集美大学航海学2教案：天文航海-2

航海学(2),天文航海,内容,球面三角与船位误差理论基础,测天定位天测罗经差,天文航海的优点使用的设备是六分仪，简单可靠；不发射任何电波，隐蔽性好；观测目标是天体，不受任何人控制；可能是大洋航行中唯一的全球导航系统。,球面三角与船位误差理论基础,第一章球面三角第二章内插法第二章船位误差基础第二章等精度观测平差,第一章球面三角,球面三角,主要研究球面上由三个大圆弧相交围成的球面三角形及其性质、解算等问题。1.1球面几何1.2球面三角形1.3球面三角形的边角函数关系,1.1球面几何,1.1.1球、球面在空间与一定点等距离的点的轨迹称为球面。包围在球面中的实体称为球,这一定点称为球心。球心和球面上任意一点间的距离称为球半径R。过球心与球面相交的直线段称为球直径。,同球的半径和直径都相等。同理,半径或直径相等的球全等。所以,球面又可定义为半圆周绕它的直径旋转一周的旋转面。,1.1.2球面上的圆任意一平面和球面相截的截痕是圆。当平面通过球心时，所截成的圆称为大圆，它的一段圆周叫大圆弧。截面不通过球心的圆称为小圆，它的一段圆周叫小圆弧。,1.1.3大圆的性质,1大圆的圆心与球心重合。2大圆的直径等于球直径，半径等于球半径。3同球或等球上的大圆的大小相等。4大圆等分球面和球体。5同球上的两个大圆平面一定相交，交线是它们的直径，并且两大圆互相平分。,6过球面上不在同一直径两端上的两个点，能作且仅能作一个大圆，却能作无数个小圆。若在同一直径两端上的两个点，则能作无数个大圆而不能作小圆。7小于180的大圆弧（劣弧）是球面上两点间的最短球面距离。因此，两点间的球面距离应用大圆弧度量。,1.1.4轴、极、极距、极线,垂直与任意圆面的球直径称为该圆（大圆或小圆）的轴。轴的两个端点称为极。垂直于同一轴可有数个平行圆，其中只有一个通过球心的是大圆，其余的都是小圆。从极到圆(大圆或小圆)弧上任一点沿大圆弧的球面距离叫极距,又叫球面半径。同一个圆的极距或球面半径都相等。极距为90的大圆弧又称为该极的极线。,大圆弧是它的极的极线。反之，极线必定是大圆弧。,1.1.5球面角及其度量,球面上两大圆弧相交构成的角称为球面角，其交点叫球面角的顶点，两大圆弧称为球面角的边。,球面角的三种度量方法：1切于顶点大圆弧的切线夹角CPD；2顶点的极线被其两边大圆弧所截的弧长AB；3大圆弧AB所对的球心角AOB。,1.2球面三角形,1.2.1定义在球面上由三个大圆弧围成的三角形称为球面三角形。围成三角形的大圆弧称为球面三角形的边。由大圆弧相交所成的球面角称为球面三角形的角。,三个角A、B、C和三个边a、b、c合称为球面三角形的六要素。航海上讨论的球面三角形的六要素均大于0，而小于180，又称其为欧拉球面三角形。,1.2.2球面三角形分类球面三角形分为直角、直边、等腰、等边、初等和任意三角形。1球面直角三角形和球面直边三角形至少有一个角为90的三角形称为球面直角三角形。至少有一个边为90的三角形称为球面直边三角形。,2球面等腰三角形和球面等边三角形有两边或两角相等的三角形称为球面等腰三角形。若三边或三角都相等的三角形称为球面等边三角形。3球面初等三角形三个边相对其球半径甚小的三角形称为球面小三角形。只有一个角及其对边相对球半径甚小的三角形称为球面窄三角形。两者统称为球面初等三角形。4球面任意三角形不具备上述特殊条件的球面三角形,1.2.3极线球面三角形球面三角形ABC三个顶点的极线所构成的球面三角形ABC称为原球面三角形ABC的极线球面三角形。,1原三角形与其极线三角形是互为极线三角形。原球面三角形ABC的三个边，也就是其极线球面三角形ABC三个顶点的极线。换句话说，若画极线三角形ABC的极线三角形，则所得到的就是原球面三角形ABC。所以，它们之间的关系是相互的。2原球面三角形的边与其极线三角形对应角互补,1.3球面三角形的边角函数关系,1.3.1任意球面三角形1.3.2球面直角和直边三角形1.3.3球面初等三角形,余弦公式,边的余弦公式记忆口诀：一边的余弦等于其它两边余弦的乘积，加上这两边正弦及其夹角余弦的乘积。,边的余弦公式应用：已知两边及其夹角求对边；已知三边求三角。,Bytransposingtheformula,wegetaforminwhichitmaybeused,giventhreesides,tofindanyangle.,角的余弦公式记忆口诀：一角的余弦等于其它两角余弦的乘积冠以负号加上这两角正弦及其夹边余弦的乘积。cosA-cosBcosCsinBsinCcosa,角的余弦公式应用：已知两角及其夹边求对角；已知三角求三边。,正弦公式记忆口诀：边的正弦与其对角的正弦成比例。,正弦公式应用：已知两角及其一对边，求另一边；已知两边及其一对角，求另一角。,Itsbigdisadvantageistheambiguityabouttheactualvalueofthepartfound,since,Inshort,whenwehavetakenouttheanti-sineandobtained(say)42,thequestionarises-istheanswer42or138?,Thisdifficultyariseswheneveranangleisfoundthroughitssine.Therewillsometimesbetwosolutions,sometimesone,andsometimesnosolutionatall.Disregardingthethirdrathertheoreticalpossibility,someprogresscanbemadeonthefirsttwobyrememberingthat,inanytriangle,A-Banda-bmustbeofthesamesign.,Forexample,atrianglePAB,inwhichb=2621,B=5222,A=10444.Tofinda,wehaveSin(a)=0.542039a=3249.4ora=14710.6AB,ab.Bothvaluesforasatisfythisrequirement;thusbotharesolutionsofthedataasgiven.,余切公式（四联公式）记忆口诀：外边余切内边正弦乘积等于外角余切内角正弦乘积加上内边内角余弦积。,余切公式用于：在球面三角形中，已知相连的四个要素中的三个，求另一个要素。即已知两边及夹角求相连的角或已知两角及夹边求相连的边。,解球面任意三角形,根据球面三角形已知要素求解其余要素的方法称为解球面三角形。,使用计算器解算球面三角时注意事项计算器有三种角度单位供选择即：度DEG、弧度RAD、公制度GRAD。球面三角中，角或边都是以六十进制的度、分及分的小数给出的，利用计算器计算，在DEG状态下，一定要将角或边转换成以十进制的度为单位输入。,Examples,inatrianglePAB,P=6630,a=4700,b=6700,findA.Weshallusethefour-partsformula.A=5232.6,1.3.2球面直角和直边三角形,球面直角三角形公式有一个或一个以上的角为直角的球面三角形称为球面直角三角形。设球面直角三角形ABC中，C90。因为sin901，cos900，可由球面任意三角形的基本公式，导出相应的球面直角三角形十个公式：,sinasinAsinccoscctgActgBsinactgBtgbcosAsinBcosasinbsinBsinccosActgctgbsinbctgAtgacosBsinAcosbcosccosacosbcosBctgctga,球面直角三角形公式的纳比尔记忆法则在球面三角形ABC中，C90。先画“大”字图形，大字上部竖线代表直角C，相邻两侧为夹直角的两边a和b，大字下面三个空格依次填入相对应元素边或角的余数（a边对90A，b边对90B，C角对90c)。,任一要素的正弦等于相邻两要素正切乘积或相对两要素余弦乘积,NapiersRulesareusuallystatedasfollows:Sinemiddlepart=productoftangentsofadjacentpartsSinemiddlepart=productofcosinesofoppositepartsThereisonefinalpointtobenoted,namely,thatinwritingdowntheequationsbymeansoftheserules,thetwopartsnexttotherightanglearewrittendownastheyare,theothers(awayfromtherightangle)arewrittendownascomplements.,(2)NapiersRulesapplytoquadrantaltriangleswithoneimportantmodification,whichmustbenoted:Inaquadrantaltriangle,ifbothadjacentsorbothoppositesarebothsidesorbothangles,putinaminussign(i.e.putaminussigninfrontoftheproduct).,InthetriangleABC,letABbethequadrant.ThenAandBarenexttothequadrant,theotherthreepartsarewrittendownascomplements.Forexample,givenbandC,findAByNapiersRules:,1.5.3PropertiesofRightangledandQuadrantalTriangles,(1)Inanyrightangledorquadrantaltriangle,anangleanditsoppositesidearealwaysofthesameaffection.(“ofthesameaffection”,i.e.bothgreaterthan90,orbothlessthan90)ThisfollowsfromtheformulaSincethesineis+veforallvaluesupto180,itfollowsthatctg(a)andctg(A)mustbeofthesamesign,i.e.aandAmustbeofthesameaffection.,Properties-cont,(2)Inanyrightangledtriangle,wemusthaveeither,allthreesideslessthan90or,twosidesgreaterandonelessthan90.ThisfollowsfromtheformulaSincecos(c)musthavethesamesignastheproductcos(a)cos(b).Forthistohappenwemusthaveeitherallthreeconsines+veoronlyone+ve.,1.6SmallSphericalTriangle&NarrowSphericalTriangle,Asphericaltrianglewiththreesidesmuchsmallerthantheradiusofthesphereiscalledasmallsphericaltriangle.Inthiscasewewilltreatthissmallsphericaltriangleasaplantriangle.Asphericaltrianglewithonesidemuchsmallerthantheradiusofthesphereiscalledanarrowsphericaltriangle.InanarrowsphericaltriangleABCwithsideabesmallerthantheothertwosides,knownc,Banda,find(c-b)andA.,NarrowSphericalTriangle,Chapter2Interpolation,2.1Introduction,2.2SingleInterpolation,2.3DoubleInterpolation,2.1Introduction,Ifonequantityvarieswithchangingvaluesofasecondquantity,andthemathematicalrelationshipofthetwoisknown,acurvecanbedrawntorepresentthevaluesofonecorrespondingtovariousvaluesoftheother.,2.1Introduction-cont1,Tofindthevalueofeitherquantitycorrespondingtoagivenvalueoftheother,onefindsthatpointonthecurvedefinedbythegivenvaluesandreadstheansweronthescalerelatingtotheotherquantity.Thisassumesthatforeachvalueofonequantity,thereisonlyonevalueoftheotherquantity.,2.1Introduction-cont2,Informationofthiskindcanalsobetabulated.Eachentryrepresentsonepointonthecurve.ThefindingofavaluebetweentabulatedentriesiscalledInterpolation.,2.1Introduction-cont3,Thus,theNauticalAlmanactablesvaluesofdeclinationoftheSunforeachhourofGreenwichMeanTime.Thefindingofdeclinationforatimebetweentwowholehoursrequiresinterpolation.Sincethereisonlyoneargument,SingleInterpolationisinvolved.,2.1Introduction-cont4,Thereisonetablethatgivesthedistancetraveledinvarioustimesatcertainspeeds.Inthistable,therearetwoenteringarguments.Ifbothgivenvaluesarebetweentabulatedvalues,DoubleInterpolationisneeded.,2.1Introduction-cont5,InPub.229,azimuthanglevarieswithachangeinanyofthethreevariableslatitude,declination,andlocalhourangle.Withintermediatevaluesofallthree,TripleInterpolationisneeded.,2.1Introduction-cont6,Interpolationcansometimesbeavoided.,Herearetwokindsofmethod.Trytothinkofthem?-,2.1Introduction-cont7,Atablehavingasingleenteringargumentcanbearrangedasacriticaltable.Interpolationisavoidedthroughdividingtheargumentintointervalssochosenthatsuccessiveintervalscorrespondstosuccessivevaluesoftherequiredquantitytherespondent.Foranyvalueoftheargumentwithintheseintervals,therespondentcanbeextractedfromthetablewithoutinterpolation.,Iftheheightofeyeisbetween10.2and10.6meters,thedipwillbe5.7.Iftheheightofeyeisequalto10.6meters,thedipwillbe5.7.,Exampleofacriticaltable,2.1Introduction-cont8,Thelowerandupperlimits(criticalvalues)oftheargumentcorrespondtohalf-wayvaluesoftherespondentand,byconvention,arechosensothatwhentheargumentisequaltooneofthecriticalvalues,therespondentcorrespondingtotheproceeding(upper)intervalistobeused.,2.1Introduction-cont9,Anotherwayofavoidinginterpolationwouldbetoincludeeverypossibleenteringargument.,2.2SingleInterpolation,2.2.1ProportionalInterpolation2.2.2InterpolationbyRateofChange,2.2.1ProportionalInterpolation,Theaccuratedeterminationofintermediatevaluesrequiresknowledgeofthenatureofthechangebetweentabulatedvalues.Thesimplestrelationshipislinear,thechangeinthetabulatedvaluebeingdirectlyproportionaltothechangeintheenteringargument.,2.2.1ProportionalInterpolation.1,EntriesFunctionx1y1x2y2,xy=?,2.2.1ProportionalInterpolation.2,Whenthecurverepresentingthevaluesofatableisastraightline.TheprocessoffindingintermediatevaluesinthemannerdescribedaboveiscalledLinearInterpolation.Iftabulatedvaluesofsuchalineareexact(notapproximations),theinterpolationcanbecarriedtoanydegreeofprecisionwithoutscarifyingaccuracy.,2.2.1ProportionalInterpolation.3,Manyofthetablesofnavigationarenotlinear.Tobestrictlyaccurateininterpolatinginsuchatable,oneshouldconsiderthecurvatureoftheline.,2.2.1ProportionalInterpolation.4,However,inmostnavigationaltablesthepointonthecurveselectedfortabulationaresufficientlyclosethattheportionofthecurvebetweenentriescanbeconsideredasastraightlinewithoutintroducingasignificanterror.Thisissimilartoconsideringthelineofpositionfromacelestialobservationasapartofthecircleofequalaltitude.,2.2.2InterpolationbyRateofChange,Ifweknowthederivativeofafunction,thetangentlinecanbetreatedasthecurve.Thenwherexisnearertox0thantoanyotherentry.,Forexample,givenatableasfollows,obtain2.73.,x0=3,y0=27,y0=27,y=27+27*(2.7-3)=18.9Thetruevalueis19.7,Ifwetakex0=2,y0=8,y0=12,y=8+12*2.7-2)=16.4Theerrorisbiggerthanthatabove.,2.3DoubleInterpolation,Inadouble-entrytable,itmaybenecessarytointerpolateforeachenteringargument.Ifoneenteringargumentisanexacttabulatedvalue,thefunctioncanbefoundbysingleinterpolation.However,ifneitherenteringargumentisatabulatedvalue,doubleinterpolationisneeded.,Combinedmethod:,Selectatabulated“base”value,preferablythatnearestthegiventabulatedenteringarguments.,Next,findthecorrectiontobeapplied,withitssign,forsingleinterpolationofthisbasevaluebothhorizontallyandvertically.,Finally,addthesetwocorrectionsalgebraicallytothebasevalue.,Forexample,givenanextracttablefromAmplitudes,latitudeis45.7anddeclinationis21.8,findtheamplitude.,Thebasevalueis32.6,fordeclination22(21.8isnearer22than21.5)andlatitude46.Thecorrectionfordeclinationis=-0.3.Thecorrectionforlatitudeis=-0.2.Theinterpolatedvalueisthen32.6-0.3-0.2=32.1.,Chapter3NavigationalErrors,3.1ErrorofObservation3.2RandomErroranditsCriteria3.3ProbabilityDistributionofRandomError3.4ArithmeticMeanandLeastSquareMethod3.5ErrorPropagation,3.1ErrorofObservation,3.1.1Introduction3.1.2Definitions,3.1.1Introduction,Ascommonlypracticed,navigationisnotanexactscience.Anumberofapproximations,whichwouldbeunacceptableincarefulscientificwork,areusedbythenavigator,becausegreateraccuracymaynotbeconsistentwiththerequirementsortimeavailable,orbecausethereisnotalternative.,3.1.1Introductioncont1,Thus,whenthenavigatorinterpolateinsightreductionorlatticetables,heisassumesalinear(constantrate)changebetweentabulatedvalues.Whenhemeasuresdistancebyradar,ordepthbyechosounder,heassumesthattheradioorsoundwavehasconstantspeedunderallconditions.,3.1.1Introductioncont2,Theseareonlyafewoftheapproximationscommonlyappliedbyanavigator.Therearesomanythatthereisanaturaltendencyforsomeofthemtocancelothers.Thus,underfavorableconditions,apositionatsea,determinedfromcelestialobservationbyanexperiencedobserver,shouldseldombeinerrorbymorethan2miles.However,ifthevarioussmallerrorsinaparticularobservationallhavethesamesign,theerrormightbeseveraltimesamount,withoutanymistakehavingbeenmadebythenavigator.,3.1.1Introductioncont3,Greateraccuracycouldbeattained,butataprice.Thenavigatorisapracticalindividual.Inthecourseofordinarynavigation,hecouldratherspend10minutesdeterminingapositionhavingaprobableerrorofplusorminus2miles,thantospendseveralhourslearningwherehewastoanaccuracyofafewmeters.Butifhecandeterminearecentorpresentpositiontogreateraccuracy,thedecreaseinerrorisattractivetohim.,3.1.1Introductioncont4,Anunderstandingofthekindsoferrorsinvolvedinnavigation,andoftheelementaryprinciplesofprobability,shouldbeofassistancetoanavigatorininterpretinghisresults.,3.1.2Definitions,Erroristhedifferencebetweenaspecificvalueandthecorrectorstandardvalue.Asusedhere,itdoesnotincludemistakes,butisrelatedtolackofperfection.Mistakeisablunder,suchasanincorrectreadingofaninstrument,thetakingofawrongvaluefromatable,ortheplottingofareciprocalbearing.,3.1.2Definitionscont1,Standardissomethingestablishedbycustom,agreement,orauthorityasabasisforcomparison.Frequently,astandardissochosenthatitservesasamodelwhichapproximatesameanoraveragecondition.However,thedistinctionbetweenthestandardvalueandtheactualvalueatanytimeshouldnotbeforgotten.,3.1.2Definitionscont2,Accuracyisthedegreeofconformancewiththecorrectvalue,whileprecisionisthedegreeofrefinementofavalue.Thus,analtitudedeterminedbymarinesextantmightbestatedtothenearest0.1,andyetbeaccurateonlytothenearest1ifthehorizonisindistinct.,3.1.2Definitionscont3,Systematicerrorsarethosewhichfollowsomelawbywhichtheycanbepredicted.Theaccuracywithwhichasystematicerrorcanbepredicteddependsupontheaccuracywhichthegoverninglawisunderstood.Anerrorwhichcanbepredictedcanbeeliminated,orcompensationcanbemadeforit.Thesimplestformofsystematicerrorisoneofunchangingmagnitudeandsign.Thisiscalledaconstanterror.,3.2RandomErroranditsCriteria,Randomerrorsarechanceerrors,unpredictableinmagnitudeorsign.Theyaregovernedbythelawsofprobability.Ifthealtitudeofacelestialbodyisobserved,thereadingmaybe(1)toogreat,(2)correct,or(3)toosmall.Ifanumberofobservationsaremade,andthereisnosystematicerror,theprobabilityofapositiveerrorisexactlyequaltotheprobabilityofanegativeerror.,Thisdoesnotmeanthateverysecondobservationhavinganerrorwillbetoogreat.However,thegreaterthenumberofobservations,thegreateristheprobabilitythatthepercentageofpositiveerrorswillequalthepercentageofnegativeones,andthattheirmagnitudeswillcorrespond.,Supposethat500observationsaremade,withtheresultsshownintable3.1.Acloseapproximationoftheplotoftheseerrorsisshowninfigure3.1.Theplothasbeenmodifiedslightlytoconstitutethenormalcurveofrandomerrors,whichisthesameastheactualcurveexceptthatthenormalcurveapproacheszeroastheerrorincrease,whiletheactualcurvereacheszeroat+10and10.,Figure3.1,1.8,Figure3.1,1.8,Figure3.1,1.8,Figure3.1,Thehighofthecurveatanypointrepresentsthepercentageofobservationsthatcanbeexpectedtohavetheerrorindicatedatthatpoint.Theprobabilityofanysimilarobservationhavinganygivenerroristheproportionofthenumberofobservationshavingthiserrortothetotalnumberofobservations,orthepercentageexpectedasadecimal.Thus,theprobabilityofanobservationhavinganerrorof3is40/500=8%.,Iftheerrorunderthecurverepresents100percentoftheobservations,halfthearearepresents50percentoftheobservations.Thevalueoftheerroratthelimitsisoftencalledthe“50percenterror”,orprobableerror,meaningthat50percentoftheobservationscanbeexpectedtohavelesserror,and50percentgreatererror.,Similarly,thelimitswhichcontainthecentral95percentoftheareadenotethe95percenterror.Thepercentageoferrorisfoundmathematically.Foranormalcurve,eacherrorissquared,thesumofthesquaresisdividedbyonelessthanthenumberofobservations,andthesquarerootofthequotientisdetermined.,Thisvalueiscalledthestandarddeviationorstandarderror(,theGreeklettersigma).Intheillustration,thestandarddeviationis,Thestandarddeviationis68.27%error.Theprobabilityoftheoccurrenceofanerroroforlessthanaspecificmagnitudemaybeapproximatelydeterminedbythefollowingrelationship(withtheanswerfortheillustrationgiven):50%error=2/3=2(approx.)68%error=1=3(approx.)95.4%error=2=6(approx.)99.7%error=3=10(approx.),3.3ProbabilityDistributionofRandomError,Thedistributiondensityanddistributionfunctionofrandomerrorareasfollows:,3.4ArithmeticMeanandLeastSquareMethod,Letlibetheobservations,arithmeticmeanofthesevalueswouldbethemostprobablevalue:,LeastSquareMethodLetbethemostprobablevalue,theresidualerrorsoftheobservationswouldbe:i=1,2,nisobtainedundertheconditionthatistheminimum.,3.5ErrorPropagation,Givenanerrorfunction:,Example:TB=CB+Var+Dev,known,Then=0.6,Chapter4TreatmentofObservationswithEqualAccuracy,4.1TreatmentofDirectObservationswithEqualAccuracy4.2TreatmentofIndirectObservationswithEqualAccuracy4.3TheAccuracyoftheMostProbablePosition,4.1DirectObservationswithEqualAccuracy,(1)Themostprobablevalue,(2)Thestandarderrorofsingleobservation,4.1DirectObservationswithEqualAccuracycont1,(3)Thestandarderrorofthemostprobablevalue,4.1DirectObservationswithEqualAccuracycont2,(4)ThefinalresultWith95%uncertainty,thefinalresultis:,4.2IndirectObservationswithEqualAccuracy,(1)Observedequationsi=1,2,n(2)Residualerrorequationsi=1,2,n,4.2IndirectObservationswithEqualAccuracycont1,(3)Normalequations,4.2IndirectObservationswithEqualAccuracycont2,(4)Themostprobablevalues,4.3TheAccuracyoftheMostProbablePos