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Efficient variance component estimation for large-scale least-squares problems in satellite geodesy

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Abstract

Efficient Variance Component Estimation (VCE) is significant to optimal data combination in large-scale least-squares problems as those encountered in satellite geodesy, where millions of observations are jointly processed to estimate a huge number of unknown parameters. In this paper, an efficient VCE algorithm with rigorous trace calculation is proposed based on the local–global parameters partition scheme in satellite geodesy, which is directly applicable to both the simplified yet common case where local parameters are unique to a single observation group and the generalized case where local parameters are shared by different groups of observations. Moreover, the Monte-Carlo VCE (MCVCE) algorithm, based on the stochastic trace estimation technique, is further extended in this paper to the generalized case. Two numerical simulation cases are investigated for gravity field model recovery to evaluate both the accuracy and efficiency of the proposed algorithm and the extended MCVCE algorithm in terms of trace calculation. Compared to the conventional algorithm, the relative trace calculation errors in the efficient algorithm are all negligibly below 10–7%, while in the MCVCE algorithm they can vary from 0.6 to 37% depending on the number of adopted random vector realizations and the specific applications. The efficient algorithm can achieve computational time reduction rates above 96% compared to the conventional algorithm for all gravity field model sizes considered in the paper. In the MCVCE algorithm, however, the time reduction rates can change from 61 to 99% for different implementations.

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Data availability

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

References

  • Alkhatib H, Schuh WD (2007) Integration of the Monte Carlo covariance estimation strategy into tailored solution procedures for large-scale least squares problems. J Geod 81(1):53–66

    Article  Google Scholar 

  • Altamimi Z, Rebischung P, Mtivier L, Collilieux X (2016) ITRF2014: a new release of the international terrestrial reference frame modeling nonlinear station motions. J Geophys Res Solid Earth 121(8):6109–6131

    Article  Google Scholar 

  • Amiri-Simkooei AR (2007) Least squares variance component estimation: theory and GPS applications. PhD dissertation, Delft Univ. Technol., Delft, The Netherlands

  • Amiri-Simkooei AR (2016) Non-negative least-squares variance component estimation with application to GPS time series. J Geod 90(5):451–466

    Article  Google Scholar 

  • Bähr H, Altamimi Z, Heck B (2007) Variance component estimation for combination of terrestrial reference frames. Universitätsverlag Karlsruhe, Karlsruhe, Germany

    Google Scholar 

  • Bettadpur S, McCullough C (2017) The classical variational approach. In: Flury J, Naeimi M (eds) Global gravity field modeling from satellite-to-satellite tracking data. lecture notes in earth system sciences. Springer, Cham

    Google Scholar 

  • Beutler G, Jäggi A, Mervart L, Meyer U (2010) The celestial mechanics approach: theoretical foundations. J Geod 84(10):605–624

    Article  Google Scholar 

  • Borko A, Even-Tzur G (2021) Stochastic model reliability in GNSS baseline solution. J Geod 95(2):841

    Article  Google Scholar 

  • Brockmann JM (2014) On high performance computing in geodesy—applications in global gravity field determination. Ph.D. thesis, Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany

  • Brockmann JM, Roese-Koerner L, Schuh WD (2014) A concept for the estimation of high-degree gravity field models in a high performance computing environment. Studia Geophys Et Geod 58(4):571–594

    Article  Google Scholar 

  • Chen J, Tapley B, Wilson C, Cazenave A, Seo KW, Kim JS (2020) Global ocean mass change from GRACE and GRACE follow-on and altimeter and argo measurements. Geophys Res Lett 47(22):1526

    Article  Google Scholar 

  • Chen Q, Shen Y, Chen W, Francis O, Zhang X, Chen Q, Li W, Chen T (2019) An optimized short-arc approach: Methodology and application to develop refined time series of Tongji-Grace2018 GRACE monthly solutions. J Geophys Res Solid Earth 124(6):6010–6038

    Article  Google Scholar 

  • Chen Q, Shen Y, Francis O, Chen W, Zhang X, Hsu H (2018) Tongji-Grace02s and Tongji-Grace02k: high-precision static GRACE-only global Earth’s gravity field models derived by refined data processing strategies. J Geophys Res Solid Earth 123(7):6111–6137

    Article  Google Scholar 

  • Förstner W (1979) Ein Verfahren zur Schätzung von Varianz- und Kovarianzkomponenten. Allg Vermess-Nachr 86:446–453

    Google Scholar 

  • Girard DA (1989) A fast ‘Monte-Carlo cross-validation’ procedure for large least squares problems with noisy data. Numer Math 56:1–23

    Article  Google Scholar 

  • Helmert F (1924) Die Ausgleichungsrechnung nach der Methode der kleinsten Quadrate, 3rd edn. Teubner, Leipzig

    Google Scholar 

  • Hutchinson MF (1990) A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun Statist Simulat Comput 19:433–450

    Article  Google Scholar 

  • Jin S, van Dam T, Wdowinski S (2013) Observing and understanding the Earth system variations from space geodesy. J Geodyn 72:1–10

    Article  Google Scholar 

  • Kim J (2000) Simulation study of a low-low satellite-to-satellite tracking mission. Ph.D. thesis, The University of Texas at Austin, Austin, USA

  • Klees R, Ditmar P, Broersen P (2003) How to handle colored observation noise in large least-squares problems. J Geod 76(11–12):629–640

    Article  Google Scholar 

  • Koch KR (1986) Maximum likelihood estimate of variance components. Bull Géod 60:329–338

    Article  Google Scholar 

  • Koch KR (1999) Parameter estimation and hypothesis testing in linear models, 2nd edn. Springer, Berlin Heidelberg New York

    Book  Google Scholar 

  • Koch KR, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. J Geod 76(5):259–268

    Article  Google Scholar 

  • Koch KR, Kuhlmann H, Schuh WD (2010) Approximating covariance matrices estimated in multivariate models by estimated auto- and cross-covariances. J Geod 84(6):383–397

    Article  Google Scholar 

  • Kusche J (2003) A Monte-Carlo technique for weight estimation in satellite geodesy. J Geod 76(11–12):641–652

    Article  Google Scholar 

  • Kusche J, Springer A (2017) Parameter estimation for satellite gravity field modeling. In: Naeimi M, Flury J (eds) Global gravity field modeling from satellite-to-satellite tracking data. lecture notes in Earth system sciences. Springer, Berlin

    Google Scholar 

  • Landerer FW, Flechtner FM, Save H et al (2020) Extending the global mass change data record GRACE follow-on instrument and science data performance. Geophys Res Lett 47(12):8306

    Article  Google Scholar 

  • Li B, Shen Y, Lou L (2011) Efficient estimation of variance and covariance components: a case study for GPS stochastic model evaluation. IEEE Trans Geosci Remote Sens 49(1):203–210

    Article  Google Scholar 

  • Li X, Ge M, Dai X, Ren X, Fritsche M, Wickert J, Schuh H (2015) Accuracy and reliability of multi-GNSS real-time precise positioning: GPS, GLONASS, BeiDou, and Galileo. J Geod 89(6):607–635

    Article  Google Scholar 

  • Loomis BD, Nerem RS, Luthcke SB (2012) Simulation study of a follow-on gravity mission to GRACE. J Geod 86(5):319–335

    Article  Google Scholar 

  • Lucas JR, Dillinger WH (1998) MINQUE for block diagonal bordered systems such as those encountered in VLBI data analysis. J Geod 72:343–349

    Article  Google Scholar 

  • Noomen R, Springer TA, Ambrosius BAC, Herzberger K, Kuijper DC, Mets GJ, Overgaauw B, Wakker KF (1996) Crustal deformations in the Mediterranean area computed from SLR and GPS observations. J Geodyn 21(1):73–96

    Article  Google Scholar 

  • Pail R (2015) It’s all about statistics: global gravity field modeling from GOCE and complementary data. In: Freeden W, Nashed M, Sonar T (eds) Handbook of geomathematics. Springer, Berlin, Heidelberg, pp 2345–2372

    Chapter  Google Scholar 

  • Pukelheim F (1976) Estimating variance components in linear models. J Multivariate Analysis 6:626–629

    Article  Google Scholar 

  • Rao CR (1973) Linear statistical inference and its applications. Wiley, New York

    Book  Google Scholar 

  • Reigber C (1989) Gravity field recovery from satellite tracking data. In: Sansò F, Rummel R (eds) Theory of satellite geodesy and gravity field determination Lecture Notes in Earth Sciences. Springer, Heidelberg

    Google Scholar 

  • Reigber C, Schmidt R, Flechtner F, König R, Meyer U, Neumayer KH, Schwintzer P, Zhu SY (2005) An Earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S. J Geodyn 39:1–10

    Article  Google Scholar 

  • Rodell M, Famiglietti JS, Wiese DN, Reager JT, Beaudoing HK, Landerer FW, Lo MH (2018) Emerging trends in global freshwater availability. Nature 557:651–659

    Article  Google Scholar 

  • Sahin M, Cross PA, Sellers PC (1992) Variance component estimation applied to satellite laser ranging. Bull Géod 66:284–295

    Article  Google Scholar 

  • Savcenko R, Bosch W (2012) EOT11a-Empirical ocean tide model from multi-mission satellite altimetry. DGFI Report No. 89, Deutsches Geodätisches Forschungsinstitut, München

  • Schuh WD (1996) Tailored Numerical Solution Strategies for the Global Determination of the Earth’s Gravity Field. Tech. Rep. 81, TU Graz, Graz, Austria

  • Schuh WD (2003) The processing of band-limited measurements; filtering techniques in the least squares context and in the presence of data gaps. Space Sci Rev 108(1–2):67–78

    Article  Google Scholar 

  • Schwintzer P (1990) Sensitivity analysis in least squares gravity modelling by means of redundancy decomposition of stochastic prior information. Internal Report, Deutsches Geodätisches Forschungsinstitut, München

  • Siemes C (2008) Digital filtering algorithms for decorrelation within large least squares problems. Ph.D. thesis, Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany

  • Sneeuw N (2000) A semi-analytical approach to gravity field analysis from satellite observations. Deutsche Geodätische Kommission, Reihe C, Heft Nr. 527, München

  • Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins MM (2004) GRACE measurements of mass variability in the earth system. Science 305(5683):503–505

    Article  Google Scholar 

  • Tehranchi R, Moghtased-Azar K, Safari A (2021) Fast approximation algorithm to noise components estimation in long-term GPS coordinate time series. J Geod 95(2):1–16

    Article  Google Scholar 

  • Teunissen PJG, Amiri-Simkooei AR (2008) Least-squares variance component estimation. J Geod 82(2):65–82

    Article  Google Scholar 

  • Travelletti J, Malet JP (2012) Characterization of the 3D geometry of flow-like landslides: a methodology based on the integration of heterogeneous multi-source data. Eng Geol 128:30–48

    Article  Google Scholar 

  • Xu P (2009) Iterative generalized cross-validation for fusing heteroscedastic data of inverse ill-posed problems. Geophys J Int 179:182–200

    Article  Google Scholar 

  • Xu P (2021) A new look at Akaike’s Bayesian information criterion for inverse ill-posed problems. J Franklin Inst 358(7):4077–4102

    Article  Google Scholar 

  • Xu P, Shen Y, Fukuda Y, Liu Y (2006) Variance component estimation in linear inverse ill-posed models. J Geod 80(2):69–81

    Article  Google Scholar 

  • Xu P, Liu Y, Shen Y, Fukuda Y (2007) Estimability analysis of variance and covariance components. J Geod 81:593–602

    Article  Google Scholar 

  • Yang Y, Xu T, Song L (2005) Robust estimation of variance components with application in global positioning system network adjustment. J Surv Eng 131(4):107–112

    Article  Google Scholar 

  • Yang Y, Zeng A, Zhang J (2009) Adaptive collocation with application in height system transformation. J Geod 83(5):403–410

    Article  Google Scholar 

  • Yi W (2012) The Earth’s gravity field from GOCE. Ph.D. thesis, Technische Universität München, München, Germany

Download references

Acknowledgments

This study is sponsored by the National Natural Science Foundation of China (41731069, 41974002). The first author acknowledges the support of the China Scholarship Council (CSC). The editors and three reviewers are greatly appreciated for their constructive comments and suggestions.

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Authors and Affiliations

  1. College of Surveying and Geo-Informatics, Tongji University, Shanghai, 200092, China

    Yufeng Nie, Yunzhong Shen & Qiujie Chen

  2. Institute of Astronomical and Physical Geodesy, Technical University of Munich, Arcisstr. 21, 80333, Munich, Germany

    Roland Pail

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  1. Yufeng Nie
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  2. Yunzhong Shen
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  3. Roland Pail
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  4. Qiujie Chen
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Contributions

YN proposed the idea, designed and conducted the numerical experiments, analyzed the data and wrote the manuscript. YN and YS developed the theory and algorithms. YS and RP supervised the research and revised the manuscript. QC provided the post-fit residuals files. All authors joined discussions throughout the development.

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Correspondence to Yunzhong Shen.

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Nie, Y., Shen, Y., Pail, R. et al. Efficient variance component estimation for large-scale least-squares problems in satellite geodesy. J Geod 96, 13 (2022). https://doi.org/10.1007/s00190-022-01599-9

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