⑦ Time and Frequency (revised)

2015/4/141AtmosphericAbsorptionAntennaPointingLossTimeandFrequencyN.Kawaguchi⑦April14,13:30-15:00LecturePlanDateTitleContents1Jan.20VLBIFundamentalHistoricalReview,RadioAstronomyandVLBI2Jan.27VLBIApplication(1)Geodesy,Astronomy,Astrometry,SpaceSciences3Feb.3VLBIApplication(2)VLBITechnologies(1)RotationintheNatureDigitalSignalProcessing,ADConverter4March10VLBITechnologies(2)ADConversion5Marｃｈ17VLBITechnology(3)NobleIdentity,CorrelationProcessingAtmospherictransparency,CoherenceLoss6March24VLBITechnology(4)RadioTelescopeSensitivity7April14VLBITechnology(5)AtmosphericabsorptionlossAntennaPointingLossTimeandFrequency8April21VLBITechnology(6)FrequencyInstabilityandthecoherenceloss2015/4/142Thelargesterrorsource:AtmosphericLoss•Atmosphericabsorption•Howtomeasure?FeedPowerMeterＴｓｋｙＴｈＴｒLoss3KLossMeasurementbyMoonLossFeedPowerMeterＴｓｋｙＴｈＴｃＴｒ~200K3K
=+ +
=+ + → →  = +1−⟶ (!)=1.12×$−50 ∶()**+ (,-*.2015/4/143LossMeasurementbySec-ZMethod)ln(sec(z))*ln(0rhsysTTTTh++=+⋅τ[]{}hrhrhrskysyshsysTTTTTTTT1YTT-+⋅=+--=+⋅==-=)()τexp()τexp(1)τexp()()τexp()exp(*τ/0/∗ ：SystemnoisemeasuredbyR/Skymethodα0：Transparencyofthezenithdirection,=2345skyhrskyhrskyrhhhsysTTTTTTTTTTYTT-+=-++=-=)(11*[].:)τexp(1assumedisTTTThambhsky=--⋅=definitionbyTTTrskysys:+=α０isderivedbyalinearregression.sec(z)0⋅=ττAQuestionablePointonSec-Z[])exp(10α--⋅+=ambskyTTT[])exp(1α--⋅=hskyTT•T0is3Kandnot0.•TambisnotalwaysTh.assumed2015/4/144ExactExpressionfortheSkyNoise{}τ21τfffτfτffαα)(αα)α1(αα)α-(1α)α1(⋅-=--⋅+-⋅=+⋅⋅+-⋅=⋅⋅⋅⋅TTTTTTTTTTcambambhcambhskysec(Z))τexp(1α021τ21⋅-⋅+-++=⋅-++=++=rhrhrrrhrskyrhTTTTTTTYTTTTTTTTTYcsambhTTTTTT-=⋅+-⋅=2ff1α)α1(ObservationEquation())()exp()()0exp()()exp(00000ccxcxbbbxcaaCOxcbaOiiii-⋅-+-⋅⋅---=-⋅-⋅-=⋅⋅⋅-----⋅--⋅--⋅--⋅--⋅--=⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅)exp()exp()exp()exp()exp()exp()exp()exp()exp()exp(11111000030302020101000302010NNiiNixcxbxcxbxcxbxcxbxcxbxcxcxcxcxcP)δe()()()(⋅==-PRCOパーシャルマトリックス2015/4/145SensitivityMatrixandParameterEstimation[])()()()()(T)()δe(1RWPPWPT⋅⋅⋅⋅=-)(P：SensitivityMatrix)(R：ResidualVector)δe(：CorrectionVector=22222Ni321/σ1/σ1/σ1/σ1/σ1)(00W計測誤差（σｉ）での重み付け行列FormalErrorofParameterEstimation()[]WPWPT⋅-1Diagonalelementsareformalerrorsofestimationparameters.2015/4/146QRDecomposingand‘Observability’),(),(),(MMMNMNRQP⋅=Partialmatrix,P,isdecomposedintoanormalorthogonalmatrix,Q,andatriangularmatrix,R.Ameasureof‘Observable’isdefinedbyanequationbelow.N:AnumberofdataM:Anumberofparameters∑==ijijiiirrf12Thefiaboveindicatestheindependenceofthei-thparameter.WeneedtoassignenoughnumberofdigitsinthecalculationlargerthanLibelow.210logiifL-=ＳｅｃＺMeasurementsinMizusawaＳｅｃＺmeasureddataTatm2932002年5月29日＠水沢／４３ＧＨｚ帯ElSKYHOTsecZYrskyTsys*ln(Tsys*+Th)90-3.83028-1.8193111.588901497.53696.67271236275-3.83134-1.810311.0352761.592587494.44246.6687902760-3.80508-1.802661.1547011.585776500.19136.67606444645-3.71473-1.795551.4142141.555671527.29076.70965883430-3.5367-1.7887421.495531591.28486.7847792115-3.10719-1.79493.8637031.352786830.53277.0242332085-2.38123-1.7932211.473711.144992020.8357.7466617812015/4/147MeasuredTransparencyinMizusawabyaconventionalSec-ZmethodSecZ法による４３ＧＨｚ帯大気吸収係数y=0.1255x+6.5376.66.656.76.756.86.856.96.9577.050246sec(Z)ln(Tsys*+Th)ln(Tsys*+Th)線形(ln(Tsys*+Th))ln(Trx+Th)Trx6.537397.2５°Elevation（secZ=11.5）wasomitted.6=0.882Non-linearanalysis解析結果物理パラメータ解析パラメータδeeτ０0.1255a01-0.08018660.919813aαf0.01b00.41847-0.07149850.346972bTr400c00.12550.03159320.157093cTs293P01Th293P1-exp(-c0*secZ)T03P2b0・secZ・exp(-c0・secZ)Tr397.2T1293T2290ElSKYHOTsecZ1/YrskyP1P2CO-C90-3.83028-1.8193110.62937-0.88210.3691140.63088576-0.0015275-3.83134-1.810311.0352760.62791-0.87820.3804470.63251627-0.0046160-3.80508-1.802661.1547010.63061-0.86510.4180210.63798296-0.0073845-3.71473-1.795551.4142140.64281-0.83740.4955640.64958352-0.0067730-3.5367-1.7887420.66866-0.7780.6511590.67442065-0.0057615-3.10719-1.79493.8637030.73922-0.61580.9955920.74232182-0.003115-2.38123-1.7932211.473710.87337-0.23691.1376430.90084788-0.027482015/4/148MatricCalculationObs0.6293658250.6279093090.6306061410.6428095710.6686587940.7392154260.873370423:=Calc0.6308857550.6325162710.6379829590.6495835160.6744206530.7423218210.900847885:=RObsCalc-:=R0.00151993-0.00460696-0.00737682-0.00677394-0.00576186-0.00310639-0.02747746-=P01111111:=P10.882055764-0.878159393-0.865095894-0.837374563-0.778022372-0.615761993-0.236939366-:=P20.3691142450.3804471510.4180212720.4955637440.6511586950.9955920321.13764294:=PaugmentP0P1,P2,():=δePTP⋅()1-PT⋅R⋅:=δe0.08018663-0.0714985-0.03159318=AtmosphericTransparencymeasuredbyanexactsolution0.550.60.650.70.750.80.850.90.95051015Sec(Z)!/Y初期値非線形解析後実測値TransparencybySec-Zmethod0.874Transparencybyexactmethod0.7432015/4/149PhysicalParametersobtainedinanexactsolutionrhTTTa+=1rhTTTb+=2（１）（２）ambhTTT⋅+-⋅=ff1α)α1(（４）（３）02TTTamb-=1.Trisprerequisitetobemeasured.2.Sec-Zdataissolv