Improvement of the accuracy of the quasigeoid model VIGAC2017

Vietnam Journal of Earth Sciences, 40(1), 39-46, Doi: 10.15625/0866-7187/40/1/10914 39 (VAST) Vietnam Academy of Science and Technology Vietnam Journal of Earth Sciences Improvement of the accuracy of the quasigeoid model VIGAC2017 Ha Minh Hoa Vietnam Institute of Geodesy and Cartography (VIGAC) Received 14 June 2017; Received in revised form 25 October 2017; Accepted 10 November 2017 ABSTRACT A national spatial reference system will be constructed based on a highly accurate national quasigeoid model with accuracy more than 4 cm. In Vietnam at the present stage there isn’t a detailed gravimetric measurement in mountainous regions and marine area. So with the purpose of improvement of accuracy of the national quasigeoid model VIGAC2017, we only can solve the task of fitting this model to national quasigeoid heights obtained from heights GPS/first, second orders levelling quasigeoid heights through least squares collocation. This scientific article will introduce a first research result for improvement of accuracy of the quasigeoid model VIGAC2017 on the base of it’s fitting to 194 national quasigeoid heights by the least squares collocation. Research results show that accuracy of the quasigeoid model VIGAC2017 will be obtained at level of ±0,058 m and increased to 20,69 %. Keywords: National spatial reference system; national quasigeoid height; least squares collocation; covariance matrix; semivariogram; semivariance function. ©2018 Vietnam Academy of Science and Technology 1. Introduction1 A wide application of GNSS technology with GNSS data processing in ITRF and a combined usage of detailed gravimetric data and more accurate with every passing day Earth Gravity Model (EGM) for the construc- tion of a highly accurate national quasigeoid model naturely lead to a bulding of a national spatial reference system. Ha Minh Hoa, 2017 had found that the most impotant base for the bulding of the national spatial reference sys- tem is the national quasigeoid model with ac- *Corresponding author, Email: minhhoavigac@gmail,com curacy more than ±4 cm, which is the guarantee that the national geodetic height of every point on the national territory is equal to the sum of the it’s national normal height and national quasigeoid height. At present, many countries had constructed the highly accurate national quasigeoid/geoid models, for example, OSGM2002 (United Kingdom) with accuracy at level ± 3,2 cm (Iliffe J.C., Ziebart M,, Cross P.A., Forsberg R., Strykowski G., Tscherning C.C., 2003), USGG2009 (United States) with accuracy at level ± (3-4) cm (Roman D. R., Y.M. Wang, J. Saleh, X. Li, 2010), CGG2013 (Canada) with accuracy more ±3 cm on the 80% Ha Minh Hoa/Vietnam Journal of Earth Sciences 40 (2018) 40 continent part (Huang J., Véronneau M., 2013), GCG16 (Germany) with accuracy more ±1 cm (Alps max 2 cm, marine area 2-6 cm) (Quasigeoid of the Federal Republic of Germany GCG2016). The fit of gravimetric geoid/quasigeoid model to GPS/levelling geoid/quasigeoid heights through the least squares collocation had been accomplished in many countries. For example, the geoid model OSGM2002 had been fitted to the 179 GPS/levelling geoid heights cm (Iliffe J.C., Ziebart M., Cross P.A., Forsberg R., Strykowski G., Tscherning C.C., 2003). In (Metin Soycan, 2014) had been presented results of fitting EGM2008 de- rived geoid heights to the 87 GPS/leveling geoid heights in Turkey. (Ha Minh Hoa, 2017) has presented results of construction of the initial national spatial referense system on base of orientation of the WGS84 ellipsoid to best fit it to the Hon Dau local quasigeoid at tide gauge Hon Dau with using the most stable 164 co - located GPS observations first and second orders bench marks. When the national quasigeoid heights ζ have been calculated from the GPS/first and second orders levelling quasigeoid heights levelingGPS /ζ by formula: ,. 0 0 0 /           += dZ dY dX AlevelingGPSζζ (1) while national quasigeoid heights *ζ from the inital national quasigeoid model VIGAC2017 have been determined by following formula: ,. 0 0 0 **           += dZ dY dX Aζζ (2) where the GPS/first and second orders levelling quasigeoid height levelingGPS /ζ has been calculated by formula: ,/ γζ zzlevelingGPS HH −= zH - geodetic height of the first (or second) order bench mark obtained from the GPS data processing in ITRF and converted to the zero - tide system; γzH - first (or second) order national normal height converted to the zero - tide system; *ζ - mixed quasigeoid height of point got from the mixed quasigeoid model VIGAC2014 and converted to the zero - tide system; matrix ,)sinsincoscos(cos BLBLBA ⋅⋅= LB , - geodetic latitude and longitude of point according to the WGS84 ellipsoid; co- ordinate transformation parameters from ITRF to the VN2000-3D: .417880,111,192468,42,511083,204 000 mdZmdYmdX === In (Ha Minh Hoa, 2017) with purpose of comparision of an accuracy of series of the national quasigeoid heights ζ (1) with an accuracy of according series of the quasigeoid heights *ζ (2) on the 164 GPS/first order levelling points, the both those series of the quasigeoid heights had been considered to be the equal accuracy at level of .062,0 m± However, in practice the both above mentioned series of the quasigeoid heights don’t have the same accuracy. In (Ha Minh Hoa, 2017) RMS of the differencies *ζζ −=Z is equal to: .088,0 164 265,1 164 164 1 2 2 * 2 m Z mmm i i Z ±=±=±=+±= ∑ = ζζ Meanwhile in (Ha Minh Hoa et al., 2016) based on co - located GPS observations first order bench marks and global quassigeoid heights from the EGM2008 model on those bench marks. RMS of series of the quasigeoid Vietnam Journal of Earth Sciences, 40(1), 39-46 41 heights *ζ had been established at level of .070,0* mm ±=ζ When contribution portion of RMS ζm of series of the 164 national quasigeoid heights ζ to the RMS value mmZ 088,0±= is equal to .053,0 m± As such for following usage in this article, we accept that the RMS of the national quasigeoid height ζ calculated by formula (1) from the corresponding GPS/first (or second) order levelling quasigeoid height levelingGPS /ζ on the stable first (or second) order bench mark is equal to ,053,0 m± while the RMS of the national quasigeoid height *ζ from the quasigeoid model VIGAC2017 calculated by formula (2) is equal to: .070,0* mm ±=ζ (3) With the purpose of improvement of accuracy of the quasigeoid model VIGAC2017 this scientific article will introduce results of fitting this model to the 194 GPS/first, second orders levelling quasigeoid heights by the least squares collocation. 2. Data Apart from the 164 GPS/first, second orders leveling quasigeoid heights ζ for solving abovementioned task had been added 30 GPS/first order levelling quasigeoid heights in the zero - tide system on the stable first order bench marks obtained by Vietnam Institute of Geodesy and Cartography (VIGAC) in period 2012 - 2013 (Ha Minh Hoa, et al., 2012; Ha Minh Hoa, Nguyen Ba Thuy, Phan Trong Trinh, et al, 2016), Stability of the first order benchmarks had been controled by Smirnov’s criteria (Smirnov N.V., Belugin D.A., 1969), The abovementioned 30 GPS/first order levelling quasigeoid heights had been converted to the national WGS84 reference ellipsoid by formula (1). On the 30 first order bench marks had been determined quasigeoid heights *ζ according to the quasigeoid model VIGAC2017 by formula (2). The total 194 first and second orders bench marks have been distributed relatively regularly on whole territory of Vietnam. 3. Applied methods We symbolize Q as a set of n GPS/first and second orders leveling bench marks (in our case n = 194), P as a set of points whose quasigeoid heights will be determined by the least squares collocation. In the set Q had been calculated the differencies ,194,..,2,1,* =−= iZ iii ζζ where for point i the national quasigeoid height iζ had been determined by formula (1), while the quasigeoid height *iζ from the quasigeoid model VIGAC2017 had been determined by formula (2). In addition the accuracy of the national quasigeoid height iζ is considered equal to .053,0 m± On base of the least squares collocation, at a point ,Pp∈ a national quasigeoid height *~pζ will be determined by formula: ,~ *** ppp δζζζ += (4) where quasigeoid height *pζ from the quasigeoid model VIGAC2017 is calculated by formula (2), correction *pδζ is determined by formula (Moritz, H,, 1980): ,.. 1* ZKC ZpQp −=δζ (5) )( 21 pnppPQ CCCC = is the cross - covariance matrix between the differences Ha Minh Hoa/Vietnam Journal of Earth Sciences 40 (2018) 42 ),194,..,2,1(* =−= iZ iii ζζ in the set Q and the estimated quasigeoid height at the point ,Pp∈ Z is column - vector containing the differences ),194,..,2,1(* =−= iZ iii ζζ covariance matrix has form: ,ZZZZ CCK += (6) ZC is the auto - covariance matrix of vector Z, ZZC is the covariance matrix, which reflects the spatial dependencies of the all differences )194,..,2,1(* =−= iZ iii ζζ in the set Q. For the 194 differences ),194,..,2,1(* =−= iZ iii ζζ their RMS is equal to: .090,0008149,0 194 580915,1 mmZ =±=±= (7) When the auto - covariance matrix ZC has the form: ,.008149,0. 22 ><== mEEmC nxnnxnZZ (8) where nxnE - unit matrix of order 194. The covariance matrix ZZC , which reflects the spatial dependencies of the all differences )194,..,2,1(* =−= iZ iii ζζ in the set Q, will be determined based on a covariance function ),()( 2 dmdC Z γ−= (9) where )(dγ is a semivariance; d is a distance between any two points in the set Q. As such in our case the spatial dependence of quassigeoid heights in the set Q will be studied using semivariogram, The experimental semivariance )(hγ at lag distance h is calculated by formula (Cressie N.A.C., 1993; Schabenger O., Gotway C.A., 2005; Marcin Ligas, Marek Kulczycki, 2014): ( ) , 1 2)()(. 2 1)( ∑ = +−= hn i ii h hxZxZ n hγ where )( ixZ is the difference *ζζ −=Z of the point at position ,ix )( hxZ i + is the difference *ζζ −=Z of the point at position hxi + separated from position ix by a distance not more than lag distance h; hn is the number of pairs ).( ixZ By such way in the set Q we must create groups of points, in addition in every group the distances between points not more than lag distance h. Based on an experimatal semivariogram we will determine form of theoretical semivariance, which in general case has following form: ,.)( 10      += a dfCCdγ (10) where 0C is the nugget effect; 1C is the structural variance; a is the range of spatial dependence; function       a df will be selected in relation to distribution of the semivariogram corresponding to standaed models of semivariance functions (Gaussian, spherical, exponential, linear models). Value 10 CC + is the sill and determined from the semivariogram. 4. Results From the 194 most stable co - located GPS observations first and second orders bench marks covering the whole territory of Vietnam had been constructed the set Q, which contains the 194 differences .*ζζ −=Z In the set Q had been created 58 groups of points with change of the distances from 25 km to 1475 km. The lag distance h = 25 km. For the semivariogram of the experimatal semivariances, shown in Figure 1, the sill Vietnam Journal of Earth Sciences, 40(1), 39-46 43 ,007928,0 210 mCC =+ the range of spatial dependence .1475 kma = Next analysis results show that the nugget effect ,002706,0 20 mC = the structural variance .005222,0 21 mC = From the semivariogram of the experimatal semivariances we realize that distribution of the experimatal semivariances corresponds to spherical model. So the theoretical semivariance (10) has form: .. 2 1 .2 .3.005222,0002706,0)( 2 3 ><              −+= m a d a ddγ (11) On account of the formulas (7), (11), the covariance function (8) gets form: .. 2 1 .2 .3.005222,0005443,0)( 2 3 ><              −−= m a d a ddC (12) Figure 1. The semivariogram of the experimatal semivariances After determination of the covariance matrix ZZC baded on the the covariance function (12), on account of the auto - covariance matrix ZC (8), we had calculated the covariance matrix ZK (6), The correction *pδζ to the quasigeoid height * pζ of any point Pp∈ was calculated by formula (5) and the corrected quasigeoid height *~ pζ of this point was determined by formula (4). With purpose of accuracy estimation of the 194 corrected quasigeoid heights * ~ζ of the quasigeoid model VIGAC2017 at the 194 first and second orders bench marks, we had calculated 194 differences ),194,...,2,1(~* =−= iZ iii ζζ where iζ is the national quasigeoid height of bench mark i calculated by formula (1) (see Table 1). d (km) C0 + C1 0.007 0.006 0.005 0.004 C0 0 200 400 600 800 1000 1200 1400 Vietnam Journal of Earth Sciences, 40(1), 39-46 45 Table 1. The differences Z on the 194 first and second orders bench marks No Points Differences Z (m) No Points Differences Z (m) No Points Differences Z (m) 1 IBH-LS97 0,0543 66 IVL-HT71 0,0523 131 IILC-TG15 0,0427 2 IBH-TH122A 0,0049 67 IBH-TH59 0,0627 132 IILC-TG19A -0,0469 3 IBH-TH119 0,0246 68 IVL-HT173-2 0,0860 133 IILC-TG31 0,0422 4 IBH-HN33 -0,0141 69 IBH-TH70A 0,0665 134 IIMC-XM7-1 -0,0825 5 IBH-HN39 -0,0123 70 IHN-VL50 0,1029 135 IIMT-TH25 -0,1431 6 IBH-HN42 -0,0410 71 IVL-HT123 0,0804 136 IIMT-TH4 -0,0217 7 IHN-HP7 0,0344 72 ILS-HN12 0,0415 137 IIMT-TH7 -0,1424 8 IHN-VL10A -0,1006 73 IHP-MC4-1 0,0550 138 IIMT-TV11 -0,0902 9 IHN-VL4-1 -0,0039 74 IBH-LS80 0,0470 139 IIMX-DC34 -0,1341 10 IHN-VL6-1 -0,0206 75 IDN-BT86 0,0950 140 IINB-HN11-1 0,0281 11 IDN-BMT16 -0,0646 76 IVL-HT320A 0,1044 141 IINB-HN15 -0,0019 12 IDN-BMT28 -0,0582 77 IBMT-APD49-1 0,1158 142 IINB-HN24 0,0397 13 IVL-HT150 -0,0686 78 IHP-NB14A -0,1340 143 IINB-HN27-1 0,0055 14 IVL-HT152-1 -0,0192 79 ILS-HN36 0,0140 144 IINB-HN32-1 0,1176 15 IHN-VL34- -0,0504 80 ILS-HN22 -0,1483 145 IINK-PT10 0,0268 16 IHP-MC48A -0,0945 81 ILS-HN29 -0,0746 146 IINK-PT13 0,0887 17 IBH-TH3-1 -0,0572 82 IBH-HN16A 0,0509 147 IINK-PT6-1 -0,2096 18 IVL-HT181 -0,0485 83 IHN-VL28-1 0,0222 148 IIPLK-PL12 -0,0317 19 ILS-TY4 -0,0933 84 IBH-HN48 0,0954 149 IIPLK-PL16 -0,0667 20 IVL-HT309A -0,0278 85 IHN-HP2A 0,0859 150 IIPLK-PL2 0,0641 21 IVL-HT317 -0,0323 86 IHN-HP5 0,1210 151 IIPLK-PL24 -0,1687 22 IVL-HT187 -0,0337 87 IVL-HT73 0,1703 152 IIPLK-PL8 -0,0346 23 IVL-HT170-1 -0,0414 88 IVL-HT95 0,1522 153 IISC-PL29 -0,0922 24 IHP-MC41 -0,0684 89 IIDK-TM41 0,0320 154 IISC-VT3-1 0,0001 25 IHN-VL56 0,0631 90 IIAB-CL5 -0,0628 155 IITL-TV5-1 -0,0861 26 IBH-TH11 0,0272 91 IIAS-KS10 -0,1188 156 IITL-TV7 -0,0792 27 IHN-VL40-1 0,0619 92 IIAS-KS16 -0,0715 157 IITT-TK29 -0,1479 28 IVL-HT130 -0,0353 93 IIAS-KS22 -0,1120 158 IITX-TL14 -0,0624 29 IBH-LS77 0,0036 94 IIAS-KS32 -0,0971 159 IITX-TL20-1 -0,0886 30 IBH-TH5 -0,0512 95 IIAS-KS35 -0,1490 160 IITX-TL25 -0,0068 31 IHN-VL38-1 -0,0157 96 IIBH-XL11-1 -0,0204 161 IITX-TL6 -0,0214 32 IVL-HT197 -0,0177 97 IIBH-XL17 0,0250 162 IIYB-CN18 -0,0811 33 IBMT-APD63 -0,0186 98 IIBH-XL6 0,1134 163 IIYB-CN24-1 -0,1574 34 IVL-HT127-3 -0,0283 99 IIBMT-DT12 -0,0944 164 IDN-BT18-1 -0,0764 35 IBMT-APD59-1 -0,0199 100 IIBMT-DT14 -0,1441 165 IBMT-APD46 -0,0854 36 IVL-HT278-1 0,0208 101 IIBMT-DT4 0,1568 166 IVL-HT305 -0,0510 37 IVL-HT108 -0,0264 102 IIBN-QT11-1 0,1120 167 IVL-HT159-3 0,1423 38 IDN-BT77 -0,0083 103 IIBS-CD12 -0,0333 168 IVL-HT262A 0,1721 39 IBMT-NH17-1 -0,0103 104 IIBS-CD14 0,1611 169 IHN-VL76 0,1302 40 IVL-HT83 -0,0326 105 IIBS-CD3 0,0155 170 IVL-HT113 0,1196 41 IBH-HN17 -0,0392 106 IIBS-CD7-1 0,0832 171 ILS-HN10 0,0748 42 IHN-VL45-1 0,0611 107 IICD-HN6 0,1058 172 IBH-HN19-1 0,1009 43 IBH-TH65 -0,0178 108 IICD-VC4 -0,1091 173 IBMT-NH11-1 0,1350 44 IVL-HT178 0,0113 109 IICD-VC4-1 0,0054 174 IBH-HN20-1 0,1026 45 IVL-HT103 -0,0079 110 IICT-GD1 0,1305 175 TB01 0,1079 46 IHN-VL64 0,0259 111 IICT-GD10 0,0103 176 QN01 -0,0246 47 IVL-HT141-3 0,0082 112 IICT-GD15-1 -0,0216 177 QNG1 -0,1084 48 IVL-HT329A 0,0175 113 IICT-GD4 0,1442 178 BP01 0,0219 49 IHN-VL72 0,0225 114 IICF-VT1 0,0049 179 22A1 -0,0264 50 IVL-HT158 0,0264 115 IIDK-TM29 -0,0886 180 38A1 -0,0757 51 IVL-HT121 0,0765 116 IIDK-TM45 -0,1262 181 VL48 0,0401 52 IDN-BT74 0,0485 117 IIDL-PR31 -0,1293 182 IHN-VL59 0,0123 Ha Minh Hoa/Vietnam Journal of Earth Sciences 40 (2018) 46 53 IBH-LS88-1 -0,0155 118 IIGD-AB12 -0,0212 183 VL73 0,1348 54 IVL-HT98 0,0110 119 IIGD-AB3-1 -0,0451 184 HT73 0,1263 55 IBH-LS85-1 -0,0117 120 IIGD-AB9-1 -0,0068 185 HT84 0,0415 56 IBH-LS93 -0,0133 121 IIGD-APD2-1 0,1062 186 HT94 0,0882 57 IBH-LS71 -0,0074 122 IIGD-APD6-1 -0,0193 187 HT106 0,0137 58 IBT-APD56 0,0382 123 IIHN-AB11 -0,0317 188 HT121 -0,0415 59 IVL-HT87 0,0281 124 IIHN-AB17 -0,0880 189 HT127-4 0,0117 60 IVL-HT247A 0,0574 125 IIHN-AB20 -0,0542 190 IVL-HT141-3 0,0622 61 ILS-TY1 0,0040 126 IIHN-AB23 -0,0333 191 HT159-1 -0,0326 62 IVL-HT325-1 0,1074 127 IIHN-AB3 -0,0346 192 HT173-3 -0,0243 63 IDN-BT83 0,0552 128 IIHN-AB7 -0,1025 193 HT197 0,0932 64 IVL-HT78 0,0298 129 IIHN-MT15 -0,0598 194 IHP-MC45 0,0950 65 ILS-HN7 0,0170 130 IIHN-MT5 0,0092 The RMS of the differences )194,...,2,1(~* =−= iZ iii ζζ is equal to: .078,0 194 1750,1 194 194 1 2 m Z m i i Z ±=±=±= ∑ = Because the RMS of the national quasigeoid heights ζ calculated by formula (1) got equal to ,053,0 mm ±=ζ the contribution portion of RMS *~ζm of the quasigeoid heights *~ζ of the corrected quasigeoid model VIGAC2017 to the RMS value mmZ 078,0±= is equal to .058,0 m± From the RMS values mm 058,0*~ ±=ζ and *ζm (3) we realize that in comparison with the initial quasigeoid model VIGAC2017, the corrected quasigeoid model VIGAC2017 has been more accurate than 20,69 %. 5. Discussions Research results show that after fitting the initial quasigeoid model VIGAC2017 to 194 national quasigeoid heights at the first and second orders bench marks by the least squares collocation, accuracy of the corrected quasigeoid model VIGAC2017 had been increased to 20,69 %. That has been obtained taking into account the spatial dependences of the quasigeoid heights in the Earth gravity field on territory of Vietnam. However, the corrected quasigeoid model VIGAC2017 still does not obtain accuracy more than 4 cm. The next increase of accuracy of the national quasigeoid model in Vietnam will be accomplished in the future on base of using detailed gravimetric data. 6. Conclusions Above represented research results show, that on the base of solving the task of fitting the initial quasigeoid model VIGAC2017 to the 194 national quasigeoid heights got from the 194 GPS/first and second orders levelling quasigeoid heights by the least squares collocation, the accuracy of the this model has been increased to to 20,69 %. That had been obtained due to taking into account the spatial dependences of the quasigeoid heights in the Earth gravity field on territory of Vietnam, With obtained accuracy of ± 0,058 m the cor- rected quasigeoid model VIGAC2017 may be used for solving of some tasks related to phys- ical geodesy in the initial spatial reference system VN2000-3D. A perfection of the national spatial reference system in relation to step by step ac- curacy improvement of the national quasigeoid model is iterative process. 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