^{. Satellite}

s 3

^then disappears. At ea h epo h,thelarge-sizednumbersdenetheedgesof

G st,k

^,the

sele tedspanningtreeof

G k

^{;seeFig.5furtheron.}

Equation(1) anthenbewrittenin theform











B κ β κ + R ^e _κ N = b κ

for

κ = 1, . . . , k

(5)

where

B κ

^{isthefollowingbiasoperator:}

(B κ β κ )(i, j) ^def = β rκ (i) − β sκ (j)

^(forall

(i, j) ∈ H κ

^{) (6)}

In what follows, wewill assumethat Re eiver

1

^denes

thereferen eforthere eiverand satellitebiases:

β rκ (1) = 0

⁽

κ = 1, . . . , k

^{) (7)}

This is ommonly used by the GNSS investigators for

removingtherankdefe tof operators su h as

B κ

^{. The}

number

n bκ

^{of phasebiasesof epo h}

κ

^{to beestimated}

isthereforeequalto

n vκ − 1

^where

n vκ

^{isthenumberof}

verti esof

H κ

^:

n bκ = n vκ − 1 (n vκ = n rκ + n sκ )

⁽⁸⁾

Withregardtoitsfun tionalvariables

β 1 , . . . , β k

^and

N

^,

Eq. (5) proves to have a basi rank defe t. We now

spe ifythispoint.

2.2 Identi ation of the rank defe t

For larity,letusset

G ^def = G k

⁽⁹⁾

Atepo h

k

^{,the numberof}ambiguities

N (i, j)

^involved

intheproblemisequaltothenumberofedgesof

G

^(for

exampletwentyin Fig.1for

k = 3

^{). Again, for larity,}

thisnumberissimplydenoted by

n e

^{. Wethenset}

n st = n v − 1 (n v = n r + n s )

⁽¹⁰⁾

where

n v

^{isthenumberofverti esof}

G

^;

n r

^and

n s

^arethe

numberofre eiversandsatellites(respe tively)involved

in that graph(four and eightin Fig. 1for

k = 3

^{). As}

spe ied in Se t.A2,

n st

^{is thenumberof edges ofany}

spanningtree

G st

^of

G

^{. Thetotalnumberofphasebiases}

tobeestimatedatepo h

k

^,

P k

κ=1 n bκ

^{,isgenerallymu h}

largerthan

n st

^{; see Eqs. (8) and(10). The partplayed}

bythe verti esof

G

^{is notobvious. Wenowshowthat}

n st

^{denesthe`size'oftherankdefe tin question.}

Letusdenoteby

B

^{theoperatorfrom}

R ^{n st}

^into

R ^{n e}

^de-

nedbytherelation

(Bα)(i, j) ^def = α r (i) − α s (j)

^(forall

(i, j) ∈ G

^{) (11)}

Denotingby

µ

^{anyinteger-valuedfun tiontakingitsval-}

uesontheverti esof

G

^{otherthanthereferen ere eiver,}

wehave

R ^e _κ Bµ = B _κ R ^v _κ µ

⁽¹²⁾

where

R ^v _κ µ

^{is the} restri tionof

µ

^{to theverti esof}

H κ

(otherthan thereferen ere eiver). Note that

µ

^anbe

regardedasave torof

Z ^{n st}

^{. ItthenfollowsfromEq.(5)}

thatforany

µ

ⁱⁿ

Z ^{n st}

^,











B _κ (β _κ + R ^v _κ µ) + R ^e _κ (N − Bµ) = b κ

for

κ = 1, . . . , k

(13)

Via theoperators

B _κ

^,

R ^v _κ

^,

R ^e _κ

^and

B

^{, any variationof}

the`vertex-ambiguity've tor

µ

^anthusbe ompensated by a variation of the `edge-ambiguity' ve tor

N

^{. As a}

result,with regardto thebiasand ambiguityvariables,

Eq. (5) is notof full rank. The dimension of the rank

defe tisequaltothatofve tor

µ

^,i.e.,

n st

^.

2.3 Parti ular LS solutions

InGNSS,forthereasonsspe iedinRemark 2.2(atthe

endof this se tion),ea h lo k-phasebiasis to be esti-

mated upto a onstantinteger. As aresult,the hoi e

of

µ

ⁱⁿ

Z ^{n st}

^{does not ae t the signi ant part of the}

valuesofthebiasfun tions

w κ

def = β _κ + R ^v _κ µ

⁽

κ = 1, . . . , k

^{) (14)}

tobeestimated; seeEq. (13). The ambiguityve torto

beretrieved

v ^def = N − Bµ

⁽¹⁵⁾

isof ourseae tedbythis hoi e,butthishasnoa tual

GNSS impa t. As aresult, theGNSS methods provid-

ing estimatesofthe lo k-phasebiasesmustremovethe

rank defe t of Eq. (5) by hoosing

µ

ⁱⁿ

Z ^{n st}

^somehow,

impli itlyorexpli itly.

In pra ti e, as laried in the remainder of the paper,

removingthisrankdefe tamountstoimposing

n st

^on-

straints on somevalues of the biasesor ambiguities to

beretrieved. Inother words

µ

^{isdened viathese on-}

straints. Theminimum- onstrainedproblemtobesolved

in theLSsense isthereforeoftheform











B κ w κ + R ^e _κ v = b κ

⁽

κ = 1, . . . , k

⁾

subje tto

n st

onstraintson

w κ

^or

v

(16)

Withregardtoaparti ularsetofsu h onstraints,where

v

^isaninteger-valuedfun tionfrom Eq.(15),theLSso- lutionofEq.(16),

( ˇ w 1 , . . . , ˇ w k ; ˇ v)

⁽¹⁷⁾

isthenunique. Forexample,thesolutionprovidedbythe

CAAmethoddenedinSe t.4istheparti ularLSsolu-

tionobtainedbyimposing theapriori onstraint

v = 0

onaspanningtreeof

G

^{( hosen}arbitrarily). Theparti - ularLSsolutionintrodu edinSe t.2.5isdenedbyim-

posing,aposteriori,

n st

onstraintsonsomebiasvalues.

Inouranalysis,thisparti ularsolutionplaystheroleof

referen esolution;itisdenotedby

( ¯ w 1 , . . . , ¯ w k ; ¯ v)

^.

Remark2.2. Thesatellite omponentsofthebiasesthus

obtained (for example those of the referen e solution)

anbebroad asted tothenetwork usersforPPP appli-

ations. Thefa tthat

w ˇ sκ (j)

^{isanLSestimateof}

β sκ (j)

uptosomeunknown onstantintegerdoesnotraiseany

di ulty. Oneisthensimplyledto redenetheinteger

ambiguities involvedin the PPP problem to be solved;

see, e.g.,Se t. 9in LannesandTeunissen2011

·

2.4 Equivalent LS solutions

Givensomeparti ularLSsolutionsu has(17),wehave

B _κ w ˇ _κ + R ^e _κ v ˇ ^LS = b κ

LikeforEq.(13),itthenfollowsfromEq. (12)that

B _κ ( ˇ w _κ + R ^v _κ µ) + R ^e _κ (ˇ v − Bµ) ^LS = b κ

TheLSsolutionsof Eq.(5)arethereforeoftheform

( ˇ w ^(µ) ₁ , . . . , ˇ w ^(µ) _k ; ˇ v ^(µ) )

⁽¹⁸⁾

with

ˇ

w ^(µ) _κ ^def = ˇ w κ + R _κ ^v µ, v ˇ ^{(µ) def} = ˇ v − Bµ

⁽¹⁹⁾

where

µ

^{isanyve torof}

Z ^{n st}

^.

ThemethodsprovidingLSestimatesofthephasebiases

generallydierbythe hoi eoftheimposed onstraints.

To omparetheirresults,itis onvenienttorepresentthe

equivalent solutions (18)-(19) by a referen e parti ular

solution. ThisisdoneinSe t.2.5.

Remark2.3. Foranyxed

µ

ⁱⁿ

Z ^{n st}

^{,thetemporalvari-}

ations of the estimated phase biases make sense. For

example,ifsatellite

s j

^{remainsintheeldofviewofthe}

network fromepo h1to

κ

^,wehave

(R ^v _κ µ) s (j) = (R ^v ₁ µ) s (j) = µ s (j)

hen efrom Eqs.(19)and(14),

ˇ

w sκ ^(µ) (j) − ˇ w ^(µ) _s1 (j) = ˇ w sκ (j) − ˇ w s1 (j)

≃ β sκ (j) − β s1 (j)

A similar result of ourse holds for the re eiver lo k-

phasebiases

·

2.5 Referen e solution

Wehere on entrateon thefamilyof equivalent LSso-

lutions(18)-(19)generatedbyaparti ularsolutionsu h

as (17):

( ˇ w 1 , . . . , ˇ w k ; ˇ v)

^{. In ouranalysis, the referen e}

solutionofthisfamilyistheparti ularsolution

( ¯ w 1 , . . . , ¯ w k ; ¯ v)

⁽²⁰⁾

denedasfollows:

w ¯ κ

^and

v ¯

^{areoftheform (19)}

¯

w κ

def = ˇ w κ + R ^v _κ µ, ˇ v ¯ ^def = ˇ v − B ˇ µ

⁽²¹⁾

in whi h

µ ˇ

^{is dened by imposing spe i} onstraints on

n st

^{biasvalues;note that here,these} onstraintsare imposedaposteriorionthesolution

( ˇ w 1 , . . . , ˇ w k ; ¯ v)

^pro-

videdbyanymethod. Werstrequirethephasebias

w ¯

to be small at epo h 1. More pre isely, weimpose the

ondition

| ¯ w 1 | ≤ 1/2

^,i.e. expli itly,











| ¯ w r1 (i)| ≤ 1/2

^for

i = 2, . . . , n r1

| ¯ w s1 (j)| ≤ 1/2

^for

j = 1, . . . , n s1

(22)

Thefollowingvaluesof

µ ˇ

^aredeneda ordingly:











ˇ

µ r1 (i) := − ⌊ ˇ w r1 (i)⌉

^for

i = 2, . . . , n r1

ˇ

µ s1 (j) := − ⌊ ˇ w s1 (j)⌉

^for

j = 1, . . . , n s1

(23)

Here,

⌊x⌉

^{denotes the integer losest to}

x

^{. Likewise,}

at ea h epo h

κ

^{when some} satellite(s)

s j

^{appear(s) in}

the eld of view of the network (see Fig. 1), we then

imposethe ondition(s)

| ¯ w sκ (j)| ≤ 1/2

⁽²⁴⁾

bysetting

ˇ

µ sκ (j) := − ⌊ ˇ w sκ (j)⌉

⁽²⁵⁾

(Inthe asewherenewre eiverswouldbea tivated,sim-

ilar onditionswouldbeimposed.) Atepo h

k

^,wehave

thus ompletely denedsomeve tor

µ ˇ

^of

Z ^{n st}

^;

v ¯

^isthen

obtainedviatherelation

v := ˇ ¯ v − B ˇ µ

^;seeEq.(21).

Remark 2.4. When some LS solution

( ˇ w 1 , . . . , ˇ w k ; ˇ v)

hasbeenfound, forinstan ethat provided bytheCAA

methoddenedinSe ts.4to6,thereferen esolutionof

its equivalentsolutions is obtained asdes ribedin this

se tion. Clearly,this analsobedonefortheLSsolution

of anymethod providing estimates ofthe phase biases;

see Se t. 7 together with, e.g., Blewitt 1989; Ge et al.

2005; Lauri hesseandMer ier2007;Collinset al. 2010;

and Loyeretal. 2012. To ompareandvalidate there-

sults provided byallthese methods (andmanyothers),

onemayinspe ttheambiguitysetsoftheirreferen eso-

lutions. These referen e ambiguity solutionsshould be

identi alonalltheedgesof

G

^{forallmethods;otherwise,}

thiswouldbeanindi ationthatthemethodsareindis-

agreement, and that some of those results are wrong.

The omparison of the referen e solutions is therefore

a good diagnosis for testing the ompatibility of these

methods

·

Remark2.5. Fromate hni alpointofview, onemight

trytosolveEq.(16)intheLSsensebyimposingthenon-

linearbias onstraints(22)and(24)on

w 1

^andsome

w κ

^,

fromtheoutset. Itisnoteasyatalltosolvetheproblem

that way. Moreover,thenumberofedge ambiguitiesto

be xed would then be equal to

n e

^{, whereasthe num-}

berofambiguitiestobexedin theCAAapproa h(for

example) isequalto

n e − n st

^{;seeSe ts. 4and6}

·

3 Algebrai framework

ThepreliminaryanalysisdevelopedinSe t.2showsthat

graph

G

^{, operator}

B

^{, and}

Z ^{n st}

^{play a key role in the}

formulationof theproblem and thedenition of its so-

lutions;see, in parti ular, Eqs. (13)and (18)-(19). The

aim of this se tion is to dene the orresponding alge-

brai framework.

We rst dene related spa es of fun tions (Se t. 3.1).

Thekeypropertyonwhi houranalysis isbasedispre-

sentedinSe t.3.2. Therelatednotionsof losuredier-

en e, CDambiguity(also alled losureambiguity),and

losure matrixare spe ied in Se ts. 3.3 and 3.4. Se -

of theUD-CD and UD-DD relationships. Theanalysis

on erningtheoperator

D

^{introdu edintheappendixB}

ofBlewitt(1989)isthus ompleted. Wenowdrawfreely

fromtheelementarynotionsintrodu edin AppendixA.

3.1 Referen e spa es

Given some graph

G ≡ G(V, E)

^{, with vertex set}

V

^and

edgeset

E

^{(seeSe t.A1),weintrodu esome}fun tionals spa eswhi h playakeyrolein thealgebrai analysisof

theproblem. Inwhatfollows,theGNSS gridasso iated

with

G

^isdenotedby

G

^;seeFig.A1.

3.1.1 Vertex-biasspa e

Let

V b

^{bethespa eof}real-valuedfun tions

α ^def = (α r , α s )

⁽²⁶⁾

taking theirvalues onthe verti es of

G

^with

α r (1) = 0

^.

Thisspa e,whi hisreferredtoasthevertex-biasspa e,

is asso iated with the denition of (virtual) phase bi-

ases

α

^{on the verti es of}

G

^{(other than the referen e}

re eiver). FromEq. (10),

V b ∼ = R ^{n st}

⁽²⁷⁾

Here, the symbol

∼ =

^means `isomorphi to.' Note that

Z ^{n st}

^{is the `integer latti e' of}

V b

^:

V b (Z) ∼ = Z ^{n st}

^{. The}

integerve tor

µ ^def = (µ r , µ s )

^{isapointofthislatti e.}

3.1.2 Edge-delayspa e

A real-valued fun tion

ϑ

^{taking its values on}

G

^{, and}

thereby on

E

^{, anbe regardedasa ve tor of theedge-}

delayspa e

E ∼ = R ^{n e}

⁽²⁸⁾

Thevalues of

ϑ

^on

G

^{are thenregarded asthe ompo-}

nentsof

ϑ

^{in thestandardbasisof}

E

^;

Z ^{n e}

^{isthe`integer}

latti e' of

E

^:

E(Z) ∼ = Z ^{n e}

^{. The} integer-ambiguity ve - tor

N

^{isapointofthislatti e.}

3.1.3 Spanning-tree delay spa e.

Closure-delay spa e

Given somespanning tree

G st

^of

G

^{, grid}

G

^{an be de-}

omposed into twosubgrids:

G st

^and

G c

^{; seeSe t. A2.}

These gridsin lude

n st

^and

n c

^points,respe tively(see Fig.A2):

n c = n e − n st

⁽²⁹⁾

Thefun tions of

E

^{that vanish on}

G c

^{form asubspa e}

of

E

^{denoted by}

E st

^{: the} spanning-tree delay spa e.

Likewise, the fun tions of

E

^{that vanish on}

G st

^form

asubspa eof

E

^denotedby

E c

^{: the} losure-delayspa e;

thisterminologyisjustiedinSe t.3.3. The orrespond-

ingintegerlatti esaredenotedby

E st (Z)

^and

E c (Z)

^,re-

spe tively. AsillustratedinFig.2,theEu lideanspa e

E

istheorthogonalsumof

E st

^and

E c

^{. Clearly,}

dim E st = n st , dim E c = n c

⁽³⁰⁾

Theorthogonal proje tionsof

ϑ

^on

E st

^and

E c

^{are re-}

spe tivelydenoted by

Q st ϑ

^and

Q c ϑ

^.

3.1.4 Edge-biasspa e

Bydenition,thebiasoperator istheoperatorfrom

V b

into

E

^{dened by Eq. (11). The range of}

B

^{, whi h is}

denotedby

E b

^{(seeFig.2), anbereferredtoastheedge-}

biasspa e. Itsfun tionsareof theform

α r (i) − α s (j)

^.

Theoperatorfrom

V b

^into

E st

^{indu edby}

B

^{is denoted}

by

B st

^{. Likewise,theoperator from}

V b

^into

E c

^{indu ed}

by

B

^{isdenoted by}

B c

^.

Thematrixof

B

^{isgenerally expressedin the standard}

bases of

V b

^and

E

^{. Forexample,letussorttheedgesof}

thegraphshowninFig.A1intheorderobtainedviathe

appli ation oftheKruskalalgorithm;see Se t.A2. The

pointsof

G

^{arethenorderedasfollows:}

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

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

Wethenhave

[B][α] =





























0 0 −1 0 0 0

0 0 0 0 −1 0

0 0 0 0 0 −1

1 0 −1 0 0 0

1 0 0 −1 0 0

0 1 0 −1 0 0

1 0 0 0 0 −1

0 1 0 0 −1 0

0 1 0 0 0 −1

















































α r (2)

α r (3)

α s (1)

α s (2)

α s (3)

α s (4)





















The olumnsof

[B]

^{thendenethestandardbasisof}

E b

^.

Clearly,

[B st ] =

















0 0 −1 0 0 0

0 0 0 0 −1 0

0 0 0 0 0 −1

1 0 −1 0 0 0

1 0 0 −1 0 0

0 1 0 −1 0 0

















(31)

and

[B c ] =





1 0 0 0 0 −1

0 1 0 0 −1 0

0 1 0 0 0 −1





⁽³²⁾

The ondition

B st α = 0

^,i.e.,

Bα = 0

^ontheedgesof

G st

^,

implies that

α

^{is onstant on}

V

^;as

α r (1) = 0

^{, this on-}

stantiszero. Thenullspa eof

B st

^{is thereforeredu ed}

to

{0}

^{. As}

Bα = 0

^implies

B st α = 0

^{,thenullspa eof}

B

isalsoredu edto

{0}

^{. Wethus have}

ker B = ker B st = {0}

⁽³³⁾

Asaresult,

B

^{isoffullrank,hen efromEq.(27),}

dim E b = n st

⁽³⁴⁾

Theedge-biasspa e

E b

^{anditsambiguitylatti e}

E b (Z) =

BV b (Z)

^{areisomorphi tothe} vertex-biasspa e

V b

^and

itsintegerlatti e

V b (Z)

^,respe tively;seeSe t.3.1.1.

3.2 Key property

As

ker B st = {0}

^{(Eq.(33)), and}

dim E st = dim V b

^(see

Eqs.(30)and(27)),

B st

^maps

V b

^onto

E st

^;

B st

^isthere-

foreinvertible. As spe iedin this se tion,ouranalysis

derivesfromthisproperty.

Letus on entrateonthevertex-biasfun tion

α ^(ϑ) _st ^def = B ⁻¹ _st Q _st ϑ (α ^(ϑ) ≡ α ^(ϑ) _st )

⁽³⁵⁾

When no onfusion may arise, subs ript st is omitted.

A ordingtoitsdenition(whi hisillustratedinFig.2),

Q st ϑ

^{is the fun tion of}

E st

^{whose valuesare those of}

ϑ

onsubgrid

G st

^.

The valuesof

α ^(ϑ)

^{anbe obtainedfrom those of}

Q st ϑ

in a very simple manner; the orresponding re ursive

pro essis des ribed in Se t.5of Lannesand Teunissen

(2011). The olumn ve torsof

[B st ] ^{− 1}

^{anthus beeas-}

ily obtained. In fa t,

[B st ]

^{is a parti ular unimodu-}

lar 2

matrix whose inverse an be obtained via another

integer-programmingte hnique;seeSe t.A1.4inLannes

and Teunissen (2011). Forexample, theinverse ofma-

trix(31)is

[B st ] ⁻¹ =

















−1 0 0 1 0 0

−1 0 0 1 −1 1

−1 0 0 0 0 0

−1 0 0 1 −1 0

0 −1 0 0 0 0

0 0 −1 0 0 0

















(36)

Letusnow onsiderthefollowingedge-biasfun tion:

ϑ b

def = Bα ^(ϑ)

⁽³⁷⁾

A ordingto Eq. (35), the valuesof

ϑ b

^and

ϑ

^{oin ide}

on

G st

^{. Thefun tion}

ϑ c

^{dened bytherelation}

ϑ c

def = ϑ − ϑ b

⁽³⁸⁾

2

Bydenition,aunimodularmatrixisasquareintegermatrix

withdeterminant

±1

^.

            

 

C /

     

r

Q c ϑ

r ϑ

r

ϑ b = Bα ^(ϑ)

Q st ϑ r

r

ϑ c

E

(n e )

E st

(n st )

E b

(n st )

0 E c

(n c )

Figure 2: Geometri al illustration of Property 1.

In this geometri al representation of the edge-delay

spa e

E ∼ = R ^{n e}

^,

E st

^isthespanning-treedelayspa e.

This spa e is isomorphi to the vertex-bias spa e

V b ∼ = R ^{n st}

^{. Theorthogonal omplementof}

E st

^inthe

Eu lideanspa e

E

^isthe losure-delayspa e

E c

^.The

rangeof thebiasoperator

B

^{,theedge-bias spa e,is}

asubspa eof

E

^denotedby

E _b

^{.Thisspa eisisomor-}

phi to

E st

^{and thereby to}

V b

^{. (The dimensions of}

thesespa esarewrittenwithinparentheses.) Asillus-

tratedhere,

E

^{istheobliquedire tsumof}

E _b

^and

E _c

^.

The losureoperator

C

^{istheobliqueproje tionof}

E

onto

E _c

^along

E _b

^{;forfurtherdetailsseeProperty1 .}

thereforeliesin

E c

^{. Wethushavethefollowingproperty}

(seeFig.2):

Property 1. Anyedge fun tion

ϑ

^of

E

^{anbe de om-}

posed in theform

ϑ = ϑ b + ϑ c

^with

ϑ b

def = Bα ^(ϑ)

^and

ϑ c

in

E c

^{. Foragivenspanning tree, this} de omposition is unique. Asa orollary,

E

^{istheobliquedire tsumof}

E b

and

E c

^:

E = E b + E c

^with

E b ∩ E c = {0}

^.

AsillustratedinFig.2,

ϑ c

^{istheobliqueproje tionof}

ϑ

on

E c

^along

E b

^{. The} orrespondingoperatoristhe` lo- sureoperator'

C

^:

ϑ c = Cϑ

⁽³⁹⁾

Itsnullspa e(i.e.,itskernel) istherangeof

B

^:

ker C = E b

⁽⁴⁰⁾

with

dim E b = n st

^(Eq.(34)).

A ordingtoProperty1,anyfun tion

N

^oftheambigu-

itylatti e

E(Z) ∼ = Z ^{n e}

^{anbede omposedin theform}

N = N b + N c

⁽⁴¹⁾

with

N b

def = Bµ ^{(N )} _st

^{where(fromEq.(35))}

µ ^{(N )} _st ^def = B _st ⁻¹ Q _st N (µ ^{(N )} ≡ µ ^{(N )} _st )

⁽⁴²⁾

As

B st

^isunimodular,

µ ^{(N )}

^isaninteger-valuedfun tion;

N b

def = Bµ ^{(N )}

^and

N c

def = CN

^{are therefore points of the}

integerlatti es

E b (Z) ∼ = Z ^{n st}

^and

E c (Z) ∼ = Z ^{n c}

^{, respe -}

tively. Asaresult,theintegerlatti e

E(Z)

^istheoblique

dire tsumoftheintegerlatti es

E b (Z)

^and

E c (Z)

^:

E(Z) = E b (Z) + E c (Z) E b (Z) ∩ E c (Z) = {0}

⁽⁴³⁾

3.3 Closure delays ( losure dieren es)

and losure ambiguities

A ordingtoEqs.(38)and(37),thequantities

ϑ c (i ℓ , j ℓ )

^,

for

ℓ = 1, . . . , n c

^{, anbe omputedviatheformula}

ϑ c (i ℓ , j ℓ ) = ϑ(i ℓ , j ℓ ) − α ^(ϑ) r (i ℓ ) − α ^(ϑ) _s (j ℓ ) 

(44)

where

α ^(ϑ)

^{isdeterminedviaEq.(35). As lariedinthis}

se tion,thesequantities anbereferredtoasthe` losure

delays'orthe` losuredieren es'of

ϑ

^;the

N c (i ℓ , j ℓ )

^'sare

therefore `CD ambiguities,' also simply alled ` losure

ambiguities.'

IntheexampleofFig.A2,letus onsiderthese ondloop,

i.e.,theloopasso iatedwiththe losurepoint

(i 2 , j 2 ) =

(3, 3)

^{. In}

G

^{,thesu essivepointsofthislooparethefol-}

lowing:

(3, 3)

^,

(3, 2)

^,

(2, 2)

^,

(2, 1)

^,

(1, 1)

^,and

(1, 3)

^{. Sin e}

ϑ b (i, j) = α ^(ϑ) r (i) − α ^(ϑ) s (j)

^{,wethen have,inateles op-}

ingmanner,

ϑ b (3, 3) − ϑ b (3, 2) + ϑ b (2, 2) − ϑ b (2, 1)

+ ϑ b (1, 1) − ϑ b (1, 3) = 0.

Furthermore,as

ϑ c

^vanisheson

G st

^,

ϑ c (3, 3) − ϑ c (3, 2) + ϑ c (2, 2) − ϑ c (2, 1)

+ ϑ c (1, 1) − ϑ c (1, 3) = ϑ c (3, 3)

Sin e

ϑ = ϑ b + ϑ c

^{fromProperty1,itfollowsthat}

ϑ(3, 3) − ϑ(3, 2) + ϑ(2, 2) − ϑ(2, 1)

+ ϑ(1, 1) − ϑ(1, 3) = ϑ c (3, 3)

This expli itlyshowsthat

ϑ c (i 2 , j 2 )

^{anbe regardedas}

the losuredieren eof

ϑ

^{onthese ondloop. Thegen-}

eralizationisstraightforward. IntheexampleofFig.A2,

wethushave





























ϑ c (2, 4) = ϑ(2, 4) − ϑ(2, 1) + ϑ(1, 1) − ϑ(1, 4)

ϑ c (3, 3) = ϑ(3, 3) − ϑ(3, 2) + ϑ(2, 2) − ϑ(2, 1)

+ ϑ(1, 1) − ϑ(1, 3)

ϑ c (3, 4) = ϑ(3, 4) − ϑ(3, 2) + ϑ(2, 2) − ϑ(2, 1)

+ ϑ(1, 1) − ϑ(1, 4)

(45)

More generally, owing to the teles oping stru ture of

their onstru tion, the losure dieren es

ϑ c (i ℓ , j ℓ )

^are

asso iated with loopswhose order is even, and greater

than or equal to

4

^{. In this limit ase, the notion of}

losure dieren e (CD) redu es to that of double dif-

feren e(DD). A ordingtoEq.(44),the

ϑ c (i ℓ , j ℓ )

^{'s an}

howeverbe omputedwithoutknowingtheedgesoftheir

loop. Howtoidentifytheseedges,ifneedbe,isspe ied

in Se t.3.4. Subje tto some ondition,theseCD's an

beexpressedaslinear ombinationsofDD's. Therelated

matteris analysed in Se t.10 ofLannes and Teunissen

3.4 Closure matrix

A ordingtothedenitionsof

B st

^and

B c

(introdu edin Se t. 3.1.4),theve tor

ϑ b

def = Bα ^(ϑ)

^anbeorthogonally de omposedintheform

ϑ b = B st α ^(ϑ) + B c α ^(ϑ) = Q st ϑ + B c α ^(ϑ)

Likewise,

ϑ = Q st ϑ + Q c ϑ

where

Q c ϑ

^{isthe orthogonalproje tionof}

ϑ

^on

E c

^{; see}

Fig.2. Itthenfollowsfrom Eq.(35)that

ϑ c = ϑ − ϑ b = Q c ϑ − B c α ^(ϑ) = Q c ϑ − B _c B _st ⁻¹ Q _st ϑ

Denotingby

[C]

^thematrixof

C

^{expressed inthe stan-}

dardbasesof

E

^and

E c

^{,wethushave,fromEq. (39),}

[C][ϑ] = −[B c ][B st ] ^{− 1} [Q st ϑ] + [Q c ϑ].

The olumnve torsof

[C]

orrespondingtothespanning- treeedges (onwhi h

Q c ϑ

^{vanishes)are therefore those}

of

−[B _c ][B st ] ⁻¹

^{. It is also lear that the olumn ve -}

torsof

[C]

orrespondingtothe losureedges(onwhi h

Q st ϑ

^{vanishes) are those of the identity matrix on}

E c

^.

Consequently,with regardto theorthogonaldire tsum

E st ⊕ E c

^,

[C] =  −[B c ][B st ] ⁻¹ [I c,c ] 

(46)

IntheexampleofFig.A2,wethushave,fromEqs.(31),

(36),and(32),withthesameedgeordering,

[B c ][B st ] ⁻¹ =

2

4

1 0 0 0 0 −1

0 1 0 0 −1 0

0 1 0 0 0 −1

3

5

×

2

6

6

6

6

6

6

4

−1 0 0 1 0 0

−1 0 0 1 −1 1

−1 0 0 0 0 0

−1 0 0 1 −1 0

0 −1 0 0 0 0

0 0 −1 0 0 0

3

7

7

7

7

7

7

5

Asaresult,

[C] =

" _{1 0 −1 −1 0 0 1 0 0}

1 −1 0 −1 1 −1 0 1 0

1 0 −1 −1 1 −1 0 0 1

#

Applied to

[ϑ]

^{, this matrix of ourse yields Eq.(45).}

More generally, the edges of a ` losure loop' are iden-

tied via the nonzero entries of the orresponding row

of

[C]

^{. In fa t,this is the moste ient way of identi-}

fying the loops in question. Note howeverthat in the

CAAmethodpresentedthroughSe ts.4to6,thea tion

3.5 On some generalized inverse

of the UD-CD relationship

The losureoperator

C

^{,whi his anoblique}proje tion, is not of full rank. The simplest way of removing its

rankdefe tisto introdu etheoperator

C ⋄

^from

E

^into

E st × E c

C ⋄ ϑ ^def = (Q st ϑ, Cϑ)

⁽⁴⁷⁾

A ordingtoProperty1,

C ⋄

^isinvertible;this anbeim- mediatelyunderstoodfrom Fig.2forexample;

C ⋄ ⁻¹

^an

thenberegarded assomegeneralizedinverse of

C

^{. We}

nowspe ifythis point, expli itly,in matrix terms. The

orrespondingdevelopmentisaimedatanalysingtheap-

proa hes of Blewitt (1989)and Collins et al. (2010)in

anelementarymanner;seeSe ts. 3.6and7further on.

Inthestandardbasesof

E = E st ⊕ E c

^and

E st × E c

^,the

matrixof

C ⋄

^{anbewrittenintheform(seeEq. (46))}

[C ⋄ ] ^def = " [Q st ]

[C]

#

=

"

[I st,st ] [0 st,c ]

−[B c ][B st ] ⁻¹ [I c,c ]

#

(48)

Itisreadilyveriedthat

[C ⋄ ] ⁻¹ =

"

[I st,st ] [0 st,c ]

[B c ][B st ] ⁻¹ [I c,c ]

#

(49)

Given somepoint

N ˘ st

arbitrarilyxed in

E st (Z)

^{, letus}

now onsidertheambiguitypoint

N ˘

^of

E(Z)

^{dened by}

therelation

[ ˘ N ] ^def = [C ⋄ ] ^{− 1} " [ ˘ N st ]

[N c ]

#

(N c

def = CN )

⁽⁵⁰⁾

Inthefollowingproperty,

E _b ^{(N )} (Z) ^def = N + E b (Z)

⁽⁵¹⁾

is the`ane latti e' passingthrough

N

^{and parallel to}

theintegerlatti e

E b (Z)

^{ofthe edge-biasspa e}

E b

^{; see}

Se t.3.1.4andFig.3.

Property 2. Theambiguitypoint

N ˘

^{isthepointofthe}

ane latti e

E _b ^{(N )} (Z)

^{whose proje tionon}

E st

^{is equal}

to

N ˘ st

^{. Morepre isely,}

N = N ˘ c + Bµ ^{( ˘ N st )}

^{. As a orol-}

lary,inthespe ial asewhere

N ˘ st

^issetequalto

0

^,

N ˘

^is

nothingelsethan

N c

^.

Forreasonsof larityandbrevity,theproofislefttothe

reader. Notethat this property analso beunderstood

withintheS-systemframework;seeforinstan ethetable

giveninSe t.1.6ofTeunissen(1984).

3.6 On the Blewittgeneralized inverse

of the UD-DD relationship

Wenowapply theresultsofthepreviousse tionto the

UD-DD relationship, and thus make the link with the

approa hofBlewitt(1989).

# # # # # # # # # # #

\ \

\ \

\ \

\ \

\ \

# # # # # # # # ##

˘ s

N st

E c

s N

6

Bµ ^{( ˘ N st )}

N ˘

s

s N c

E

E st

E b E _b ^{(N )} (Z)

s

0

Figure 3: Geometri al illustration of Property 2.

In this symboli representation of the edge-delay

spa e

E

^,

E _b ^(N) (Z)

^{is the ane latti e passing}

through

N

^{andparalleltotheintegerlatti e}

E b (Z)

oftheedge-biasspa e

E _b

^{(here,for larity,thever-}

ti alaxis);

N ˘

^{istheUDambiguityobtainedviathe}

relationship (50) inwhi h

N ˘ st

^is arbitrarily xed in

E st (Z)

^,and

N c

^{istheCDambiguitypointof}

N

(the losure ambiguity of

N

^{). In the important}

spe ial asewhere

N ˘ _st

^issetequalto

0

^,

N ˘

^{redu es}

to

N c

^.

A ording to Eq. (68) ofLannes and Teunissen (2011),

the maximumnumberofindependentDD's islessthan

or equal to

n c

^:

n ^m _d ≤ n _c

^{. For larity, let us now set}

n _d := n ^m _d

^{. Intheimportantspe ial asewhere}

n c = n d

⁽⁵²⁾

the information ontained in the DD data is equiva-

lent to that ontained in the losure data. Let us then

denote by

D d,e

^{the operator providing a maximum set}

of

n d

^{DD's. Bydenition,}

D d,e

^{is an operator from}

E

into

R ^{n d}

^{, i.e. then,}

R ^{n c}

^{. By sorting the edges of}

G

^as

spe iedin Se t. 3.1.4, thematrixof

D d,e

^hasthenthe

followingblo kstru ture:

[D d,e ] =  [D d,st ] [D d,c ] 

(53)

Here,matrix

[D d,e ]

^{isexpressedinthestandardbasesof}

E = E st ⊕ E c

^and

R ^{n d}

^{. The olumnsof}

[D d,st ]

^and

[D d,c ]

therefore orrespondto the edgesof

G st

^{and to the lo-}

sure edges, respe tively. Provided that Condition (52)

is satised,

[D d,c ]

^is invertible; moreover, the entries of

[D d,c ] ⁻¹

^{are then equal to}

±1

^or

0

^{; see Lannes and}

Teunissen2011.

Likefor

C

^{(seeEq.(48)),wethenintrodu etheoperator}

[D ⋄ ] ^def = " [Q st ]

[D d,e ]

#

(54)

As

N = N b + CN

^{fromProperty1,and}

[D d,e ][N b ] = 0

^,