GPS ( ) L1 L2 LC ( ) PCV GNSS GNSS PCO v = Ax b. (1) A (3) x PCO b v x = (A T PA) 1 (A T Pb), (2) P [1] 90
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1 30 4 Vol. 30, No PROGRESS IN ASTRONOMY Nov., (2012) GNSS 1,2 2 1 ( ) GNSS LEO GPS GNSS LEO P128 A 1 GNSS GNSS, ARP (PCO) (PCV) GNSS GNSS GNSS GNSS 4 GNSS ( ) (06DZ22101)
2 GPS ( ) L1 L2 LC ( ) PCV GNSS GNSS PCO v = Ax b. (1) A (3) x PCO b v x = (A T PA) 1 (A T Pb), (2) P [1] 90
3 4 GNSS i ( x i, y i ) ARP (x 0, y 0 ) (x 1, y 1 ) (x 2, y 2 ) x 1 = x 0 x 1 + x 2 y 1 = y 0 y 1 + y 2 x 5 = x 0 + x 2 + y 1 y 5 = y 0 + y 2 x 1 x 2 = x 0 x 1 y 2 y 2 = y 0 y 1 + x 2 x 6 = x 0 + x 2 + x 1 y 6 = y 0 + y 2 + y 1 x 3 = x 0 x 1 x 2 y 3 = y 0 y 1 y 2 x 7 = x 0 x 2 y 1 y 7 = y 0 + y 2 + x 1 (3) x 4 = x 0 x 1 + y 2 y 4 = y 0 y 1 x 2 ( x i, y i ) (i =1,2,,7) (x 0, y 0, x 1, y 1, x 2, y 2 ) (1) (2) (x 1, y 1 ) (x 2, y 2 ) 1 [1] min
4 [2] 2 [2] A B H A H B A B h A h A δh A δh B (U A ) i,j (U B ) i,j A B i = 1, 2, 3, 4 j = 1, 2 H AB = H B H A = [(U B ) 1,1 h B δh B ] [(U A ) 1,1 h A δh A ], (4) H AB = H B H A = [(U B ) 1,2 h B δh A ] [(U A ) 1,2 h A δh B ], (5) δh B δh A = 1 2 {[(U B) 1,1 (U A ) 1,1 ] [(U B ) 1,2 (U A ) 1,2 ]}. (6) h AB = δh B δh A ( U AB ) i,j = (U B ) i,j (U A ) i.j ( U AB ) i,j i A B h AB = δh B δh A = 1 2 ( 1)i 1 [( U AB ) i,1 ( U AB ) i,2 ]. (7) 4 h AB 2.3
5 4 GNSS [3] 3 [3] 2.4 [4] (ARP) h L1 L2 LC PCO
6 ARP (PCV) GNSS PCO PCV 3 GNSS 3 (PCV) 70% 3.1 [5,6] GPS GNSS GPS PCV AOAD/M T PCVs 0 [7] ( ) PCV PCV PCV PCV PCV L1 L2 LC PCV [4] PCO PCV i = τ i + α 1 θ i + α 2 θ 2 i + α 3 θ 3 i + α 4 θ 4 i, (8) i θ i α i τ i
7 4 GNSS 507 PCV PCV PCV z 90 0 PCV(z = 90 ) = 0. (9) RMS PCV 24 h GNSS [8] PCV PCV [4,8] L1 t 1 ( 1) i Φ i 1(t 1 ) =ρ i 1(t 1 ) c dt 1 + c dt i + c f L1 N i 1(1) i 1,ion(t 1 )+ i 1,trop(t 1 ) + d 1 (t 1 ) + d i (t 1 ) + PVC 1 (θ i (t 1 )), Φ i 1 t 1 ρ i 1(t 1 ) c dt 1 (GPS ) dt i (GPS ) f L1 L1 N i 1(1) L1 i 1,ion(t 1 ) L1 i 1,trop(t 1 ) L1 d 1 (t 1 ) d i (t 1 ) PCV 1 (θ i (t 1 )) L1 PCV θ i (t 1 ) i t 1 ( 2) Φ i 2(t 1 ) =ρ i 2(t 1 ) c dt 2 + c dt i + c f L1 N i 1(1) i 2,ion(t 1 )+ i 2,trop(t 1 ) + d 2 (t 1 ) + d i (t 1 ) + PVC 2 (θ i (t 1 )). (10) (11) SD SD i 1 2(t 1 ) = ρ i 1 2(t 1 ) c dt (10) (11) c f L1 N i 1 2(1) + d 1 2 (t 1 ) + PVC 1 2 (θ i (t 1 )). (12) dt i d i (t 1 ) ( i 1,ion (t 1) i 2,ion (t 1)) ( i 1,trop(t 1 ) i 2,trop(t 1 )) PCV PCV 1 2 (θ i (t 1 )) = α 0 + α 1 (θ i (t 1 )) + α 2 (θ i (t 1 )) 2 + α 3 (θ i (t 1 )) 3 + α 4 (θ i (t 1 )) 4. (13) ( 1) ( 2) j j
8 SD j 1 2(t 1 ) =ρ j 1 2(t 1 ) c dt c f L1 N j 1 2(1) + d 1 2 (t 1 )+ α 0 + α 1 (θ j (t 1 )) + α 2 (θ j (t 1 )) 2 + α 3 (θ j (t 1 )) 3 + α 4 (θ j (t 1 )) 4. (14) ρ i 1 2(t 1 ) ρ j 1 2(t 1 ) t i j DD1 2(t i j 1 ) = + c N f 1 2(1) i j + α 1 [(θ i (t 1 )) (θ j (t 1 ))] + α 2 [(θ i (t 1 )) 2 (θ j (t 1 )) 2 ]+ L1 α 3 [(θ i (t 1 )) 3 (θ j (t 1 )) 3 ] + α 4 [(θ i (t 1 )) 4 (θ j (t 1 )) 4 ]. (15) t i j DD1 2(t i j 2 ) = + c N f 1 2(1) i j + α 1 [(θ i (t 2 )) (θ j (t 2 ))] + α 2 [(θ i (t 2 )) 2 (θ j (t 2 )) 2 ]+ L1 α 3 [(θ i (t 2 )) 3 (θ j (t 2 )) 3 ] + α 4 [(θ i (t 2 )) 4 (θ j (t 2 )) 4 ]. (16) L1 (16) (15) T D i j 1 2(t 2 1 ) = +α 1 A i j 1 (t 2 1 ) + α 2 A i j 2 (t 2 1 ) + α 3 A i j 3 (t 2 1 ) + α 4 A i j 4 (t 2 1 ), (17) A i j 1 (t 2 1 ) = [(θ i (t 2 )) (θ j (t 2 ))] [(θ i (t 1 )) (θ j (t 1 ))], (18) A i j 2 (t 2 1 ) = [(θ i (t 2 )) 2 (θ j (t 2 )) 2 ] [(θ i (t 1 )) 2 (θ j (t 1 )) 2 ], (19) A i j 3 (t 2 1 ) = [(θ i (t 2 )) 3 (θ j (t 2 )) 3 ] [(θ i (t 1 )) 3 (θ j (t 1 )) 3 ], (20) A i j 4 (t 2 1 ) = [(θ i (t 2 )) 4 (θ j (t 2 )) 4 ] [(θ i (t 1 )) 4 (θ j (t 1 )) 4 ]. (21) (17) K N 4 = TD N 1 = K N 4 α 4 1, A i j 1 (t 2 1 ) A i j 2 (t 2 1 ) A i j 3 (t 2 1 ) A i j 4 (t 2 1 ).. A i j 1 (t n (n 1) ) A i j 2 (t n (n 1) ) A i j 3 (t n (n 1) ) A i j 4 (t n (n 1)).., (22) α 4 1 = [α 1 α 2 α 3 α 4 ] T, (23) α = (K T K ) 1 (K T TD). (24)
9 4 GNSS 509 PCV PCV PCV PCV 1 cm [8] IGS PCVs 0 Wubbena [9,13] Geo++ [10,11] GPS ( ) GPS, ( PCV ) ( PCV [12] ) PCVs Φ j i = ρj i (t) c dt i + c dt i + λ N j i j ion,i + j trop,i + j MP,i + dj PCV,i + ε. (25) [10,13] δ SID δ SID Φ j i = c (δsid dt i δ SID dt j ) + λ δ SID N j i δsid j ion,i + δsid j trop,i + δsid ε. (26) PCV δ SID Φ j,m i,k = λ δsid N j,m i,k + δsid ε. (27) 180 δ SID Φ j i = c (δsid dt i δ SID dt j )+λ δ SID N j i δsid j ion,i +δsid j trop,i +d(0,0),j PCV,i d(α,z),j PCV,i +δsid ε, (28)
10 δ SID Φ j,m i,k = λ δsid N j,m i,k d PCV PCV + δsid ε + d j,m PCV,i,k, (29) α j = α i + α z j = z i + z d PCV (α i, z i, α, z) = d PCV (α i, z i ) d PCV (α j, z j ), (30) PCV PCO PCV ANTEX ( 4.1 ) 3.4 PCV IGS 11 6 IGS GFZ TUM 10 GPS 1994 IGS PCVs AOAD/M T PCV GPS VLBI SLR GPS [5,14] cm [14] IGS 5 10 IGS VLBI SLR [5] 90 0 PCVs PCVs PCVs 10 PCVs PCVs
11 4 GNSS mm [24] PCVs PCVs PCVs IGS GNSS GNSS PCV [15] 4.1 PCV PCO PCV PCO 4 4 PCVs PCVs 4 z z [16] sin(z ) = R sin(z), (31) r
12 R r ( ) z max Rmax = arcsin sin(z max ). (32) r min PCO PCV [16] φ (z ) = φ(z). (33) φ (z ) = φ(z). (34) z ( ) r φ (z ) = r(1 cos z ). (35) PCO PCV PCV PCO PCV PCO PCV PCO PCO APR [17] PCO PCV GPS PCO PCV PCO PCV raw PCV [18,19] 4.2 PCO PCV [16,20] IGS ( ) [21] PCO PCV [21] IGS PCO PCV LC ( ) w = cos 2 z(z ), PCV
13 4 GNSS 513 ( raw PCV) GFZ TUM [19] 14 6 ( Neill 3 h ) 15 PCV ( ) raw PCV PCO PCVs PCV PCV raw PCV [16] [16] 14 z =0 PCV raw (z ) = 0. (36) PCV raw (z = 0 ) = 0. (37) raw PCV PCV PCO PCV PCV [18,19] PCV PCO z PCV raw (z ) = PCV(z ) + z (1 cos z ). (38) PCV z [PCV raw (z ) a i z (1 cos z )] 2 = min (j = 1, 2,, 14). (39) j a i PCV raw (z ) i PCV PCV PCO PCV PCO PCV PCO PCV PCV PCV 5 [16] PCV PCV PCV [16,21] PCV (1) GNSS (2) PCV ( ) PCV n max n PCV(α, z) = (a nm cos mα + b nm sin mα)p nm (cos z), (40) n=0 m=0
14 PCV Block IIR PCV [16] PCV(α, z) α z p nm a nm b nm n m PCV n m a nm b nm n m( n max = m max = 8) a nm b nm PCV x y z 4.3 LEO GPS LEO GPS km GPS LEO GPS GPS LEO LEO GPS igs08.atx [22] GPS PCO PCV 14 GPS [23] 14 LEO GPS PCVs LEO PCVs 5 5 LEO LC GPS PCV [24]
15 4 GNSS 515 (Φ orb Φ model ) = PCV true (e) PCV model (e) = PCV(e), (41) e LEO GPS LEO GPS 14 PCV igs08.atx PCV 2 3 mm GPS PCV 1 mm LEO PCV igs08.atx 14 PCV LEO GPS LEO [23] LEO GPS 5 GNSS 3 PCVs PCVs PCV PCV GPS 14 LEO GPS km 17 SVN714 L2 [20] PCV [20,25] R714 PCV PCV [26] PCV PCVs GNSS ( GPS ) GPS [1],., 2000, (12): 15 [2]., 2000, 2(3): 23 [3] Bányai L. Journal of Geodesy, 2005, 79(4): 222 [4] Mader G L. GPS Solutions, 1999, 1: 55 [5] Rothacher M, Schaer S, Mervart L, et al. SPECIAL TOPICS AND NEW DIRECTIONS, Gendt G, Dick G, eds. Germany GFZ
16 [6] Rothacher M. GPS Solutions, 2001, 4(4): 55 [7] Rothacher M, Mader G. Estimation and validation of IGS absolute antenna phase center variations, 01.txt, 1996 [8] Daniel N A, Andrew R, Barbara A O, et al. IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, 2005, 54(5): 1820 [9] Schmitz M, Wubbena G, Boettcher G, et al. GPS Solutions, 2002, 6: 18 [10]., 2010, (3): 7 [11] Dach R, Schmid R, Schmitz M, et al. GPS Solutions, 2011, 15: 49 [12] Akrour B, SanterreR, Geiger A. GPS world, 2005, 16(2): 49 [13] Wubbena G, Menge F, Schmitz M, et al. Proc Int Tech Meet ION GPS-96, Kansas City, Missouri, 1996: 1205 [14] Zhu Y, Massmann H, Yu Y,et al. J Geod, 2003, 76(11): 668 [15] Mader G L, Czopek F M. GPS World, 2002, 13(5): 40 [16] Schmid R, Rothacher M. Journal of Geodesy, 2003, 77(7): 440 [17] Rothacher M, Schaer S, Mervart L, et al. IGS Workshop, Germany:GFZ Potsdam, 1995: 205 [18] Ge M, Gendt G, Meindl M. eds. Bern: Proceedings of the IGS Workshop and Symposium 2004, 2005 [19] Schmid R, Steigenberger P, Gendt G, et al. J Geod, 2007, 81: 781 [20] Dilssner F, Springer T, Flohrer C, et al. Journal of Geodesy, 2010, 84(8): 467 [21] Dach R, Hugentobler U, Fridez P, et al. Bernese GPS Software, version 5.0, University of Berne: Astronomical Institute, 2007: 355 [22] ftp://garner.ucsd.edu/pub/gamit/tables/igs08.atx, 2012 [23] Jaggi A, Dach R, Bock H, et al. Combining terrestrial and LEO data to extend the GPS satellite antenna patterns to nadir angles beyond 14, [24] Montenbruck O, Garcia-Fernandez M, Yoon Y, et al. GPS Solutions, 2009, 13(1): 23 [25] ftp://garner.ucsd.edu/pub/gamit/tables/antmod.dat, 2012 [26] Schmitz M, Wubbena G, Boettcher G. GPS Solutions, 2002, 6: 18 Research on Calibration Methods of GNSS Antenna Phase Center Offsets and Variations LI Xiao-bo 1,2, WANG Xiao-ya 2, REN Jin-wei 1 (1. Institute of Earthquake Science China Earthquake Administration, Beijing , China; 2. Shanghai Astronomical Observatory, Chinese Academy of sciences, Shanghai , China) Abstract: GNSS measurement, including pseudo-range and carrier phase observation, is the distance between phase center of satellite antenna and phase center of receiver antenna. Satellite mass center and receiver antenna reference point (ARP) are used as reference in GNSS data processing. Consequently, the antenna phase center correction should be considered in GNSS data processing. GNSS antenna phase center correction including mean phase center offset with respect to the ARP or the mass of satellite, i.e. PCO, and the variation to the mean phase center with respect to the satellite elevation and azimuth angle, i.e. PCV. In general, the same type
17 4 GNSS 517 of antennas have similar PCO and PCV. This paper summarizes the methods of calibrating GNSS antenna phase center offsets and variations. In addition, the calibration of on-orbit GNSS satellite antenna phase center variation is described. PCVs are highly correlated with PCOs, the so-called daily raw PCVs are normally estimated together with other parameters including site coordinates, site-specific troposphere parameters, orbit parameters and Earth rotation parameters etc. Afterwards, the raw PCV estimates are separated into PCOs and PCVs under the assumption that PCV is the flattest by the least-square fit. The calibration of the on-orbit GPS satellite antenna PCV by LEO observations is described as well. It extends the maximum nadir angle up to 17 by establishing the un-differential carrier phase residuals. Key words: GNSS; calibration; antenna phase center offset; antenna phase center variation; LEO CN /P 4-819
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