A Variational Bayesian Approach to Self-Tuning Robust Adjustment for Joint Inversion of Nonlinear Volcano Source Model with -Distributed Random Errors
Publication: Journal of Surveying Engineering
Volume 148, Issue 2
Abstract
Variance component estimation (VCE), herein called joint inversion, is a widely used approach to weigh the contributions of different data sets. Traditionally, the random errors of observations in VCE are modeled as Gaussian. However, in many geodetic measurements and sensor technologies, the observation data are non-Gaussian; therefore, the joint inversion with a more general heavy-tailed error model is preferred. Another issue is that the VCE deduced from the existing approaches may be not an interior solution, which means that the estimates may lie outside of the parameter space. Although there are some works on VCE in the Gaussian error model with equality or inequality constraints to mitigate this effect, to the best of our knowledge, there does not exist any work addressing the interior solutions of variance components for the heavy-tailed error model. In this article, we consider these issues for the first time and describe the behavior of multiple data sets using the joint functional model, which allows for the nonlinear modeling through nonlinear (differentiable) observation functions, where the random errors are modeled as Student’s -distributed. To address the estimation problem, an iteratively reweighted least squares (LS) approach to self-tuning robust estimation of joint functional model parameters, the variance components, and the degree of freedom () of the Student’s -distribution is derived based on a variational generalized expectation-maximization (GEM) algorithm. The proposed algorithm is computationally cheap and easy to implement. The performance of the algorithm is evaluated by means of Monte Carlo simulations of the joint volcano source model. Furthermore, the suitability of the research model and the proposed variational GEM algorithm is investigated within a numerical experiment involving the multisource modeling and adjustment of real data sets of the 2015 Calbuco volcano eruption.
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Data Availability Statement
Some data, models, or code generated or used during the study are available from the corresponding author by reasonable request. The types of the raw data (e.g., the data simulated by the Mogi model, 2015 Calbuco volcano real data) can be provided by the MAT file in MATLAB.
Acknowledgments
The authors are grateful to all of the anonymous reviewers and editors for their careful review and valuable suggestions, which improved the quality of this paper. The authors thank Wenbin Xu for providing the raw data of the 2015 Calbuco volcano and his patient help. In this article, some images are drawn using the open source software GMT, and this research is supported by the National Natural Science Foundation of China, (Grant Nos. 42174011, 41874001, and 41664001); and Jiangxi Provincial Natural Science Foundation, (No. 20202BABL212015).
References
Alkhatib, H., B. Kargoll, and J. A. Paffenholz. 2018. “Further results on robust multivariate time series analysis in nonlinear models with autoregressive and t-distributed errors.” In Time series analysis and forecasting, 25–38. Berlin: Springer.
Amiri-Simkooei, A. R. 2016. “Non-negative least-squares variance component estimation with application to GPS time series.” J. Geod. 90 (5): 451–466. https://doi.org/10.1007/s00190-016-0886-9.
Amiri-Simkooei, A. R., C. C. J. M. Tiberius, and P. J. G. Teunissen. 2007. “Assessment of noise in GPS coordinate time series: Methodology and results.” J. Geophys. Res. Atmos. 112 (7): B07413. https://doi.org/10.1029/2006JB004913.
Amiri-Simkooei, A. R., F. Zangeneh-Nejad, and J. Asgari. 2013. “Least-squares variance component estimation applied to GPS geometry-based observation model.” J. Surv. Eng. 139 (4): 176–187. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000107.
Anderson, S. C., T. A. Branch, and A. B. Cooper. 2017. “Black-swan events in animal populations.” Proc. Natl. Acad. Sci. U.S.A. 114 (12): 3252–3257. https://doi.org/10.1073/pnas.1611525114.
Badiu, M. A., T. L. Hansen, and B. H. Fleury. 2017. “Variational Bayesian inference of line spectra.” IEEE Trans. Signal Process. 65 (9): 2247–2261. https://doi.org/10.1109/TSP.2017.2655489.
Beal, M. J. 2003. “Variational algorithms for approximate Bayesian inference.” Ph.D. thesis, Gatsby Computational Neuroscience Unit, Univ. College London.
Bifulco, I., G. Raiconi, and R. Scarpa. 2009. “Computer algebra software for least squares and total least norm inversion of geophysical models.” Comput. Geosci. 35 (7): 1427–1438. https://doi.org/10.1016/j.cageo.2008.11.005.
Castellaro, M., G. Rizzo, and M. Tonietto. 2017. “A variational Bayesian inference method for parametric imaging of pet data.” Neuroimage 150 (Apr): 136–149. https://doi.org/10.1016/j.neuroimage.2017.02.009.
Chappell, M. A., A. R. Groves, and B. Whitcher. 2009. “Variational Bayesian inference for a nonlinear forward model.” IEEE Trans. Signal Process. 57 (1): 223–236. https://doi.org/10.1109/TSP.2008.2005752.
Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. “Maximum likelihood from incomplete data via the EM algorithm.” J. R. Stat. Soc. Ser. B 39 (1): 1–38.
El Leithy, H. A., Z. A. Abdel Wahed, and M. S. Abdallah. 2015. “On non-negative estimation of variance components in mixed linear models.” J. Adv. Res. 7 (1): 59–68. https://doi.org/10.1016/j.jare.2015.02.001.
Fuhrmann, T., and M. C. Garthwaite. 2019. “Resolving three-dimensional surface motion with InSAR: Constraints from multi-geometry data fusion.” Remote Sens. 11 (3): 241. https://doi.org/10.3390/rs11030241.
Fukuda, J., and K. M. Johnson. 2008. “A fully Bayesian inversion for spatial distribution of fault slip with objective smoothing.” Bull. Seismol. Soc. Am. 98 (3): 1128–1146. https://doi.org/10.1785/0120070194.
Grafarend, E. R. 1984. “Variance-covariance component estimation of Helmert type in the Gauss–Helmert model.” Zeitschrift für Vermessungswesen 109 (1): 34–43.
Helmert, F. R. 1907. Die Ausgleichungsrechnung nach der Methode der kleinsten Quadrate. Berlin: Zweite Auflage, Teubner.
Huber, P. J. 1964. “Robust estimation of a location parameter.” Ann. Math. Stat. 35 (Jan): 73–101. https://doi.org/10.1214/aoms/1177703732.
Jin, B., and J. Zou. 2010. “Hierarchical Bayesian inference for ill-posed problems via variational method.” J. Comput. Phys. 229 (19): 7317–7343. https://doi.org/10.1016/j.jcp.2010.06.016.
Kargoll, B., G. Kermarrec, and J. Korte. 2020. “Self-tuning robust adjustment within multivariate regression time series models with vector-autoregressive random errors.” J. Geod. 94 (5): 51. https://doi.org/10.1007/s00190-020-01376-6.
Kargoll, B., M. Omidalizarandi, and I. Loth. 2018. “An iteratively reweighted least-squares approach to adaptive robust adjustment of parameters in linear regression models with autoregressive and t-distributed deviations.” J. Geod. 92 (3): 271–297. https://doi.org/10.1007/s00190-017-1062-6.
Khodabandeh, A., A. R. Amiri-Simkooei, and M. A. Sharifi. 2012. “GPS position time-series analysis based on asymptotic normality of M-estimation.” J. Geod. 86 (1): 15–33. https://doi.org/10.1007/s00190-011-0489-4.
Koch, K. R. 1999. Parameter estimation and hypothesis testing in linear models. Berlin: Springer.
Koch, K. R. 2013. “Comparison of two robust estimations by expectation maximization algorithms with Huber’s method and outlier tests.” J. Appl. Geod. 7 (2): 115–123. https://doi.org/10.1515/jag-2013-0050.
Koch, K. R. 2017. “Expectation maximization algorithm and its minimal detectable outliers.” Stud. Geophys. Geod. 61 (1): 1–18. https://doi.org/10.1007/s11200-016-0617-y.
Koch, K. R., and B. Kargoll. 2015. “Outlier detection by the EM algorithm for laser scanning in rectangular and polar systems.” J. Appl. Geod. 9 (3): 162–173. https://doi.org/10.1515/jag-2015-0004.
Lange, K. L., R. J. A. Little, and J. M. G. Taylor. 1989. “Robust statistical modeling using the t distribution.” J. Am. Stat. Assoc. 84 (408): 881–896.
Li, Y., J. Kong, and Y. Xu. 2012. “Prediction of surface subsidence above salt rock gas storage using Mogi model.” [In Chinese.] Chin. J. Rock Mech. Eng. 31 (9): 1737–1745.
Lim, K. L., and H. Wang. 2018. “Fast approximation of variational Bayes Dirichlet process mixture using the maximization-maximization algorithm.” Int. J. Approx. Reason. 93 (Feb): 153–177. https://doi.org/10.1016/j.ijar.2017.11.001.
Liu, C., and D. B. Rubin. 1995. “ML estimation of the t distribution using EM and its extensions, ECM and ECME.” Stat. Sin. 5 (1): 19–39.
Luo, X., M. Mayer, and B. Heck. 2012. “Analysing time series of GNSS residuals by means of AR(I)MA processes.” In Proc., 7th Hotine-Marussi Symp. on Mathematical Geodesy Int. Association of Geodesy Symposia. Berlin: Springer.
McGonigle, A., A. Aiuppa, G. Giudice, G. Tamburello, A. J. Hodson, and S. Gurrieri. 2008. “Unmanned aerial vehicle measurements of volcanic carbon dioxide fluxes.” Geophys. Res. Lett. 35 (6): L06303. https://doi.org/10.1029/2007GL032508.
McLachlan, G. J., and T. Krishnan. 2008. The EM algorithm and extensions. 2nd ed. New York: Wiley.
Moghtased-Azar, K., R. Tehranchi, and A. R. Amiri-Simkooei. 2014. “An alternative method for non-negative estimation of variance components.” J. Geod. 88 (5): 427–439. https://doi.org/10.1007/s00190-014-0693-0.
Mogi, K. 1958. Relation between the eruptions of various volcanoes and deformations of the ground surfaces around them, 111–123. Tokyo: Univ. of Tokyo.
Nadarajah, S. 2009. “The product t density distribution arising from the product of two Student’s t PDFs.” Stat. Pap. 50 (3): 605–615. https://doi.org/10.1007/s00362-007-0088-x.
Nassar, S., K. P. Schwarz, and N. Elsheimy. 2004. “Modeling inertial sensor errors using autoregressive (AR) models.” Ann. Navig. 51 (4): 259–268. https://doi.org/10.1002/j.2161-4296.2004.tb00357.x.
Nikkhoo, M., T. R. Walter, and P. R. Lundgren. 2017. “Compound dislocation models (CDMs) for volcano deformation analyses.” Geophys. J. Int. 208 (2): 877–894.
Paffenholz, J. A., and K. H. Bae. 2012. “Geo-referencing point clouds with transformational and positional uncertainties.” J. Appl. Geod. 6 (1): 33–46. https://doi.org/10.1515/jag-2011-0010.
Penny, W., S. Kiebel, and K. Friston. 2003. “Variational Bayesian inference for FMRI time series.” Neuroimage 19 (3): 727–741. https://doi.org/10.1016/S1053-8119(03)00071-5.
Pereira, A., J. Antoni, and Q. Leclère. 2015. “Empirical Bayesian regularization of the inverse acoustic problem.” Appl. Acoust. 97 (Oct): 11–29. https://doi.org/10.1016/j.apacoust.2015.03.008.
Pritchard, M. E., J. Biggs, C. Wauthier, E. Sansosti, D. W. D. Arnold, F. Delgado, S. K. Ebmeier, S. T. Henderson, C. Cooper, and K. Wnuk, et al. 2018. “Towards coordinated regional multi-satellite InSAR volcano observations: Results from the Latin America pilot project.” J. Appl. Volcanol. 7 (1): 5. https://doi.org/10.1186/s13617-018-0074-0.
Rachev, S. T. 2003. Handbook of heavy tailed distributions in finance: Handbooks in finance. Amsterdam, Netherlands: Elsevier.
Sjöberg, L. E. 2011. “On the best quadratic minimum bias non-negative estimator of a two-variance component model.” J. Geod. Sci. 1 (3): 280–285. https://doi.org/10.2478/v10156-011-0006-y.
Solaro, G., V. Acocella, S. Pepe, J. Ruch, M. Neri, and E. Sansosti. 2010. “Anatomy of an unstable volcano from InSAR: Multiple processes affecting flank instability at Mt. Etna, 1994-2008.” J. Geophys. Res. Solid Earth 115 (10): B10405. https://doi.org/10.1029/2009JB000820.
Tarami, B., and M. Pourahmadi. 2003. “Multi-variate t autoregressions: Innovations, prediction variances and exact likelihood equations.” J. Time Ser. Anal. 24 (6): 739–754. https://doi.org/10.1111/j.1467-9892.2003.00332.x.
Tiku, M. L., W. Wong, and D. C. Vaughan. 2000. “Time series models in non-normal situations: Symmetric innovations.” J. Time Ser. Anal. 21 (5): 571–596. https://doi.org/10.1111/1467-9892.00199.
Tsay, R. S. 2005. Analysis of financial time series. 2nd ed. Hoboken, NJ: Wiley.
Valente, F., and C. Wellekens. 2004. “Variational Bayesian feature selection for Gaussian mixture models.” In Proc., IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, 973–976. New York: IEEE.
Verwaeren, J., P. V. D. Weeën, and B. D. Baets. 2015. “A search grid for parameter optimization as a byproduct of model sensitivity analysis.” Appl. Math. Comput. 261 (Jun): 8–27. https://doi.org/10.1016/j.amc.2015.03.064.
Wang, J., W. Shao, and Z. Song. 2019a. “Robust inferential sensor development based on variational Bayesian Student’s-t mixture regression.” Neurocomputing 369 (Dec): 11–28. https://doi.org/10.1016/j.neucom.2019.08.039.
Wang, L., J. Sun, and Q. Wu. 2021. “Nonlinear total least-squares variance component estimation for GM(1,1) model.” Geod. Geodyn. 12 (3): 211–217. https://doi.org/10.1016/j.geog.2021.02.006.
Wang, L., G. Wen, and Y. Zhao. 2019b. “Virtual observation method and precision estimation for ill-posed partial EIV model.” J. Surv. Eng. 145 (4): 04019010. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000282.
Wang, L., and Q. Wu. 2020. “Non-negative variance component estimation for the partial EIV model by the expectation maximization algorithm.” Geomatics Nat. Hazards Risk 11 (1): 1278–1298. https://doi.org/10.1080/19475705.2020.1785955.
Wang, L., F. Yu, and Z. Li. 2020. “Jackknife method for variance components estimation of partial EIV model.” J. Surv. Eng. 146 (4): 04020016. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000327.
Wen, R., and Y. Wu. 1993. “A grid approach to peaks.” [In Chinese.] J. Hebei Min. Civ. Eng. Inst. 1 (May): 6–10.
Xu, C. 2001. “Progress of joint inversion on geodesy and geophysics.” [In Chinese.] Geomatics Inf. Sci. Wuhan Univ. 26 (6): 555–561.
Zhou, X. 2017. “Markov chain Monte Carlo method used to invert for fault slip from geodetic data.” [In Chinese.] J. Geod. Geodyn. 37 (10): 996–1002.
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Journal of Surveying Engineering
Volume 148 • Issue 2 • May 2022
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© 2021 American Society of Civil Engineers.
History
Received: Apr 14, 2021
Accepted: Oct 15, 2021
Published online: Dec 17, 2021
Published in print: May 1, 2022
Discussion open until: May 17, 2022
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