Encyclopedia of Geodesy

286 Pages·19.893 MB·English

by  Erik Grafarend (eds.)

A
Attitude Determination Attitudedetermination,whichistheprocessofdetermin-
ing the attitude parameters, is accomplished by employing
GabrieleGiorgi oneormultipleattitudesensors.Theseeithersensechanges
InstituteforCommunicationsandNavigation,Technische in the object’s rotational dynamic (e.g., gyroscopes) or
Universita¨tM€unchen,Munich,Germany exploit directional measurement of an external signal or
forcefield(e.g.,startrackers,magnetometers).Thesesensors
provide the raw measurements necessary to extract an esti-
Definition mateoftheobject’sattitude.
Attitude determination. The process of determining the ori-
entationofanobjectwithrespecttoareferenceframe. AttitudeRepresentation
Thespatialorientationisformallydescribedbythetransfor-
Introduction mation matrix R between a body-fixed orthogonal frame
F andareferenceorthogonalframeF :
body ref
Thecharacterizationofanobject’spositioninspace,beita
vehicleorameasuringinstrument,requirestheknowledgeof F ¼RF
body ref
bothitslocationandattitude(viz.orientation)withrespectto
areferenceframe. Since the transformation is applied to orthogonal frames,
Attitude determination is a prerequisite for a number of both the lengths of the canonical vectors that identify the
applicationsthatnecessitateknowingtheorientationinorder main axes and their mutual angles must be preserved. The
to execute a task. These include autonomously navigating necessaryandsufficientconditiontomaintaintheseproper-
space,air,sea,andlandvehicles,orpointinginstrumentsat ties is to employ an orthogonal rotation matrix as transfor-
desired targets, such in spaceborne and airborne altimetry, mation operator: RTR¼I . An additional constraint must
3
satellite laser ranging, radar interferometry, and optical alsobeimposed:thedeterminantoftherotationmatrixmust
tracking. bepositive.Thisisnecessarytoavoidphysicallyinadmissi-
The attitude of an object is formally identified by the ble rotations that cause axes reflections rather than pure
relativeorientationbetweenaframeofcoordinatesattached rotations.
to the object and a reference frame (cf. Figure 1). The Theorthogonalityconstraintreducesthedegreesoffree-
orientationisdescribedthroughasetofreal-valuedvariables domassociatedtoanarbitraryrotationtothree,whichisthe
knownasattitudeparameters. Differentlyfromthedescrip- minimum number of attitude parameters necessary to
tion of a spatial position, for which Euclidean coordinates describe the orientation. Attitude parameterizations
usually suffice, multiple choices of attitude parameters are employing a minimal set of attitude parameters are the
available,withdifferentanalyticalandnumericalproperties Euler angles (Eulero, 1776), matrix exponentiation (Golub
(Shuster, 1993). Examples of attitude parameters are the andVanLoan,1996),andtheRodriguesvectors(Rodrigues,
Euler angles (such as the heading, elevation, and bank 1840). These minimal set representations cannot avoid the
angles)orthequaternions. presenceofsingularities,foritisnotpossibletoparameterize
#SpringerInternationalPublishingSwitzerland2015
E.W.Grafarend(ed.),EncyclopediaofGeodesy,
DOI10.1007/978-3-319-02370-0_2-1
2 AttitudeDetermination
AttitudeDetermination,
Figure1 Theattitudeof
anobjectisidentifiedbythe
mutualorientationbetween
anorthogonalreference
frameandabody-attached
orthogonalframe.
certainparticularorientationswithoutincurringinnonfinite systems (MEMS) measure the Coriolis force generating
values or ambiguous couplings of the associated attitude from vibrating elements subject to rotations of the plane of
parameters (Stuelpnagel, 1964). An example of a singular vibration (Ghodssi and Lin, 2011); special magnetometers
configurationisthe“gimbal-lock”whenusingEulerangles, measure the variation of the rotation axis of a spinning
in which two parameters can no longer be decoupled, thus superconductor developing a magnetic moment whose axis
degenerating into a “locked” state that requires aligns with the axis of rotation (London-moment effect)
reinitialization. (Fairbanketal.,1988;Everittetal.,2007).
A larger number of parameters (four) are used in the The complementary class of contingent attitude sensors
quaternion (Hamilton, 1844–1850), the Euler axis-angle uses an external source of information to infer the spatial
(Eulero,1776)andtheCayley-Klein(Klein,1875)parame- orientation.Thisclassofsensorsincludes:startrackersand
terizations, which avoid singular configurations but may horizon sensors, which extract the object’s orientation with
increasethecomputationalloadrequiredfortheirhandling. respecttoagroupofselectedstarsorhorizonlinesofclose
celestialbodies;magnetometers,whichsensethedirectionof
anexternalmagneticfield;gyrocompasses,whichdetectthe
AttitudeSensors gravity field vector; global navigation satellite system
(GNSS) antenna arrays, which exploit the interferometry
Attitude sensors divide into two categories: self-contained principle between the received satellite signals at closely
systems that sense the variation of the rotational dynamics spacedantennas.
withbuilt-indevices,andcontingentsystemsthatuseexter- Theseattitudesensorsmakeavailabledirection-of-arrival
nalsourcestoinfertheobject’sorientation. measurements from an external source whose position is
Inertial measurement units are attitude sensors of the known. Multiple such measurements enable estimating the
formertype.Asthebodyundergoesarotationalmotion,an object’sorientationatanygivenmoment,providedtheexter-
inertial sensor reacts proportionally to the amplitude of the nalsourceofinformationisavailable.
motion.Differenttypologiesofinertialsensorsareavailable:
gimbaled gyros exploit the gyroscopic effect, maintaining
thesameangularmomentumduringrotationalmaneuvering AttitudeEstimationandCapabilities
(Wertz,1978);ringlasergyros(RLG)andfiber-opticgyros
(FOG)arebasedontheinterferometryprinciple,providinga The information collected by either independent or contin-
measureoftheperturbationstriggeredbyrotationalacceler- gent attitude sensors may not give a direct reading of the
ations (Anderson et al., 1994); micro electro-mechanical orientation (e.g., the sensed rotational variations need to be
AttitudeDetermination 3
integrated over time), and it needs further elaboration to Cross-References
obtainausableestimateoftheattitude.
Whenemployingself-containedsystems,theorientationis ▶CoordinateTransformation–Geodesy
determinedthroughdead-reckoning:aknowninitialorienta- ▶GPS,ReferenceSystems
tionispropagatedusingthesensedrotationalvariations.Due ▶HighPrecisionGNSS
to the accumulation of measurement errors over time, self- ▶INS–Geodesy
contained systems are evaluated both in terms of precision ▶MultisensorSystems
(e.g.,angularrandomwalk,ARW)andstability,withthelatter
givingaconservativevalueoftheexpectedbiasovertime.The
capabilities of marketed self-contained systems vary from a ReferencesandReading
1/10 (cid:2) 1deg/√hrARWand5 (cid:2) 30deg/hrstabilityforlower-
gradeMEMS,upto0.0001deg/√hrARWand0.0001deg/hr Anderson,R.,Bilger,H.R.,andStedman,G.E.,1994.Sagnaceffect:a
centuryofEarth-rotatedinterferometers.AmericanJournalofPhys-
stability for higher-grade RLGs and FOGs (Barbour, 2010).
ics,62(11),975–985.
Purpose-built devices are capable ofhigher performance; an
Barbour, N. M., 2010. Inertial navigation sensors. DTIC Document.
exampleistheLondonmomentgyroemployedintheGravity Cambridge,MA:CharlesStarkDraperLab.
Probe-B mission, capable of 3 (cid:3) 10(cid:4)8 deg sensitivity and Diebel,J.,2006.Representingattitude:Eulerangles,unitquaternions,
10(cid:4)11deg/hrstability(Everittetal.,2007). androtationvectors.Technicalreport,StanfordUniversity.
Eulero, L., 1776. Formulae generales pro translatione quacunque
Most contingent systems exploit the measured angular
corporumrigidorum(Generalformulasforthetranslationofarbi-
direction of the line of sight to an external source, whose trary rigid bodies). Novi Commentarii Academiae Scientiarum
positionisknown(e.g.,thelocationofanavigationsatellite Petropolitanae,20,189–207.
Everitt, F., Parkinson, B., and Kahn, B., 2007. The gravity probe
oraclusterofstars;amapofthemagneticfieldoftheEarth).
B experiment. Post flight analysis – Final report. Project report:
Oneormultiplesuchmeasurementsarethenusedtoestimate
NASA,StanfordUniversityandLockheedMartin.
the orthogonal matrix that rotates a body-attached frame to Fairbank,J.D.,Michelson,P.F.,andEveritt,C.W.,1988.NearZero:New
thereferenceframeofchoice(anotablesolutionisinWahba, FrontiersofPhysics,1stedn.NewYork:W.H.Freeman,p.959pp.
Ghodssi,R.,andLin,P.,2011.MEMSMaterialsandProcessesHand-
1965). The capabilities of dependent systems are propor-
book.Berlin:Springer.1188pp.
tional to the quality of the raw directional measurement.
Giorgi, G., Teunissen, P. J. G., Verhagen, S., and Buist, P. J., 2010.
Star trackers are among the most precise sensors, capable TestinganewmultivariateGNSScarrierphaseattitudedetermina-
of 1/10000 (cid:2) 1/1000 deg angular resolutions sensitivity tion method for remote sensing platforms. Advances in Space
Research,46(2),118–129.
(Liebe, 1995; Tapley and Bettadpur, 2004). The precision
Golub,G.H.,andVanLoan,F.C.,1996.MatrixComputations,3rdedn.
of attitude sensors based on the interferometry principle,
Baltimore,MD:TheJohnsHopkinsUniversityPress.
such as GNSS antenna arrays, also depend on the spatial Goodall,C.,Carmichael,S.,andScannell,B.,2013.Thebattlebetween
separation between the elements used to obtain the direc- MEMSandFOGsforprecisionguidance.EDNMagazine.
Hamilton,W.R.,1844–1850.OnQuaternions;oronanewsystemof
tional measurement, with wider separations yielding higher
imaginariesinAlgebra.InDavid,R.W.(ed.),TheLondon,Edin-
angular resolutions. GNSS antenna arrays are capable of
burghandDublinPhilosophicalMagazineandJournalofScience
providing angular estimates with a precision of reaching (3rdSeries).xxv-xxxvi,2000.Taylor&Francis,London
0.1degperunitlength(onemeter)(Giorgietal.,2010). Hofmann-Wellenhof,B.,Legat,K.,andWieser,M.,2003.Navigation.
PrinciplesofPositioningandGuidance.NewYork:Springer.420pp.
Attitude determination and control systems often make
Klein,F.,1875.UeberBina¨reformenmitlinearentransformationenin
use of integrated solutions, in which two or more attitude SichSelbst.MathematischeAnnalen,9,183–208.
sensor typologies are used. Typical examples are inertial Liebe,C.C.,1995.Startrackersforattitudedetermination.IEEEAES
sensors coupled with multiple GNSS antennas, with the SystemsMagazine,10(6),10–16.
Rodrigues, O., 1840. Des lois geometriques qui regissent les d´
former providing higher angular measurement precision
eplacementsd’unsystemesolidedansl’espace,etlavariationdes
andthe latter providingthe biasstabilitynecessarytocom-
coordonnees provenant de ses d´eplacements consideres
pensate for the drift of the inertial sensor (Hofmann- independamment des causes qui peuvent les produire. Journal de
Wellenhofetal.,2003). MathematiquesPuresetAppliquees,5,380–440.
Shuster,M.D.,1993.Asurveyofattituderepresentations.TheJournal
oftheAstronauticalSciences,41(4),439–517.
Stuelpnagel,J.,1964.Ontheparameterizationofthethree-dimensional
Summary rotationgroup.SIAMReview,6(4),422–430.
Tapley, B. D., and Bettadpur, S., 2004. The gravity recovery and
climateexperiment:missionoverviewandearlyresults.Geophysi-
Attitudedeterminationistheprocessofassessingtheorien-
calResearchLetters,31,L09607.
tation of an object in space. Attitude sensors provide the
Wahba, G., 1965. Problem 65-1: a least squares estimate of satellite
informationnecessarytoestimatethesetofattitudeparam- attitude.SIAMReview,7(3),409.
eters that describe the object’s orientation with respect to a Wertz, J. R., 1978. Spacecraft Attitude Determination and Control.
Dordrecht:Kluwer.858pp.
givenreferenceframe.
B
Best Integer Prediction in Mixed Models analogue of Kolmogorov–Wiener prediction (Grafarend,
1976).
P.J.G.Teunissen Theabovemethodsofpredictioncanbecastintheframe-
GNSSResearchCentre,DepartmentofSpatialSciences, workofeitherleast-squarespredictionorofbestlinearunbi-
CurtinUniversityofTechnology,Perth,Australia ased prediction (Goldberger, 1962; Moritz, 1972; Hardy,
DepartmentofGeoscienceandRemoteSensing,Delft 1977;Harville,1990;Christensen,1991;Koch,1999;Stein,
UniversityofTechnology,Delft,TheNetherlands 1999). A generalization of the above predictors is possible
when the trend parameters are permitted to be of the mixed
real-integer type (Teunissen, 2007). Such is for instance
Definitions needed to handle the models of carrier phase–based global
andregionalnavigationsatellitesystems(GNSS/RNSS),very
Best predictor: Predictor having the smallest mean square long baseline interferometry(VLBI),or interferometric syn-
predictionerror(best)ofallpredictorswithinacertainclass. thetic aperture radar (InSAR) (see, e.g., Strang and Borre,
Mixed integer model: Model of observation equations 1997; Teunissen and Kleusberg, 1998; Hanssen, 2001; and
having both real-valued and integer-valued unknown Hofmann-Wellenhof etal., 2007). Examples of such GNSS/
parameters. RNSSs are GPS (USA), Glonass (Russia), BeiDou (China),
Galileo(EU),QZSS(Japan),andIRNSS(India).
Introduction
BestLinearPrediction
Thepredictionofspatiallyand/ortemporallyvaryingvariates
findsitsapplicationinvariousspatial andEarth science dis- We speak of prediction if a function P:ℝm7!ℝr of an
ciplines. In physical geodesy, where it is used with a trend- observablerandomvectoryℝmisusedtoguesstheoutcome
signal-noisemodeltopredictfunctionalsofthegravityfield, of another random, but unobservable, vector pℝr
it is known as least-squares collocation (Krarup, 1969; (an underscore is used to denote random variables). The
Moritz,1978;SansoandTscherning,2003).Thetrend-signal- relationbetweenyandpisassumedgivenas
noisemodelalsoformsthebasisofpredictioningeostatistics,
(cid:1) (cid:3) (cid:1) (cid:3) (cid:1) (cid:3) (cid:1) (cid:3) (cid:1) (cid:3)
whereoptimallinearpredictioniscalledKriging,namedafter y A e y Q Q
¼ xþ , D ¼ yy yp (1)
Krige (1951) and further developed by Matheron (1970). p Ap ep p Qpy Qpp
When the trend is unknown, it is referred to as universal
Kriging,andwhenthetrendisabsentorsettozero,itiscalled with known coefficient matrices A and A , unknown
p
simple Kriging (Herzfeld, 1992; Wackernagel, 1995; Olea, nonrandom vector xℝn, and zero-mean random vectors e
1999). Although collocation and Kriging have been devel- ande withknowndispersion.D(.)denotesthemathematical
p
opedforspatiallyvaryingvariates,theyarecloselyconnected dispersion operator. If P(y) is a predictor of p , then
with the fundamental work of Kolmogorov (1941) and e ¼p(cid:2)PðyÞisitspredictionerror.Thepredictorisunbiased
p
Wiener (1949) on the interpolation, extrapolation, and ifE(e )=0,inwhichE(.)denotesthemathematicalexpec-
p
smoothing of stationary time series. In the absence of a tation operator. The mean squared error (MSE) E (||e ||2) is
p
trend, collocation and simple Kriging become the spatial
#SpringerInternationalPublishingSwitzerland2015
E.W.Grafarend(ed.),EncyclopediaofGeodesy,
DOI10.1007/978-3-319-02370-0_3-1
2 BestIntegerPredictioninMixedModels
often used to judge the performance of a predictor, e.g., QBLUP ¼ QBLP þA Q AT
(Sorenson,1970;AndersonandMoore,1979;Scharf,1991; ^ep^ep ^ep^ep pjy ^x^x pjy (5)
QBLP ¼ Q (cid:2)Q Q(cid:2)1Q
Koch, 1999). A predictor is c(cid:4)alled “(cid:5)best” if it (cid:4)succee(cid:5)ds in ^ep^ep pp py yy yp
minimizingtheMSE.Since E jje jj2 ¼traceE e eT ,the
p p p with Q the variance matrix of the BLUE of x and
^x^x
MSE is equal to the trace of the error variance matrix of an Apjy ¼Ap(cid:2)QpyQ(cid:2)yy1A.Since Q^BeL^eUP (cid:5)Q^BeL^eP; theBLUPpre-
p p p p
unbiased predictor. Best unbiased predictors are therefore
dictionerrorisnevermoreprecisethanthatoftheBLP.Thisis
sometimes also called ‘minimum error variance unbiased’
duetoadditionaluncertaintythatentersbyhavingtoestimate
predictors. Since one can minimize the MSE over different
xaswell.
classes of functions, there are different predictors that are
‘best’.
Best Predictor (BP): If no restrictions are placed on the
BestIntegerEquivariantPrediction
class of predictors, the best predictor of p is equal to the
conditionalmean, The BLUP is not the “best” predictor in case the parameter
(cid:4) (cid:5) vectorxofEq.1isintegervalued.Tocoverthiscase,another
j
p^ ¼E p y (2) classofpredictorsneedstobeconsidered,onethattakesthe
BP
integerness of x into account. Assume that y in Eq. 1 is
ForcomputingtheBP,informationontheprobabilitydensity perturbedbyAz,withzℤn.Thenxgetsperturbedbyzand
function(PDF)ofyandpisneeded. p getsperturbedbyApz.Whendesigningapredictorof p,it
Best linear predictor (BLP): Within the class of linear seemsreasonable torequest that any such predictor, being a
predictors(LPs),thebestpredictortakestheform function of y, behaves in the same way with regard to such
integer perturbations. Such predictors were introduced in
(cid:4) (cid:5)
^p ¼pþQ Q(cid:2)1 y(cid:2)y (3) Teunissen (2007) as integer equivariant (IE) predictors. As
py yy
BLP thisclassencompassestheclassoflinearunbiasedpredictors,
(cid:4) (cid:5) (cid:4) (cid:5)
theMSEoftheBLUPwillnever besmaller thanthatofthe
withp ¼E p andy ¼E p .IncontrasttotheBP,theBLP
bestIEP.
onlyneedsknowledgeaboutthefirsttwomoments,themean Integer Rounding Prediction (IRP): Let xˆ denote the
and the variance. The BP and BLP become identical in the
integer vector that is obtained by rounding all the entries of
Gaussiancase,i.e.,wheny andp arenormallydistributed. ^x totheirnearestinteger.Then
Best linear unbiased prediction (BLUP): The need to
know the means in Eq. 3 is not always practical. This need
pˆ = A ⎡xˆ⎦+Q Q−1(y−A⎡xˆ⎦) ð6Þ
can be circumvented by using their known relation (cf. 1) IRP p py yy
when minimizingthe MSE overthe class oflinearunbiased
predictors(LUPs).ThisgivestheBLUPas isanexampleofanIEP.Itfollowsbyreplacing^xinEq.4with
(cid:4) (cid:5) theintegervector xˆ.OtherIEPscanbeobtainedinasimilar
^p ¼Ap^x þQpyQ(cid:2)yy1 y(cid:2)A^x (4) way.Ifonereplaces xˆ byanyotherintegerestimator,e.g.,the
BLUP
integer bootstrapped estimator or the integer least-squares
estimator (Teunissen, 1999), then again an IEP of p is
inwhich^x ¼argminxℝnjjy(cid:2)Axjj2Qyyisthebestlinearunbi- obtained.
ased estimator (BL(cid:6)U(cid:7)E) of(cid:6)x(cid:7)ðthe weighted squared norm is Best integer equivariant predictor (BIEP): None of the
definedasjj:jj2M ¼ : TM(cid:2)1 : Þ.ThustheBLUPEq.4follows IEPs that follow from replacing ^x in Eq. 4 by an integer
fromtheBLPEq.3byreplacingthemeansp andy bytheir estimatorarebestintheMSEsense.Incasepandyarejoint
BLUEs. Gaussian,thebestIEPisgivenas
Since the above three classes of predictors are related as
(cid:4) (cid:5)
LarUePor(cid:3)derLePd(cid:3)asP,theminimumMSEsoftheirbestpredictors ^pBIEP ¼Ap^xBIEþQpyQ(cid:2)yy1 y(cid:2)A^xBIE (7)
MSEðBPÞ(cid:4)MSEðBLPÞ(cid:4)MSEðBLUPÞ with
Sinceallthreebestpredictorsareunbiased,thesameordering
holds true for their error variance matrices. The BLP and
BLUPerrorvariancematricesaregivenas
BestIntegerPredictioninMixedModels 3
1 1
0.5 0.5
0 0
–0.5 –0.5
–1 –1
–1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1
BestIntegerPredictioninMixedModels,Fig.1 (Left)Two-dimensionalscatterplotsof^x(green)andcorresponding^x (blue).(Right)Theblue
BIE
arrowsshowthemappingfrom^xto^x (cf.8).TheILSpull-inregionsareshownashexagons
BIE
(cid:8) (cid:9)
1 forthereal-valuedcase xℝn,butnotiftheparametersare
X exp 2jj^x (cid:2)zjj2Qx^^x integervalued,xℤn.Lettheleast-squaresobjectivefunction
^xBIE ¼ zX (cid:8)1 (cid:9) (8) forprediction,basedonthelinearmodelEq.1,begivenas
zZn exp jj^x (cid:2)zjj2
Q
2 ^x^x (cid:1) (cid:3) (cid:1) (cid:3)(cid:1) (cid:3)
zZn (cid:4) (cid:5) y(cid:2)Ax T W W y(cid:2)Ax
F x,p ¼ p(cid:2)A x Wyy Wyp p(cid:2)A x (10)
A two-dimensional example of the workings of this p py pp p
BIEestimatorisshowninFig.1.Theaboveisbasedonthe
all-integercase,xℤn.Inmanyapplications,like,e.g.,those Least-squares prediction (LSP): With the weighted(cid:4)leas(cid:5)t-
squares predictor of p defined as ^p ¼argmin F x,p ,
ofGNSSandInSAR,apartoftheunknownparameterswill LSP xℝn,Pℝr
theLSPtakestheform
be integer valued, while the other part will be real
valued. For this mixed integer/real parameter case, with (cid:4) (cid:5)
(cid:6) (cid:7)
x¼ xT1,xT2 T, x1  ℤq, x2  ℤn(cid:2)q, the class of mixed ^pLSP ¼Ap^xLS(cid:2)W(cid:2)pp1Wpy y(cid:2)A^xLS (11)
integer equivariant predictors (MIEP) is characterized by
(cid:6) (cid:7)
PðyþA1zþA2aÞ¼G y þAp1zþAp2a,8yℝm,zℤq,aℝn(cid:2)q with ^xLS ¼argminxℝnjjy(cid:2)Axjj2M, M¼W(cid:2)yy1jp, and
.Thebestpredictorwithinthisclassisgivenas Wyyjp ¼Wyy(cid:2)WypW(cid:2)pp1Wpy. If the weight matrix in Eq. 10
(cid:4) (cid:5) ischosenastheinverseofthevariancematrixinEq.1,then
^pBMIEP ¼Ap^xBMIEþQpyQ(cid:2)yy1 y(cid:2)A^xBMIE (9) Wyyjp ¼Q(cid:2)yy1 and W(cid:2)pp1Wpy ¼(cid:2)QpyQ(cid:2)yy1,andtheLSPEq.11
h i reducestotheBLUPofEq.4.
T
with ^x ¼ ^xT ,^xT , ^x ¼^x (cid:2)Q Q(cid:2)1 Integer least-squares prediction (ILSP): With the
(cid:4) BMIE(cid:5) 1BIE 2BIEP 2BIEP 2 ^x2^x1 ^x1^x1 weighted integer least-squares pred(cid:4)ictor(cid:5)of p defined as
^x (cid:2)^x , and ^x given by Eq. 8, with ^x, z, and Q
1 1BIE 1BIE ^x^x the minimizer ^p ¼argmin F x,p , the ILSP takes
replacedby^x1,z1,andQ^x1^x1,respectively. theform ILSP xℤn,Pℝr
(cid:4) (cid:5)
WeightedLeast-SquaresPrediction p^ILSP ¼Ap^xILS(cid:2)W(cid:2)pp1Wpy y(cid:2)A^xILS (12)
There is a close connection between best linear unbiased with ^xILS ¼argminxZnjjy(cid:2)Axjj2M, M¼W(cid:2)yy1jp and
estimators and weighted least-squares estimators. The Wyyjp ¼Wyy(cid:2)WypW(cid:2)pp1Wpy. The ILSP is a member of the
weighted least-squares estimatorof xbecomes theBLUE of class of IEPs, just like the LSP is a member of the class of
x, if the weight matrix is chosen as the inverse variance LUPs.However,unliketheLSP,whichbecomesidenticalto
matrix, Wyy ¼Q(cid:2)yy1. This equivalence extends to prediction the BLUP if the weight matrix is set equal to the inverse
4 BestIntegerPredictioninMixedModels
0.75 0.75
0.25 0.25
–0.25 –0.25
–0.75 –0.75
–0.75 –0.25 0.25 0.75 –0.75 –0.25 0.25 0.75
Best Integer Prediction in Mixed Models, Fig. 2 (Left) theILSpull-inregion,whilethescatterplotof^x isconcentratedin
BIE
Two-dimensional scatter plots of ^x (green) and corresponding ^x onlyafewpoints.(Right)Thebluearrowsshowthemappingfrom^xto
BIE
(blue).AsthePDFof ^x ispeaked,thescatterplotof ^x residesinside ^x
BIE
variance matrix, the ILSP does not become identical to the Cross-References
BIEPinthiscase.TheILSPwillthenstillhaveapoorerMSE
performancethantheBIEP. ▶GNSSAmbiguityResolutionandValidation
Depending on the peakedness of the PDF of ^x, the ILSP ▶GNSSMeteorology
andBLUPbecomelimitsoftheBIEP.Iftheweightmatrixis ▶GPSLeveling
chosenequaltotheinversevariancematrix,then ▶HighPrecisionGNSS
▶PrecisePointPositioning(PPP)
lim ^p ¼^p and lim ^p ¼^p (13)
s7!1 BIEP BLUP s7!0 BIEP ILSP
whereQ^x^x ¼s2G^x^x.Thedifferencebetween^pBIEPand^pBLUP ReferencesandReading
getssmaller,thesmallertheintegergridsizegetsinrelationto
Anderson,B.D.O.,andMoore,J.B.,1979.OptimalFiltering.Engle-
thesizeandextentofthePDFof^x.Similarly,thedifference
woodCliffs,NJ:Prentice-hall,Vol.11.
between ^p and ^p gets smaller, the more peaked the Christensen,R.,1991.LinearModelsforMultivariate,TimeSeries,and
BIEP ILSP
PDFof^x becomesinrelationtotheintegergridsize.Thisis SpatialData.NewYork:Springer.
Goldberger,A.S.,1962.Bestlinearunbiasedpredictioninthegeneral-
illustratedinthetwo-dimensionalexampleofFig.2.
ized linear regression model. Journal of the American Statistical
Association,57(298),369–375.
Grafarend, E., 1976. Geodetic applications of stochastic processes.
Summary PhysicsoftheEarthandPlanetaryInteriors,12,151–179.
Hanssen, R. F., 2001. Radar Interferometry: Data Interpretation and
ErrorAnalysis.SanDicgo:Springer,Vol.2.
Best prediction tailored for mixed integer models is needed
Hardy, R. L., 1977. Least squares prediction. Photogrammetric Engi-
for carrier phase–based sensors such as used in global and neeringandRemoteSensing,43(4),475–492.
regionalnavigationsatellitesystems,verylongbaselineinter- Harville, D. A., 1990. BLUP (best linear unbiased prediction) and
beyond.InAdvancesinStatisticalMethodsforGeneticImprovement
ferometry, or interferometric synthetic aperture radar. The
ofLivestock.Berlin/NewYork:Springer,pp.239–276.
class of predictors that exploits the mixed integer property
Herzfeld, U. C., 1992. Least-squares collocation, geophysical inverse
oftheirmodelsistheclassofintegerequivariantpredictors. theory and geostatistics: a bird’s eye view. Geophysical Journal
Thebestintegerequivariantpredictor(BIEP)hasthesmallest International,111(2),237–249.
Hofmann-Wellenhof, B., Lichtenegger, H., and Wasle, E., 2007.
meansquaredpredictionerrorwithinthisclassandthiserror
GNSS–GlobalNavigationSatelliteSystems:GPS,GLONASS,Gali-
isneverlargerthanthatofthebestlinearunbiasedpredictor
leo,andMore.Berlin:Springer.
(BLUP). Koch, K. R., 1999. Parameter Estimation and Hypothesis Testing in
LinearModels.Berlin:Springer.
Kolmogorov, A., 1941. Interpolation and extrapolation of stationary
randomsequences.IzvestiiaAkademiiNaukSSSR,SeriiaMatemati-
cheskiiaPubl,5,3–14.
Krarup, T., 1969. A Contribution to the Mathematical Foundation of
PhysicalGeodesy.Copenhagen:DanishGeodætiskInstitut,Vol.44.
BestIntegerPredictioninMixedModels 5
Krige, D., 1951. A statistical approach to some basic mine valuation Sorenson, H. W., 1970. Least-squares estimation: from Gauss to
problemsonthewitwatersrand.JournaloftheChemical,Metallur- Kalman.IEEESpectrum,7(7),63–68.
gicalandMiningSocietyofSouthAfrica,52,119–139. Stein, M. L., 1999. Interpolation of Spatial Data: Some Theory for
Matheron, G., 1970. The Theory of Regionalized Variables and Its Kriging.NewYork:Springer.
Applications.Fontainebleau:EcoledesMinesdeParis.Cahiersdu Strang, G., and Borre, K., 1997. Linear Algebra, Geodesy, and GPS.
CentredeMorphologieMathematique.212pp. Wellesley:WellesleyCambridgePress.
Moritz,H.1972.AdvancedLeast-SquaresMethods,Vol.175.Depart- Teunissen, P.J. G., 1999. An optimality property of theinteger least-
mentofGeodeticScience,OhioStateUniversityColumbus. squaresestimator.JournalofGeodesy,73,587–593.
Moritz, H., 1978. Least-squares collocation. Reviews of Geophysics, Teunissen,P.J.G.,2007.Bestpredictioninlinearmodelswithmixed
16(3),421–430. integer/realunknowns:theoryandapplication.JournalofGeodesy,
Olea, R. A., 1999. Geostatistics for Engineers and Earth Scientists. 81(12),759–780.
Boston:Kluwer. Teunissen,P.J.G.,andKleusberg,A.(eds.),1998.GPSforGeodesy,
Sanso,F.,andTscherning,C.C.,2003.Fastsphericalcollocation:theory 2ndedn.Berlin:Springer.
andexamples.JournalofGeodesy,77(1–2),101–112. Wackernagel, H., 1995. Multivariate Geostatistics. Berlin/Heidelberg/
Scharf, L. L., 1991. Statistical Signal Processing. Reading, MA: NewYork:Springer.
Addison-Wesley,Vol.98. Wiener,N.,1949.Extrapolation,Interpolation,andSmoothingofSta-
tionaryTimeSeries.Cambridge:MITPress.
G
GNSS Ambiguity Resolution and Validation Mixed-IntegerModel
SandraVerhagen Themixed-integermodelforGNSScanbewrittenas
DepartmentofGeoscienceandRemoteSensing,Delft
(cid:1) (cid:3)
UniversityofTechnology,Delft,TheNetherlands y(cid:1)N AaþBb,Q , aℤn,bℝp (1)
yy
whereyisthem-vectorwithGNSScodeandphaseobserv-
Definition ables, which is assumed to follow the normal distribution
denoted by (cid:1)Nð(cid:3)Þ, with mean AaþBb and variance-
GNSS ambiguity resolution. The process of estimating the covariance matrix Q . Furthermore, a is the n-vector with
yy
integerparametersofamixed-integermodel,basedontheir integer ambiguities and b the p-vector with remaining
real-valued estimates and associated variance-covariance unknownreal-valuedparameters,suchasthepositioncoor-
matrix. dinates, atmospheric delay parameters, and clock
GNSSambiguityvalidation.Applyinganacceptancetest parameters.
to decide whether or not the estimated integer ambiguities Themixed-integermodel(1)canbesolvedinthefollow-
arereliableenough. ingthreesteps:
1. The float solution together with its variance-covariance
Introduction matrixisobtainedfromstandardleast-squaresparameter
estimation:
The GNSS carrier-phase observations are ultra-precise but
(cid:4) (cid:5) (cid:6)(cid:4) (cid:5) (cid:4) (cid:5)(cid:7)
athmebseigsuoo-ucaslbleydainnteugnekrnaomwbnignuuimtibeseraroefrienstoeglveerdctyocltehse.irOcnocre- ab^^ (cid:1)N ab , QQab^^aa^^ QQab^^bb^^ (2)
rectintegervalues,thecarrier-phaseobservationsstarttoact
aspseudoranges with twoorders ofmagnitude betterpreci- In this step, the integer nature of the ambiguities is
sionthanthecodeobservations.Intheearly1990s,thetheory discarded.
forresolvingtheintegerambiguitieshasbeenlargelydevel- 2. An integer mapping I :ℝn7!ℤn is applied to the float
oped,butresearchhasbeencarriedouteversinceonseveral ambiguities:
aspects, such as the integer validation, computational effi-
ciency,andpartialambiguityresolution.Anextensiveover- a^¼ Iða^Þ (3)
viewofthetheorycanbefoundinTeunissen(2010).
suchthata^ℤn.
In addition, usually an acceptance test is applied,
which should prevent that unreliable integer solutions
arepropagated.
3. Iftheintegersolutionisaccepted,thefloatsolutionofthe
^
remainingparametersbwillbeadjusted:
#SpringerInternationalPublishingSwitzerland2015
E.W.Grafarend(ed.),EncyclopediaofGeodesy,
DOI10.1007/978-3-319-02370-0_6-1
2 GNSSAmbiguityResolutionandValidation
b^¼b^(cid:4)Qb^a^Qa(cid:4)^a^1ða^(cid:4)a^Þ (4) theprecisionofthefloatsolutioningrey.Thepositionerrors
forthe7%ofthesolutionscorrespondingtowronginteger
If the probabilitythat a^ is correct is sufficientlyhigh, solutionstendtobeofthesameorderofmagnitudeasforthe
theprecisionofb^willbeinlinewithhighprecisionofthe floatsolutionorareevenlarger.Thisdemonstratesontheone
phasedata. hand that ambiguity resolution is the key to the very high
precision and at the same time the need for appropriate
Figure 1 shows a scatter plot of the horizontal float and acceptance testing in order to prevent incorrect integer
fixed position errors based on 10,000 simulated samples of solutions.
the float solution. In 93 % ofthe cases, the integer solution
wascorrect,andthecorrespondingfixedsolutionsareshown
IntegerEstimation
ingreen.Thehighprecisionisclearlyvisible,ascomparedto
The second step of solving the mixed-integer model is to
2
apply an integer mapping, followed by an acceptance test.
The integer map is a many-to-one map, since many real-
1.5 valued vectors will be mapped to the same integer. This
meansthatasubsetofℝn,calledpull-inregion,isassigned
1 to each integer. All vectors in a subset are mapped to the
corresponding integer. The pull-in regions are translational
0.5 invariant over the integers and cover the whole space ℝn
withoutgapsandoverlaps.
0 Three two-dimensional examples of pull-in regions are
shown in Fig. 2, corresponding to integer rounding, integer
bootstrapping,andintegerleast-squares.Thesearethecom-
−0.5
monlyusedintegerestimatorsforGNSSapplications.Inthe
figure,alsosimulatedfloatsolutionsareshown.Asanexam-
−1
ple,allfloatsolutionsingreeninthiscasearemappedtothe
integer[0 0]T.
−1.5
Integerroundingisthemostsimpleintegerestimator:all
entries of a^ are simply rounded to the nearest integer. The
−2
correlationbetweentheambiguitiesisthusignored.
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
Integer bootstrapping is a generalization of integer
GNSSAmbiguityResolutionandValidation,Fig.1 Scatterplotof rounding. In fact, it is a sequential conditional rounding.
horizontalpositionerrorsforsimulatedfloatsolutions(greydots)and One starts with rounding the most precise float ambiguity,
correspondingfixedsolutions.Inthiscase,93%ofthesolutionswere then calculates the float solution of the remaining
correctlyfixed(greendots),and7%werewronglyfixed(reddots)
1.5 1.5 1.5
1 1 1
0.5 0.5 0.5
c]
y 0 0 0
c
[2
a −0.5 −0.5 −0.5
−1 −1 −1
−1.5 −1.5 −1.5
−1 0 1 −1 0 1 −1 0 1
a [cyc] a [cyc] a [cyc]
1 1 1
GNSSAmbiguityResolutionandValidation,Fig.2 Pull-inregionsforintegerrounding(left),integerbootstrapping(center),andintegerleast-
squares(right).Simulatedfloatsolutionsareshown,wheregreenindicatesacorrectintegersolutionandred,anincorrectintegersolution