JKAS Archive

Journal of the Korean Astronomical Society - Vol. 51 , No. 5

[ Article ]
Journal of the Korean Astronomical Society - Vol. 51, No. 5, pp.143-153
Abbreviation: JKAS
ISSN: 1225-4614 (Print) 2288-890X (Online)
Print publication date 31 Oct 2018
Received 07 Aug 2018 Accepted 29 Sep 2018
DOI: https://doi.org/10.5303/JKAS.2018.51.5.143

DETERMINATION OF THE INVARIANT POINT OF THE KOREAN VLBI NETWORK RADIO TELESCOPES: FIRST RESULTS AT THE ULSAN AND TAMNA OBSERVATORIES†
Sung-Moon Yoo1 ; Taehyun Jung1, 2 ; Sung-Mo Lee1 ; Ha Su Yoon1 ; Han-Earl Park1 ; Jong-Kyun Chung1 ; Kyoung-Min Roh1 ; Seog Oh Wi1 ; Jungho Cho1 ; Do-Young Byun1, 2, 3
1Korea Astronomy and Space Science Institute, 776 Daedukdae-ro, Yuseong-gu, Daejeon 34055, Korea (yoo@kasi.re.kr)
2University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Korea
3Department of Astronomy, Yonsei University, 134 Shinchon-dong, Seodaemun-gu, Seoul 03722, Korea

Correspondence to : S.-M. Yoo
Part of a Special Collection on KVN, KaVA, and EAVN


JKAS is published under Creative Commons license CC BY-SA 4.0.

Abstract

We present the first results of the invariant point (IVP) coordinates of the KVN Ulsan and Tamna radio telescopes. To determine the IVP coordinates in the geocentric frame (ITRF2014), a coordinate transformation method from the local frame, in which it is possible to survey using the optical instrument, to the geocentric frame was adopted. The least-square circles are fitted in three dimensions using the Gauss-Newton method to determine the azimuth and elevation axes in the local frame. The IVP in the local frame is defined as the mean value of the intersection points of the azimuth axis and the orthogonal vector between the azimuth and elevation axes. The geocentric coordinates of the IVP are determined by obtaining the seven transformation parameters between the local frame and the east-north-up (ENU) geodetic frame. The axis-offset between the azimuth and elevation axes is also estimated. To validate the results, the variation of coordinates of the GNSS station installed at KVN Ulsan was compared to the movement of the IVP coordinates over 9 months, showing good agreement in both magnitude and direction. This result will provide an important basis for geodetic and astrometric applications.


Keywords: instrumentation: interferometers, telescopes, reference systems

1. INTRODUCTION

Very Long Baseline Interferometry (VLBI) is a powerful tool for achieving extremely high angular resolution (milli/micro-arcsecond levels) by utilizing an aperture synthesis technique, and also for measuring the time delays between radio telescopes by observing radio sources in the universe.

The first characteristic provides a unique opportunity to study compact objects such as quasars, active galactic nuclei, and astronomical masers, and the second is particularly important for astrometry and geodesy, with great potential in astrophysics, the determination of universal time UT1, and space navigation. By measuring the time delay between two radio telescopes, we can determine either the position of radio sources (astrometry) and/or the position of telescopes (geodesy). For both measurements, it is crucial to determine an accurate reference position of the radio telescope and to monitor the variation caused by crustal movement and gravitational deformation.

This reference position is often called the invariant point (IVP), and two different definitions are typically used: (a) the intersection of the primary axis (i.e., fixed axis), with the perpendicular between the secondary axis (i.e., moving axis) and primary axis (Johnston & Dawson 2004; Johnston et al. 2004); and (b) the intersection of the primary axis with the plane perpendicular to the fixed axis that also contains the secondary axis (Ma 1978; Harvey 1991).

In general, the IVP is located at the azimuth (Az) axis of a radio telescope in the case of altitude-azimuth mounting. An offset between the elevation (El) and Az axes often exist due to the telescope type or imperfections in the telescope construction. In most cases, the IVP is not physically identified on the radio telescope. Thus, a conventional method of determining the IVP is to carry out geodetic surveys toward targets (survey markers) on the structure of a radio telescope using a total station.

In a previous study, Dawson et al. (2007) determined the IVP via an indirect method using several geometric conditions. The optimization problem was solved using geometrical models as constraints to determine the reference point of the Az-El telescope (Sarti et al. 2004). Li et al. (2014) analyzed and tested several transformation parameters for the case of different local frames to meet the required precision for the reference point coordinates. However, due to the various types of telescopes and site conditions, there is no general solution to define the IVP coordinates.

The Korean VLBI Network (KVN) is the first VLBI facility in Korea, with three 21-m radio telescopes in Seoul, Ulsan, and Jeju Island, operated by Korea Astronomy and Space Science Institute (KASI). As a dedicated millimeter VLBI (mm-VLBI) network, the KVN is designed to observe four different radio frequency bands (22, 43, 86, and 130 GHz) simultaneously (Han et al. 2013; Lee et al. 2014).

KVN has been participating in K band (22 GHz) geodetic VLBI observations together with VLBI exploration of radio astrometry (VERA) of the National Astronomical Observatory in Japan on behalf of the KaVA (KVN and VERA Array) geodesy program since 2011.1

Initilally, the IVPs of the KVN radio telescopes were roughly estimated by setting up two GNSS antennas on both sides of the elevation axis. Later, the result was updated using the KaVA K band geodesy.2 According to preliminary results for KaVA K band geodesy, we found that the IVPs obtained for the KVN radio telescopes are not stable enough to achieve desired accuracy (a few millimeters). KaVA astrometry and geodesy activities are now extended to the East Asian VLBI Network (EAVN) including Chinese telescopes. Thus, accurate determination of the reference position of the radio telescope is crucial for high-precision astrometry and geodesy applications.

To measure the IVP of KVN telescopes, we performed geodetic surveys at the KVN Ulsan and Tamna sites in a conventional way for the first time in 2017.

For the KVN IVP determination, we defined IVP as the centroid position of the line segment on the Az axis of four nearest points from the El axes. We analyzed the optical survey data to produce the IVP coordinates in the three-dimensional (3D) geocentric frame determined from the coordinate transformation described in detail in the following sections using the relative position vector of each pillar.

In this paper, we describe the overall processes used for IVP determination in Section 2. The geodetic surveys and their analysis are explained in Section 3. The final results of the IVP determination are presented in Section 4. The conclusion and discussion are followed by future prospects, in Section 5.


2. IVP DETERMINATION

The ultimate goal of the IVP survey is to determine the geocentric coordinates of the IVP of a radio telescope, which is usually not identified directly. In order to obtain the 3D geocentric coordinates of the IVP, it is necessary to transform from the local frame in which the IVP survey can be performed to the geocentric frame.

As shown in Figure 1, three different coordinate systems are introduced: (1) the 3D geocentric frame known as the Earth-centered Earth-fixed (ECEF) frame, (2) the local East-North-Up (ENU) frame, in which pillar 1 (P1) is taken as the origin referencing the WGS84 ellipsoid, and (3) the Local Ground Network (LGN) frame, in which P1 is taken as the local origin, with the direction of North-East determined by the GNSS coordinates of each pillar position as the y-x axis and a right-handed direction as the z-axis.


Figure 1. 
The relationship among different coordinate systems of geocentric, East-North-Up (ENU), and Local Ground Network (LGN) frames. λ and δ are the longitude and the latitude of Pillar 1, respectively.

The IVP determination process is divided into two steps. First, the IVP coordinates in the LGN frame are determined from the local network survey at each telescope site. Next, a coordinate transformation between the LGN and geocentric frames is adopted.

The first procedures, described in detail in Section 3, are (a) defining the LGN at each site, (b) optical surveys using a total station to measure the reference positions of pillars, (c) optical surveys of targets on a radio telescope referring to the reference position of the pillars, and (d) GNSS observations to determine the geocentric coordinates of the pillars. The coordinate transformation procedures are described in Section 4 and presented schematically in Figure 2: (e) data reduction to determine the geocentric coordinates of pillars from GNSS measurements in ITRF2014, (f) calculation of relative vectors in the LGN in ITRF2014, (g) transformation of the relative vector coordinates from the geocentric frame to the ENU frame, (h) determination of seven parameters (three transition (T), three rotation (R), and one scale (γ)) between the LGN and ENU frames, (i) coordinate transformation of IVP in the LGN to ENU frames, and (j) ENU coordinate transformation of IVP to geocentric coordinates in ITRF2014.


Figure 2. 
Flow chart of the IVP determination process.


3. LOCAL GROUND NETWORK SURVEY
3.1. Local Ground Network (LGN)

The LGN was configured for the IVP survey at both the KVN Ulsan and Tamna sites as shown in Figure 3. Four pillars (P1 to P4) were installed at each site by considering the visibility among pillars and targets marked on the telescope. P1 and P2 were placed on top of the observing building, and the other two pillars (P3 and P4) were located in the observatory yard. We defined the reference pillar as P1 in the local xyz coordinates, and the relative coordinates of other pillars to P1 are presented in Table 1. The typical uncertainty is around 1mm according to the specification of the optical instrument, which is described in Section 3.2.


Figure 3. 
Left, Satellite map of the (a) KVN Ulsan and (b) Tamna sites. The x-y axes in the LGN are presented. Right, Top view of the LGN with the projected distance among the pillars and telescope.

Table 1 
Pillar coordinates in the LGN
Pillar Ulsan Tamna
x (m) y (m) z (m) x (m) y (m) z (m)
P1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
P2 3.5938 −27.1200 0.0086 52.0888 −14.6295 −0.0102
P3 −52.5445 −34.3594 −3.9492 54.5270 −48.2284 −8.3467
P4 −46.0209 −2.1321 −3.6020 8.1343 −59.8100 −9.0733

All pillars P1 to P4 were directly visible from each other so that the relative positions could be monitored. At the Tamna site, the position of P4 was very close to the telescope, and limited optical surveys were made due to the high elevation angles of targets on the telescope.

In the LGN, the direction of the y-axis was set to a northward direction and the x-axis 90 degrees from the y-axis in the clockwise direction. The z-axis is defined by the right-hand rule.

3.2. LGN Survey

For optical surveys in the LGN, we used the Leica Nova MS60 Multistation (hereafter MS60) as an optical instrument. The observational uncertainties in angle and distance measurements for MS60 are 1 arcsecond and 1 mm+1.5 ppm, respectively. The Trimble GNSS choke ring antennas and NetR9 receivers were used to determine the geocentric coordinates of pillars in ITRF2014.

The reference point of each pillar was defined as the central point of the leveling head. Both MS60 and GNSS antennas were installed accurately at the same reference point on the pillar. Fourteen Leica reflective tape targets were stuck onto the radio telescopes with numbers as shown in Figure 4. The location of targets was almost the same for both the Ulsan and Tamna telescopes.


Figure 4. 
Positions of 14 targets on the radio telescope (a and b), and (c) the LGN survey photo using MS60. The curved arrows present the rotating direction of the telescope for optical surveys in the Az and El axes. The Leica reflective tape target (number 14) is shown as an example.

The optical surveys using MS60 were conducted from P1 to P4 in turn on 21 Nov 2017 at the Ulsan site and on 3 Dec 2017 at the Tamna site as follows: (1) Az axis (to determine the vertical axis) surveys were done by rotating the telescope to the azimuthal direction from 0◦ to 360◦ with 15◦ increments while the elevation of the telescope was fixed at 89◦. (2) El axis (to determine the horizontal axis) surveys were conducted to an elevation of 7, 10, 20, 30, 40, 50, 60, 70, 80, and 89 degrees at each pillar (P1 to P4) with four different azimuthal positions (290, 348, 130, 190 degrees at Ulsan and 232, 313, 0, 106 degrees at Tamna). A different azimuthal position at each pillar was selected as the most effective telescope position to observe targets. The total number of measurements obtained from optical surveys of the Az and El axes were 332 and 175 points at Ulsan and 361 and 163 points at Tamna, respectively. The Az and El axes were estimated from the 3D circle fitting method described in Section 4.

Four sets of GNSS antennas and receivers were placed at each pillar and GNSS observations were performed for 3 days after the optical survey to measure the geocentric coordinates of each pillar.


4. DATA REDUCTION
4.1. Determination of Pillar Position in the Geocentric Frame

The GNSS data analysis was carried out using Bernese V5.2 software (Dach et al. 2015). The International GNSS Service (IGS) stations of BJFS, CHAN, DAEJ, GMSD, MIZU, SHAO, and ULAB were used for the datum definition in the frame of ITRF2014. The reference epoch of the pillar position at the KVN Ulsan and Tamna sites was set to UT 12:00 in the middle of the 3-day GNSS observations. The formal error of coordinates is at the few millimeter level. In Table 2, the geocentric coordinates of the four pillars in the LGN at the KVN Ulsan and Tamna sites are presented.

Table 2 
Geocentric coordinates of pillars in ITRF2014
Site Pillar X (m) Y (m) Z (m) σx (m) σy (m) σz (m)
Ulsan P1 −3287264.21114 4023422.81525 3687396.09969 0.00042 0.00052 0.00049
P2 −3287282.79078 4023425.95284 3687376.26405 0.00044 0.00052 0.00051
P3 −3287250.98763 4023466.90358 3687353.23254 0.00049 0.00057 0.00057
P4 −3287235.66074 4023455.34428 3687379.92395 0.00044 0.00056 0.00053
Tamna P1 −3171696.78067 4292674.78482 3481064.33950 0.00034 0.00043 0.00038
P2 −3171741.97846 4292645.47595 3481059.31399 0.00034 0.00044 0.00039
P3 −3171754.82704 4292649.67298 3481027.35476 0.00035 0.00045 0.00039
P4 −3171725.32804 4292683.63505 3481011.12446 0.00036 0.00046 0.00040
† The epochs of GNSS observations at KVN Ulsan and Tamna are set to UT 12:00 on November 23 and UT 12:00 on December 6 in 2017, respectively.

4.2. Determination of IVP in the LGN

The IVP determination of the radio telescopes is illustrated in Figure 5. Since the IVP is the intersection of the vertical (Az) and horizontal (El) axes, it is not directly accessible in most cases. Based on the results of the optical survey described in Section 3.2, the targets form circles and arcs from the Az and El surveys, respectively.


Figure 5. 
Illustration of IVP determination. The IVP marked in bold on the Az axis is the invariant point to be determined. The circles are formed by an optical survey of targets. IV P1 and IV P2 are the nearest points on the Az axis. a1 and a2 represent the nearest distance vectors from the El axis perpendicular to the Az axis. Dr and Dy represent the distance from the measured target point to the circle plane and cylinder perpendicularly.

The Az and El axes can be determined by applying 3D line fitting to the center points of circles calculated from the 3D circle fitting of target positions. The 3D circles are fitted by applying the Gauss-Newton least-square method, which minimizes the square root of D2r and D2y, where Dr and Dy are the distance from the measured target point to the circle plane and the cylinder perpendicularly. The results of 3D circle fitting are shown in Figure 6 and Table 3 for the El survey, and in Figure 7 and Table 4 for the Az survey.

Table 3 
Properties of least-square circles around the El axis in the LGN
Target IDa
Ulsan
XYZb (m) nc σd (mm) re (m)
xc yc zc x y z
P1T2 -19.50880 -17.96697 10.10655 -0.6347 -0.7727 0.0001 0.428 1.3571
P1T3 -19.50640 -17.96603 10.10378 -0.6350 -0.7725 0.0012 0.607 2.6577
P1T6 -21.44205 -20.32353 10.10759 -0.6336 -0.7736 0.0018 0.471 3.7913
P1T10 -21.94497 -20.93538 10.10560 -0.6346 -0.7728 0.0002 0.368 4.2614
P1T12 -21.02154 -19.80471 10.10561 -0.6348 -0.7727 0.0001 0.241 3.4523
P2T2 -18.11957 -21.47434 10.10389 -0.9917 0.1285 -0.0002 0.132 1.3589
P2T3 -18.11233 -21.47200 10.10654 -0.9917 0.1288 -0.0008 0.186 2.6534
P2T6 -21.13084 -21.08025 10.10589 -0.9916 0.1295 -0.0007 0.327 3.7934
P2T10 -21.92726 -20.97829 10.10362 -0.9917 0.1288 0.0000 0.197 4.2625
P3T2 -25.32721 -22.95174 10.10660 -0.8608 -0.5090 -0.0001 0.612 1.3578
P3T3 -25.33085 -22.95581 10.10583 -0.8608 -0.5090 0.0000 0.628 2.6564
P3T6 -22.70848 -21.40516 10.10686 -0.8606 -0.5092 -0.0003 0.463 3.7933
P3T10 -22.02034 -20.99775 10.10470 -0.8606 -0.5093 0.0000 0.604 4.2634
P3T12 -23.27782 -21.73981 10.10394 -0.8607 -0.5091 0.0000 0.309 3.4489
P4T2 -25.36557 -19.06175 10.10320 -0.8710 0.4913 -0.0013 0.625 1.3600
P4T3 -25.36807 -19.06122 10.09728 -0.8710 0.4913 -0.0016 0.760 2.6636
P4T6 -22.70863 -20.55870 10.10828 -0.8721 0.4893 -0.0024 0.584 3.7910
P4T10 -22.02071 -20.94648 10.10432 -0.8713 0.4907 -0.0003 0.284 4.2620
P4T12 -23.29483 -20.23295 10.10440 -0.8712 0.4910 -0.0001 0.280 3.4525
Tamna
P1T2 18.14834 -34.74716 5.76978 0.4795 -0.8776 0.0007 0.432 1.3779
P1T3 18.15214 -34.75786 5.75977 0.4791 -0.8777 0.0042 1.186 2.6527
P1T6 19.61512 -37.44366 5.77156 0.4827 -0.8758 0.0052 1.211 3.7616
P1T10 19.99577 -38.12808 5.77272 0.4816 -0.8764 0.0004 0.406 4.2532
P2T2 23.09700 -35.78658 5.77899 -0.7916 -0.6110 -0.0002 0.481 1.3700
P2T3 23.09649 -35.78782 5.77934 -0.7916 -0.6110 0.0000 0.185 2.6358
P2T6 20.68438 -37.64949 5.77719 -0.7916 -0.6110 -0.0001 0.360 3.7587
P2T10 20.05510 -38.13919 5.77394 -0.7915 -0.6111 0.0000 0.515 4.2545
P3T2 23.85676 -38.80324 5.77701 -0.9868 0.1621 0.0000 0.697 1.3738
P3T3 23.85777 -38.79938 5.77437 -0.9868 0.1622 -0.0008 0.554 2.6420
P3T6 20.84861 -38.30166 5.77522 -0.9868 0.1618 0.0000 0.658 3.7634
P3T10 20.06477 -38.17434 5.77730 -0.9867 0.1625 0.0000 0.931 4.2604
P3T12 21.45519 -38.40900 5.77621 -0.9869 0.1614 0.0000 0.691 3.4116
P4T3 18.59333 -41.78840 5.76278 -0.3641 -0.9314 -0.0025 1.178 2.6520
P4T6 19.70961 -38.93365 5.77597 -0.3594 -0.9332 -0.0057 1.123 3.7586
P4T10 19.99130 -38.21119 5.77539 -0.3627 -0.9319 -0.0002 0.200 4.2561
a The numbers following “P” and “T” represent the pillar number and target ID, respectively.
b XYZ are the coordinates of a circle center with components (xc, yc, zc).
c n is the normal direction vector of a circle.
d σ is the standard deviation of residual errors.
e r is circle radius.

Table 4 
Properties of Least-Square Circles around the Az Axis in the LGN
Target ID
Ulsan
XYZa (m) nb σc (mm) rd (m)
xc yc zc x y z
1 -21.97810 -20.97183 12.01178 0.000039 -0.000024 1.000000 0.736 5.0067
2 -21.97825 -20.97191 11.46379 0.000188 -0.000056 1.000000 0.881 3.8897
3 -21.97831 -20.97179 11.98714 0.000108 -0.000209 1.000000 1.488 4.3222
4 -21.97829 -20.97246 10.11046 0.000124 -0.000220 1.000000 1.907 4.7255
5 -21.97773 -20.97168 10.10912 0.000187 0.000022 1.000000 1.058 4.7238
6 -21.97803 -20.97169 9.75453 0.000008 -0.000433 1.000000 0.697 3.8713
7 -21.97818 -20.97237 9.71474 0.000075 0.000120 1.000000 1.121 3.4991
8 -21.97831 -20.97158 8.52004 0.000047 0.000009 1.000000 1.312 6.8392
9 -21.97817 -20.97141 5.04031 0.000079 -0.000128 1.000000 0.873 7.3666
11 -21.97808 -20.97164 7.70459 0.000123 -0.000070 1.000000 0.833 5.3170
13 -21.97855 -20.97130 2.72509 0.000074 -0.000097 1.000000 0.968 6.8367
14 -21.97809 -20.97141 2.85736 0.000246 -0.000135 1.000000 0.809 1.8783
Tamna
2 20.01183 -38.16824 7.14698 -0.000095 0.000284 1.000000 0.854 3.8980
4 20.01182 -38.16778 5.77944 -0.000196 0.000300 1.000000 0.981 4.7341
5 20.01202 -38.16780 5.77856 -0.000163 0.000120 1.000000 0.592 4.7253
8 20.01175 -38.16796 4.15210 -0.000099 0.000119 1.000000 0.924 6.8419
9 20.01131 -38.16834 0.68840 -0.000014 0.000197 1.000000 1.003 7.3592
11 20.01165 -38.16804 3.21849 -0.000099 0.000206 1.000000 0.905 5.2910
13 20.01168 -38.16774 -1.65793 -0.000132 0.000207 1.000000 0.969 6.8331
14 20.01161 -38.16818 -1.52480 -0.000632 0.000630 1.000000 0.759 1.8700
15 20.01179 -38.16763 7.16555 -0.000452 0.000161 1.000000 1.015 3.8912
a XYZ are the coordinates of a circle center with components (xc, yc, zc).
b n is the normal direction vector of a circle.
c σ is the standard deviation of residual errors.
d r is circle radius.

In Tables 3 and 4, xc, yc and zc represent the center coordinates of circles formed by each target, and n denotes the normal direction vector of the circle. The standard deviations (σ) of the residual errors and circle radii (r) are presented. Targets with σ > 2 mm were discarded in the following analysis. A total of 6 targets (T1, T3, T6, T7, T10, T12) at Tamna and 3 targets (T10, T12, T15) at Ulsan were removed mainly due to worse meteorological conditions during the optical survey for determining the Az axis.

The Az and El axes in Figures 6 and 7 are estimated by 3D circle fitting. The dotted line in Figure 6 represents the projection of the El axis on the x-y plane. The cross points represent central points of the circles.


Figure 6. 
The results of 3D circle fitting for the El (horizontal) axis determined at four Az positions ((a) to (d) for Ulsan and (e) to (h) for Tamna) in the LGN. The dotted line represents the projection of the El axis onto the x-y plane.


Figure 7. 
The results of 3D circle fitting for the Az (vertical) axis in the LGN. The thick line represents the El axes shown in Figure 6.

In Table 5, the residual errors between cross points and each axis determined by the least-square line fitting method are presented. The directional cosine (u) of the Az and El axes and the residual distance (Rd) between the estimated axis and the center of the circles are also shown. Rd is smaller at Tamna than Ulsan because circles with large uncertainty at Tamna were removed.

Table 5 
Az and El Axes determined by least-square lines in 3D
Ulsan Az El 1 El 2 El 3 El 4
uxa 0.00002 -0.63461 -0.99166 0.86082 0.87120
uya -0.00007 -0.77283 0.12892 0.50890 -0.49093
uza 1.00000 0.00032 -0.00020 -0.00027 0.00159
Rdb (m) 0.00017 0.00133 0.00225 0.00225 0.00277
0.00013 0.00181 0.00195 0.00195 0.00327
0.00028 0.00187 0.00147 0.00147 0.00307
0.00061 0.00123 0.00116 0.00116 0.00252
0.00045 0.00304 0.00292
0.00020
0.00053
0.00023
0.00012
0.00012
0.00029
0.00019
||Rd||c (m) 0.00111 0.00439 0.00351 0.00285 0.00654
Tamna
uxa -0.00004 0.47917 -0.79124 -0.98661 0.36404
uya -0.00002 -0.87772 -0.61150 0.16310 0.93138
uza -1.00000 0.00211 -0.00112 0.00023 0.00357
Rdb (m) 0.00035 0.00514 0.00036 0.00207 0.00038
0.00014 0.00508 0.00049 0.00277 0.00190
0.00025 0.00344 0.00198 0.00294 0.00152
0.00001 0.00274 0.00157 0.00096
0.00044 0.00423
0.00009
0.00037
0.00013
0.00027
||Rd||c (m) 0.00079 0.00846 0.00260 0.00627 0.00246
a ux, uy, and uz are the components of direction cosines.
b Rd is the residual distance between the estimated axis and the center of circles.
c ||Rd|| is the norm of Rd.

In general, the Az and El axes do not intersect due to the telescope type, imperfections in construction, and gravitational deformations. Therefore, the distance from the El axis to the Az axis can be determined by calculating the magnitude of the vectors (a2 shown in Figure 5) perpendicular to each axis. In this paper, the axis offset between the Az and El axes is determined as the mean value of the magnitude of a2 . The IVP coordinate in the LGN is calculated as the centroid point of the line segment on the Az axis of the four nearest points from the El axes listed in Table 6.

Table 6 
IVP Coordinates in the LGN
Ulsan El 1 El 2 El 3 El 4
Nearest points on Az axis (m) x -21.97815 -21.97815 -21.97815 -21.97815
y -20.97187 -20.97187 -20.97187 -20.97187
z 10.10648 10.10454 10.10502 10.10673
Distance to the Az axis (m) 0.00162 0.00088 0.00068 0.00026
Axis offset (mm) 0.86 ± 0.49
IVP in LGN (m) x -21.97815
y -20.97187
z 10.10569
Tamna
Nearest points on Az axis (m) x 20.01180 20.01180 20.01180 20.01180
y -38.16792 -38.16792 -38.16792 -38.16792
z 5.77302 5.77492 5.77649 5.77705
Distance to the Az axis (m) 0.00229 0.00256 0.00180 0.00306
Axis offset (mm) 2.43 ± 0.45
IVP in LGN (m) x 20.01180
y -38.16792
z 5.77537

4.3. IVP Determination in the Geocentric Frame

Shown in Figure 8, the position vectors of pillar i and pillar j (i, j=1,2,3,4 and ij) are represented in each frame. Pi and Pj are the position vectors of pillar i and pillar j in the LGN frame in which the optical local survey was carried out. Xpi and Xpj are the position vectors of pillar i and pillar j in the geocentric frame that can be determined by GNSS observation. In order to transform from the LGN to geocentric frame, the ENU frame is introduced as an the intermediate step.


Figure 8. 
The position and relative vector of pillar i and j in the geocentric, ENU, and LGN frames

The Helmert transformation, also known as the similarity transformation, consists of seven parameters (three transition (T), three rotation (R), and one scale (γ)) and is commonly used to convert coordinates from one reference frame to another.

The pillar positions (P) in the LGN can be transformed to the ENU frame using Equation (1):

S21S31S41=T+γRP21P31P41(1) 

where

P21=P2-P1,P31=P3-P1P41=P4-P1.

Pi(i = 1, 2, 3, 4) is the position vector of the pillar in the LGN and subscripts are the pillar indices. Pij is the relative vector between two pillars. Corresponding positions of pillars in the ENU frame are expressed as S with subscripts.

The seven transformation parameters (T, γ, and R) in Equation (1) can be estimated by least-squares adjustment as provided in Table 7.

Table 7 
Seven transformation parameters
Tamna Ulsan
Value σ Value σ
Tx (mm) -0.93 1.71 1.56 1.11
Ty (mm) 2.14 1.72 -1.65 1.11
Tz (mm) 0.14 2.04 -0.60 1.30
Rx (rad) -0.0000698 0.0000447 0.0000936 0.0000539
Ry (rad) 0.0000034 0.0000441 0.0000636 0.0000316
Rz (rad) -0.1625289 0.0000317 -0.3387407 0.0000270
scale 0.9999559 0.0000312 0.9998985 0.0000267

The difference in dimensions and magnitude of scale between geocentric and geodetic frames make it difficult to obtain stable solutions for the seven parameters required for the transformation. Therefore, the relative vectors of pillars depicted in Figure 8 (Pij , Sij , Xij) are used to obtain stable solutions.

The pillar coordinates in the ENU frame can be transformed from the geocentric frame using the Euler angle as follows:

S21S31S41=Rxδ-π2Rz-λ+π2XP21XP31XP41(2) 

where

Rxθ=1000cosθ-sinθ0sinθcosθRzθ=cosθ-sinθ0sinθcosθ0001XP21=XP2-XP1,XP31=XP3-XP1,XP41=XP4-XP1,

δ and λ are the latitude and the longitude of P1 (Ulsan; λ=129.249914 and δ=35.545849 degrees, and Tamna; λ=126.459308 and δ=33.289319 degrees). Xpi(i = 1, 2, 3, 4) represents the geocentric position vector of pillars with subscripts representing the pillar index. In the same manner, Xpij represents the relative vector.

Using the seven transformation parameters obtained from Equation (1) as listed in Table 7, the IVP coordinate in the ENU frame (Sivp) can be obtained using Equation (3):

Sivp=T+γRPivp.(3) 

where Pivp is the position vector of the IVP in the LGN. Sivp can be transformed to the geocentric frame by the Euler transformation:

Xivp=Rzπ2+λRxπ2-δSivp.(4) 

Finally, the IVP position vector in the geocentric frame (XIVP ) is determined by Equation (5):

XIVP=XP1+Xivp.(5) 

The geocentric coordinates of the IVP in ITRF2014 are listed in Table 8.

Table 8 
IVP Coordinates in ITRF2014
IVP Ulsan Tamna
2017 Nov. 23, UT12: 2017 Dec. 06, UT12:00
X -3287268.72004 -3171731.72457
Y 4023450.07902 4292678.45749
Z 3687379.93904 3481038.73301


5. DISCUSSION AND CONCLUSION

We have determined the IVP coordinates of the KVN Ulsan and Tamna radio telescopes by performing an optical survey in the LGN. The 3D circle fitting method was applied to the optical survey data. The mean of the standard deviation (MSD) of residuals for the Az/El axes are 1.06 mm/0.43 mm (Ulsan) and 0.89 mm/0.66 mm (Tamna), respectively. Given the measurement accuracy of the optical instrument (MS60), this result is reasonable. The Az (vertical) and El (horizontal) axes were estimated by using the 3D least-square line fitting method that minimizes the distance between the center points of circles and the axis to be determined. The MSD of residual distances for the Az/El axes are 0.28 mm/2.10 mm (Ulsan) and 0.23 mm/2.35 mm (Tamna), respectively.

Although the number of circles generated by the optical survey for targets on the two telescopes are different due to the different observing conditions, we find that the difference in MSD of residuals between the Ulsan and Tamna telescopes obtained from both 3D circle and line fittings were similar to within 0.25 mm, implying that the precision of measurements made in the optical survey is reliable. However, the norms of residual distance on the Az/El axes are at the level of a few millimeters, and the MSD of the El axis determination is an order of magnitude larger than that of the Az axis. This is because the El axis is determined using fewer center points (3 ∼ 5) of one-fourth the total circumference data, but the gravitational deformation caused by moving the telescope in an elevation direction may also contribute. The axis-offsets were estimated as 0.86 ± 0.49 mm (Ulsan) and 2.43 ± 0.45 mm (Tamna), and then the IVP in the LGN was determined (Table 6).

For the coordinate transformation, there are several different methods for estimating the transformation parameters. In this paper we introduced the relative position vectors adjusting the magnitude of parameters to improve solution stability for the seven parameter estimation in the Helmert transformation. The final IVP in the geocentric frame was determined by applying the seven parameters to IVP in the LGN. There is a small discrepancy between the scale factors shown in Table 7 and 1.0, which could be caused by the systematic bias of MS60.

To verify the IVP coordinates, we compared the positional change of the GNSS station installed on the observation building roof (see Figure 3) between February to November 2017 with IVP coordinates in the geocentric frame as shown in Table 9. On February 22, 2017, we carried out a simplified optical survey on the ground makers as a prevalidation study in the same manner described in this paper. Although no pillar was available at that time, the IVP was obtained successfully. The standard deviations of the coordinates of GNSS stations at Ulsan ranged from 0.3−0.7 mm and the IVP coordinates had estimated standard deviations based on the uncertainty of MS60 (∼ 1 mm) ranging from 1.1 − 1.3 mm.

Table 9 
Geocentric coordinates of Ulsan GNSS station and IVP
Epoch Ulsan GNSS Station IVP at Ulsan
X Y Z X Y Z
Feb. 22, 2017 -3287287.79036 4023415.99211 3687390.62040 -3287268.69891 4023450.09048 3687379.95209
Nov. 23, 2017 -3287287.81717 4023415.98445 3687390.61734 -3287268.72004 4023450.07902 3687379.93904

Table 10 clearly shows good agreement in both the direction and magnitude of the GNSS station and IVP coordinates in the ENU frame, implying that it is mainly due to the crustal movement of the LGN over 9 months. The standard deviations of GNSS station and IVP coordinate difference are 0.7 and 1.7, respectively. Although the ‘Up’ direction shows an opposite movement with a difference of 1.1 cm, we think this discrepancy may be caused by the types of loading due to the weight of the telescope itself or by a lower accuracy of GNSS measurement in the height direction. Thus, the secular variation in IVP to the height direction needs to be checked and monitored to identify the reason for this discrepancy.

Table 10 
Magnitude and direction of the coordinate difference (from Feb. 22 to Nov. 23, 2017)
Difference (mm) N E U
IVP at Ulsan 27.4 -13.2 23.6 -3.9
Ulsan GNSS Station 28.1 -8.9 25.6 7.2

There are many factors that can affect the accuracy of IVP determination, such as the precision of the optical survey instrument, 3D circle fitting methods, the coordinate transformation methods, and the GNSS data analysis strategies for data reduction and the duration of observation etc. Therefore, it is worthwhile to try different methods and compare the results. For example the optimization problem can be applied in another way by minimizing the performance index (that is, the sum of the squares of distances from the observed target points to the 3D circles), which satisfies the constraints including geometrical models. In addition, other coordinate transformation methods can be examined (e.g., Lehmann 2014; Mercan et al. 2018).

Defining a stable IVP for the radio telescope for VLBI astrometry and geodesy applications is essential to combine different space geodetic techniques, such as Satellite Laser Ranging (SLR), Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS), and GNSS by measuring the local tie vectors at the co-locate site where more than two techniques are available (Altamimi et al. 2011).

By monitoring the IVP, deformation of the telescope structure due to gravitation, and seasonal variation in temperature and pressure can be identified so as to improve the geophysical model for VLBI astrometry and geodesy. Recent advances in VLBI applications to space navigation demonstrated by Chang’E-2 and -3 missions (Li et al. 2012; Wei et al. 2013) also require a precise radio telescope model determined by the IVP.

Based on our IVP determination of the KVN Ulsan and Tamna telescopes, the results of KaVA K band geodesy since 2011 can now be updated, and high precision astrometric and geodetic VLBI observations using KaVA and EAVN will be promoted, especially at higher frequencies (22/43 GHz). Atmospheric effects (from both the troposphere and ionosphere) on VLBI observables will be examined and benefit from geodetic VLBI observations at high frequency (>20 GHz) together with GNSS observations. In addition, our study will provide an important basis for the multi-frequency VLBI phase referencing, which is capable of constraining the amount of delay errors separately, so that a new prospect on astrometry and geodesy will be explored with the KVN.

Finally, for the KVN Yonsei radio telescope the IVP determination using the method described in this paper is not applicable because of the limited area near the site for optical survey. Therefore, we are planning to use the gimbal-mounted GNSS antennas (e.g., Abbondanza et al. 2009; Kallio & Poutanen 2012; Ning et al. 2015) as an alternative method.


Notes
1The KVN status report (http://kvn.kasi.re.kr)

Acknowledgments

The authors are grateful to Dr. Sang-Hyun Lee and Jeong-Wook Hwang for helping to operate the telescopes and install equipment at the KVN Ulsan and Tamna sites. In particular, we want to acknowledge the invaluable support received from the experts (Seong-Hoon Kim, Dong-Keun Ji, Sung-Hyun Lee, and Kun-Hyung Lee) at Leica Geosystems in South Korea. The KVN is a facility operated by the KASI (Korea Astronomy and Space Science Institute). The KVN is supported through the high-speed network connections among the KVN sites provided by the KREONET (Korea Research Environment Open NETwork), which is managed and operated by KISTI (Korea Institute of Science and Technology Information).


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