工程测量47-测量误差理论(2)评定精度指标和误差传播定律课件

测量误差理论

TheoryofErrorsinMeasurementsComparisonofErrorDistributionCurves2

-σ10-σ2+σ1+σ2IIICurveIexhibitssteeperslopeandmoreconcentrateddistributionoferrors,indicatinghigherprecision.Accuracy(准确度)vs.Precision(精确度)Accuracyreferstohowcloseameasurementistothetruevalue.Precisionreferstohowclosemeasurementsofthesamequantityaretoeachother,eveniftheyarenotclosetothetruevalue.Example:Abull’seye(thecenterofthetarget)representsthetruevalue.Theshotsonthetargetrepresentsetsofmeasurements.3NeitherPreciseNorAccurateThisisarandom-likepattern.Theshotsarenotclusteredtogether,thusthemeasurementsarenotveryprecise.Meanwhile,theshotsarenotclosetothebull’seye,sothemeasurementsarenotveryaccurate.4Precise,NotAccurate5Alltheshotsareclosetogether,thustheyareprecise.Buttheshotsdidnothittheintendedcenterofthetarget,theyarenotaccurate.PreciseandAccurate6Theshotsaretightlyclustered,sotheyareprecise.Also,theshotsareallclosetothebull’seye(truevalue),sotheyareaccurate.7NeitherPreciseNorAccuratePrecise,NotAccuratePreciseandAccurate8IndicatorsforPrecisionAssessment(评定精度的指标)IntroductionAlthougherrordistributioncurveorhistogramcanbeusedtoidentifytheprecisionofmeasurements,theyareveryinconvenientinpractice.Weusesomeindicators

toassessthequalityofmeasurementsandresultsderivedfrommeasurements.Theindicatorsaremainlyasfollows:①Variance&Standarddeviation(方差、中误差)②Allowableerror(容许误差)③Relativeerror(相对误差)9①Variance&StandardDeviationVariance

(方差)isanindicatorreflectingthedegreeofdispersionforasetofmeasurements.ItsmathematicalexpressionisStandarddeviation(SD,中误差)isdefinedasthesquarerootofthevariance.10①Variance&SD(Cont’d)Inengineeringsurveying,SDisgenerallytakenasanindicatorforprecisionassessment.Inpractice,SDcanonlybeestimatedusingthelimitednumberofmeasurements.Suchestimateisreferredtoas"StandardError

(SE)",whichisexpressedas11ExampleforSDEstimationTherearetwogroupsoftriangularmisclosure(trueerrorofthesumofthethreeinternalangles),obtainedunderdifferentmeasurementconditions.12TureError(″)Δ1Δ2Δ3Δ4Δ5Δ6A+5+2-2-10-3B+6-7-1-4+5+2FurtherInterpretationofσ

Itcanbeproventhat±σ

correspondstotwoabscissavalues(横坐标值)atthetwoinflexionpointsoftheerrordistributioncurve.13Δf(Δ)III②AllowableErrorDefinition:theabsolutemagnitudeofrandomerrorsiswithinsomelimit.Suchlimitvalueisdefinedasallowableerror.Howmuchistheprobabilityforagivenlimitorrangewithinwhichthetrueerrorsareassumedtooccur?14probability:

68.3%probability:

95.4%probability:

99.7%②AllowableError(Cont’d)Astheerrorslargerthan3σoccurhardlyprobably,3σistakenasthelimitofrandomerrors.

Δ容

=3m

Inengineeringsurveying,Δ容

=2mismoreoftenusedformorestringentrequirementstothemeasurements.Iftheerrorofameasurementislargerthantheallowableerror,suchmeasurementisproblematicandmustbere-measured.15③RelativeErrorTrueerror,standardderivationandallowableerrorarecalled“absoluteerror”,becausetheyhaveadefiniteunit.However,theabsoluteerrorcannotcompletelyreflectthequalityforsomeofthemeasurements.Insuchcase,therelativeerrorispreferred.Relativeerrorcanbeexpressedas1617ErrorPropagationLaw(误差传播定律)IntroductionInsurveying,somequantitiescannotbeobserveddirectly,buttheycanbecomputedusingdirectlyobservedquantitiesbasedonmathematicalrelationships.Forinstance:Adifferenceinheightistheresultofthedifferencebetweentwodirectobservationsofthelevelrod.Ahorizontaldistancecanbeobtainedfromdirectobservationsofaslopedistanceandaverticalangle.Duetotheexistenceoferrorsinthedirectobservations,thusthecomputedquantitiesareaffectedandcontainerrors.18误差传播定律：阐述观测值中误差与观测值函数中误差之间关系的定律。以四种常见函数来讨论误差传播的情况191.倍数函数设有函数式中k为常数，x为直接观测值，其中误差为mx。则观测值函数Z的中误差mZ

：202.和差函数设有函数式中x、y为独立观测值，它们的中误差分别为mx和my。则观测值函数Z的中误差mZ

：213.线性函数设有线性函数式中x1,

x2,…,xn为独立观测值，k1,

k2,…,kn为常数。则观测值函数Z的中误差mZ

：224.一般函数设有一般函数式中x1,

x2,…,xn为独立观测值，其中误差为mi

(i=1,2,…,n)。则观测值函数Z的中误差mZ

为：DerivationofErrorPropagationLaw23

SupposethatZisthefunctionofindependentobservationsSuppose

Δx1,

Δx2,…,Δxnarethetrueerrorsofx1,

x2,…,xnandΔZisthetrueerrorofZ.AccordingtoTaylorSeries,theequationabovecanbedeployedas24

SupposetheindependentobservationsareobservedforNtimes25