[논문]가우스-헬머트 모델 전최소제곱: 평면방정식과 측지좌표계 변환

배태석; 홍창기; 임수현

doi:10.7848/ksgpc.2022.40.4.315

가우스-헬머트 모델 전최소제곱: 평면방정식과 측지좌표계 변환
TLS (Total Least-Squares) within Gauss-Helmert Model: 3D Planar Fitting and Helmert Transformation of Geodetic Reference Frames 원문보기

한국측량학회지 = Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography, v.40 no.4, 2022년, pp.315 - 324

배태석 (Dept. Geoinformation Engineering, Sejong University) , 홍창기 (Dept. Geoinformatics Engineering, Kyungil University) , 임수현 (Geoinformation Engineering, Sejong University)

초록
AI-Helper

일반적인 조정계산에서는 독립변수의 오차는 없다고 가정하고 종속변수의 오차만을 고려하는 최소제곱해를 구한다. 그러나 지상측량에 의해 결정한 3차원 공간좌표나 GNSS (Global Navigation Satellite System) 기반 추정좌표는 성분별로 독립적으로 결정되지 않으므로 모든 성분에 오차가 있을 뿐만 아니라 공분산도 존재한다. 따라서 좌표쌍을 이용한 평면 추정이나 좌표계 변환에서는 모든 성분의 오차를 고려하는 전최소제곱을 적용해야 한다. 이를 위한 다양한 모델이 존재하며, 특별한 제약조건을 제외하면 동등한 해를 제공한다. 본 연구에서는 가우스-헬머트 모델(GHM: Gauss-Helmert Model) 기반 전최소제곱으로 VLBI 타겟이 형성하는 자취를 이용하여 평면의 법선벡터를 추정했으며, 지역좌표계를 세계측지계로 변환하는 계수 결정에도 적용했다. 평면방정식의 경우 기존 최소제곱 방법과 비교해서 법선벡터는 동일하지만 분산요소의 안정성과 타겟 위치에 따른 분산요소 특성을 명확히 확인할 수 있었다. 좌표계 변환계수는 가우스-헬머트 모델을 적용하면 변환 전후 두 좌표계에서 모두 잔차를 계산할 수 있으며, 기존 방식보다 잔차가 더 작아진다.

Abstract ▼ AI-Helper

The conventional LESS (LEast-Squares Solution) is calculated under the assumption that there is no errors in independent variables. However, the coordinates of a point, either from traditional ground surveying such as slant distances, horizontal and/or vertical angles, or GNSS (Global Navigation Satellite System) positioning, cannot be determined independently (and the components are correlated each other). Therefore, the TLS (Total Least Squares) adjustment should be applied for all applications related to the coordinates. Many approaches were suggested in order to solve this problem, resulting in equivalent solutions except some restrictions. In this study, we calculated the normal vector of the 3D plane determined by the trace of the VLBI targets based on TLS within GHM (Gauss-Helmert Model). Another numerical test was conducted for the estimation of the Helmert transformation parameters. Since the errors in the horizontal components are very small compared to the radius of the circle, the final estimates are almost identical. However, the estimated variance components are significantly reduced as well as show a different characteristic depending on the target location. The Helmert transformation parameters are estimated more precisely compared to the conventional LESS case. Furthermore, the residuals can be predicted on both reference frames with much smaller magnitude (in absolute sense).

주제어

표/그림 (7)

그림 Fig. 1. Location of targets on VLBI antenna (NGII, 2019)
그림 Fig. 2. Azimuthal rotation of the targets in 3-dimensional space
그림 Fig. 3. Comparison of variance component estimates for two different estimation models
표 Table 1. The normal vector of each plane comprised of VLBI targets (TLS within GHM)
-6mm²]" alt="Table 2. Coordinates and the variance-covariance of two frames. "FROM" represents the local frame (e,n,u) based on ground surveying, and "TO" is obtained from GNSS data processing. Unit: [m], [10^-6mm²]" onerror="this.src='/images/noimg.png'" class="objectfit" /> 표 Table 2. Coordinates and the variance-covariance of two frames. "FROM" represents the local frame (e,n,u) based on ground surveying, and "TO" is obtained from GNSS data processing. Unit: [m], [10^-6mm²]
표 Table 3. Estimated Helmert transformation parameters: TLS within GHM vs. Conventional LESS
그림 Fig. 4. Estimated residuals (in absolute sense) for both reference frames

참고문헌 (19)

Altamimi, Z., Rebischung, P., Metivier, L., and Collilieux, X. (2016), ITRF2014: A new release of the International Terrestrial Reference Frame modeling nonlinear station motions, Journal of Geophysical Research: Solid Earth, Vol. 121, pp. 6109-6131. https://doi.org/10.1002/2016JB013098

상세보기
Andrei, C.-O. (2006), 3D Affine Coordinate Transformations, Master's thesis, School of Architecture and the Built Environment, Royal Institute of Technilogy (KTH), Stockholm, Sweden.
Bae, T.-S. (2022), Adjustment Computations, Haum (in Korean).
Dach, R., Lutz, S., Walser, P., and Fridez, P. (2015), Bernese GNSS Software Version 5.2, Vol. 1, Austronomical Institute, University of Bern.
Ghilani, C.D. and Wolf, P.R. (2006), Adjustment Computations: Spatial data analysis (4th), John Wiley & Sons, Inc.
Ioannidou, S. and Pantazis, G. (2020), Helmert transformation problem. From Euler angles method to quaternion algebra, International Journal of Geo-Information, Vol. 9, No. 494, pp. 1-14.
ITRF (2020), ITRF homepage, https://itrf.ign.fr/en/solutions/ITRF2020 (last date accessed: 28 July 2022).
Jekeli, C. (2016), Geometric Reference Systems in Geodesy, Division of Geodetic Science, School of Earth Sciences, Ohio State University.
Metivier, L., Altamimi, Z., and Rouby, H. (2020), Past and present ITRF solutions from geophysical perspectives, Advances in Space Research, Vol. 65, No. 12, pp. 2711-2722. https://doi.org/10.1016/j.asr.2020.03.031

상세보기
Neitzel, F. and Schaffrin, B. (2016), On the Gauss-Helmert model with a singular dispersion matrix where BQ is of smaller rank than B, Journal of Computational and Applied Mathematics, Vol. 291, pp. 458-467. https://doi.org/10.1016/j.cam.2015.03.006

상세보기
NGII (2019), Enhancement of National Geodetic Network, National Geographic Information Institute, Suwon, Korea, 229p. (in Korean)
Schaffrin, B. (2007), Connecting the dots: The straight-line case revisited, zfv, Vol. 132, No. 1, pp. 385-394.
Schaffrin, B. and Snow, K. (2010), Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth, Linear Algebra and its Applications, Vol. 432, pp. 2061-2076.

상세보기
Schaffrin, B. and Wieser, A. (2008), On weighted total least-squares adjustment for linear regression, Journal of Geodesy, Vol. 82, pp. 415-421. https://doi.org/10.1007/s00190-007-0190-9

상세보기
Snow, K. (2002), Applications of Parameter Estimation and Hypothesis Testing to GPS Network Adjustments, OSU Report No. 465, Geodetic Science, The Ohio State University, Columbus, Ohio.
Snow, K. (2012), Topics in Total Least-Squares Adjustment within the Errors-In-Variables Model: Singular Cofactor Matrices and Prior Information, OSU Report No. 502, Geodetic Science, The Ohio State University, Columbus, Ohio.
Watson, G.A. (2006), Computing Helmert transformations, Journal of Computational and Applied Mathematics, Vol. 197, No. 2, pp. 387-394. https://doi.org/10.1016/j.cam.2005.06.047

상세보기
Zavoti, J. and Kalmar, J. (2016), A comparison of different solutions of the Bursa-Wolf model and of the 3D, 7-parameter datum transformation, Acta Geod Geophys, Vol. 51, pp. 245-256. https://doi.org/10.1007/s40328-015-0124-6

상세보기
Zhang, Z., Ma, X., Sun, Z., Zhang, A., Yuan, Y., and Sun, Z. (2021), Measuring the Deflection of the Vertical via Local Reference Point Surveying and Pointing Calibration of a VLBI Telescope: A Case Study at the Urumqi Station, Earth and Space Science, Vol. 8, No. 10, pp. 1-13. https://doi.org/10.1029/2021EA001781

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