A 3-D similarity transformation is frequently used to convert GPS-WGS84-based coordinates to those in a local datum using a set of control points with coordinate values in both systems. In this application, the Gauss-Markov (GM) model is often employed to represent the problem, and a least-squares approach is used to compute the parameters within the mathematical model. However, the Gauss–Markov model considers the source coordinates in the data matrix (A) as fixed or error-free; this is an imprecise assumption since these coordinates are also measured quantities and include random errors. The errors-in-variables (EIV) model assumes that all the variables in the mathematical model are contaminated by random errors. This model may be solved using the relatively new total least-squares (TLS) estimation technique, introduced in 1980 by Golub and Van Loan. In this paper, the similarity transformation problem is analyzed with respect to the EIV model, and a novel algorithm is described to obtain the transformation parameters. It is proved that even with the EIV model, a closed form Procrustes approach can be employed to obtain the rotation matrix and translation parameters. The transformation scale may be calculated by solving the proper quadratic equation. A numerical example and a practical case study are presented to test this new algorithm and compare the EIV and the GM models.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Akca D, Gruen A (2003) Generalized Procrustes Analysis and its applications in Photogrammetry. Internal Technical Report at IGP - ETH, Zurich, 23 pp
Arun KS, Huang TS, Blostein SD (1987) Least-squares fitting of two 3-D point sets. IEEE T Pattern Anal 9(5):698–700
Bazlov YA, Galazin VF, Kaplan BL, Maksimov VG, Rogozin VP (1999) Propagating PZ 90 to WGS 84 transformation parameters. GPS Solut 3(1):13–16. doi:10.1007/PL00012773
Crosilla F, Beinat A (2002) Use of generalized Procrustes analysis for the photogrammetric block adjustment by independent models. ISPRS J Photogramm 56(3):195–209. doi:10.1016/S0924-2716(02)00043-6
CSAT (2007) Geodetic Datums Reference System at: http://earth-info.nga.mil/GandG/coordsys/onlinedatum/index.html
Eggert DW, Lorusso A, Fisher RB (1997) Estimating 3-D rigid body transformations: a comparison of four major algorithms. Mach Vis Appl 9(5–6):272–290. doi:10.1007/s001380050048
Felus YA (2004) Applications of total least-squares for spatial pattern analysis. J Surv Eng 130(3):126–133. doi:10.1061/(ASCE)0733-9453(2004)130:3(126)
Felus YA, Schaffrin B (2005) Performing similarity transformations using the Error-In-Variables Model. In: ASPRS annual meeting, Baltimore (on CD)
Golub GH, Van Loan CF (1980) An analysis of the total least squares problem. SIAM J Numer Anal 17(6):883–893. doi:10.1137/0717073
Golub GH, Van Loan CF (1996) Matrix computations. Johns Hopkins University Press, Baltimore
Gower JC (1975) Generalized Procrustes analysis. Psychometrika 40(1):33–51. doi:10.1007/BF02291478
Gower JC, Dijksterhuis GB (2004) Procrustes problems. Oxford University Press, New York 248 pp
Grafarend EW (2006) Linear and nonlinear models: fixed effects, random effects, and mixed models. Walter de Gruyter, Berlin
Grafarend EW, Awange JL (2003) Nonlinear analysis of the three-dimensional datum transformation. J Geod 77:66–76. doi:10.1007/s00190-002-0299-9 conformal group C7(3)
Han JY, Van Gelder HWB (2006) Step-wise parameter estimations in a time-variant similarity transformation. J Surv Eng 132(4):141–148. doi:10.1061/(ASCE)0733-9453(2006)132:4(141)
Horn BKP (1987) Closed-form solution of absolute orientation using unit quaternions. J Opt Soc Am 4(4):629–642
Kashani I (2006) Application of generalized approach to datum transformation between local classical and satellite-based geodetic networks. Surv Rev 38(299):412–422
Koschat MA, Swayne DF (1991) A weighted procrustes criterion. Psychometrika 56(2):229–239. doi:10.1007/BF02294460
Kumar M (1989) A practical method to compute geoidal heights for local/regional datums. Mar Geod 13:183–187
Leick A (2004) GPS satellite surveying, 3rd edn. Wiley, Hoboken
Leick A, Van Gelder BHW (1975) On similarity transformations and geodetic network distortions based on Doppler satellite observations. Report No. 235, Dep. of Geodetic Sci., The Ohio State University, Columbus
Malys S (1988) Dispersion and correlation among transformation parameters relating two satellite reference frames. Report No. 392, Dep. of Geodetic Sci, The Ohio State University, Columbus
Markovsky I, Rastello ML, Premoli A, Kukush A, Van Huffel S (2006) The element-wise weighted total least-squares problem. Comput Stat Data Anal 50:181–209
Mikhail EM, Bethel JS, McGlone CJ (2001) Introduction to modern photogrammetry. Wiley, Chichester
Pope AJ (1972) Some pitfalls to be avoided in the iterative adjustment of nonlinear problems. In: Proceedings of the 38th annual meeting. American Society of Photogrammetry, Washington DC, pp. 479–497
Schaffrin B, Lee IP, Felus YA, Choi YC (2006) Total least-squares (TLS) for geodetic straight-line and plane adjustment. Boll Geodesia Sci Affini** 65(3):141–168
Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geodesy (in press)**
Schaffrin B, Felus YA (2008) On the multivariate total least-squares approach to empirical coordinate transformations: three algorithms. J Geod 82(6):373–383. doi:10.1007/s00190-007-0186-5
Shen YZ, Chen Y, Zheng DH (2006) A quaternion-based geodetic datum transformation algorithm. J Geod 80(5):233–239. doi:10.1007/s00190-006-0054-8
Teunissen PJG (1985) The geometry of geodetic inverse linear mapping and non-linear adjustment. Netherlands Geodetic Commission, Publications on Geodesy 8(1), Delft, 177 pp.*** http://www.ncg.knaw.nl/Publicaties/Geodesy/30Teunissen.html
Teunissen PJG (1988) The non-linear 2D symmetric Helmert transformation: an exact non-linear least-squares solution. J Geod 62(1):1–16
Umeyama S (1991) Least-squares estimation of transformation parameters between two point patterns. IEEE Trans Pattern Anal 13(4):376–380. doi:10.1109/34.88573
Van Huffel S, Vandewalle J (1991) The total least squares problem, computational aspects and analysis. Society for Industrial and Applied Mathematics, Philadelphia
Wolf PR, Ghilani CD (1997) Adjustment computations: statistics and least squares in surveying and GIS. Wiley, New York
The authors would like to thank Burkhard Schaffrin, José A. Ramos, Alfred Leick, and the anonymous reviewer for their suggestions and assistance. The authors were partially supported in this research by the National Geospatial-Intelligence Agency under contract No. HM1582–04-1-2026.
About this article
Cite this article
Felus, Y.A., Burtch, R.C. On symmetrical three-dimensional datum conversion. GPS Solut 13, 65 (2009). https://doi.org/10.1007/s10291-008-0100-5
- Datum conversion
- Total least-squares adjustment
- Procrustes algorithm