Mathematical Geosciences

, Volume 48, Issue 5, pp 487–510 | Cite as

Smoothing Three-Dimensional Manifold Data, with Application to Tectonic Fault Detection



Three-dimensional manifold data arise in many contexts of geoscience, such as laser scanning, drilling surveys and seismic catalogs. They provide point measurements of complex surfaces, which cannot be fitted by common statistical techniques, like kriging interpolation and principal curves. This paper focuses on iterative methods of manifold smoothing based on local averaging and principal components; it shows their relationships and provides some methodological developments. In particular, it develops a kernel spline estimator and a data-driven method for selecting its smoothing coefficients. It also shows the ability of this approach to select the number of nearest neighbors and the optimal number of iterations in blurring-type smoothers. Extensive numerical applications to simulated and seismic data compare the performance of the discussed methods and check the efficacy of the proposed solutions.


Bandwidth selection Kernel splines Local averaging Principal surfaces New Madrid earthquakes Sliced scanning Stopping criteria 



A sincere thank to the Editors and Reviewers for their useful comments.


  1. Aliyari-Ghassabeh Y, Linder T, Takahara G (2013) On convergence properties of the subspace constrained mean shift. Pattern Recognit 46(11):3140–3147CrossRefGoogle Scholar
  2. Biedermann S, Yang M (2015) Designs for selected nonlinear models. In: Dean A, Morris M, Stufken J, Bingham D (eds) Handbook of design and analysis of experiments. Chapman & Hall, London, pp 515–547Google Scholar
  3. Carreira-Perpiñán MÁ (2006) Fast nonparametric clustering with Gaussian blurring mean-shift. In: Proceedings of 23rd international conference on machine learning. Pittsburgh, pp 153–160Google Scholar
  4. Caumon G, Collon-Drouaillet P (2014) Special issues on three-dimensional structural modeling. Math Geosci 46(8):905–908CrossRefGoogle Scholar
  5. Chen Y (1995) Mean shift, mode seeking and clustering. IEEE Trans Pattern Anal Mach Intell 17(8):790–799CrossRefGoogle Scholar
  6. Comaniciu D, Meer P (2002) Mean shift: a robust approach toward feature space analysis. IEEE Trans Pattern Anal Mach Intell 24(5):603–619CrossRefGoogle Scholar
  7. Csontos R, Van Arsdale R (2008) New Madrid seismic zone fault geometry. Geosphere 4(5):802–813CrossRefGoogle Scholar
  8. Eberly D (1996) Ridges in image and data analysis. Kluwer, DordrechtCrossRefGoogle Scholar
  9. Einbeck J, Evers L, Powell B (2010) Data compression and regression, through local principal curves and surfaces. Int J Neural Syst 20(3):177–192CrossRefGoogle Scholar
  10. Forte AM, Mitrovica JX, Moucha R, Simmons RA, Grand SP (2007) Descent of the ancient Farallon slab drives localized mantle flow below the New Madrid seismic zone. Geophys Res Lett 34(4):L04308. doi: 10.1029/2006GL027895
  11. Genovese CR, Perone-Pacifico M, Verdinelli I, Wasserman L (2014) Nonparametric ridge estimation. Ann Stat 42(4):1511–1545CrossRefGoogle Scholar
  12. Grillenzoni C (1998) Forecasting unstable and non-stationary time series. Int J Forecast 14(4):469–482CrossRefGoogle Scholar
  13. Grillenzoni C (2007) Pattern recognition via robust smoothing, with application to laser data. Aust NZ J Stat 49(2):139–153CrossRefGoogle Scholar
  14. Grillenzoni C (2014) Detection of tectonic faults by spatial clustering of earthquake hypocenters. Spat Stat 7(2):62–78CrossRefGoogle Scholar
  15. Hastie T, Stuetzle W (1989) Principal curves. J Am Stat Assoc 84(2):502–516CrossRefGoogle Scholar
  16. Hill BJ, Kendall WS, Thönnes E (2012) Fiber-generated point processes and fields of orientations. Ann Appl Stat 6(3):994–1020CrossRefGoogle Scholar
  17. Jordan TH (2014) Faults in focus: 20 years of earthquake science accomplishments. SECS: Southern California Earthquake CenterGoogle Scholar
  18. Li X, Hu Z, Wu F (2007) A note on the convergence of the mean shift. Pattern Recognit 40(6):1756–1762CrossRefGoogle Scholar
  19. Mérigot Q, Ovsjanikov M, Guibas L (2011) Voronoi-based curvature and feature estimation from point clouds. IEEE Trans Vis Comput Graph 17(6):743–756CrossRefGoogle Scholar
  20. Ozertem U, Erdogmus D (2011) Locally defined principal curves and surfaces. J Mach Learn Res 12:1249–1286Google Scholar
  21. Park JH, Zhang Z, Zha H, Kasturi R (2004) Local smoothing for manifold learning. In: Proceedings of 2004 IEEE conference on computer vision and pattern recognition Vol 2. Washington, pp 452–459Google Scholar
  22. Prechelt L (2012) Early stopping—But when? In: Montavon G, Orr GB, Müller KR (eds) Neural networks: tricks of the trade. Springer, Berlin, pp 53–68CrossRefGoogle Scholar
  23. Silverman BW (1986) Density estimation for statistical data analysis. Chapman & Hall, LondonCrossRefGoogle Scholar
  24. Stanford DC, Raftery AE (2000) Finding curvilinear features in spatial point patterns: principal curve clustering with noise. IEEE Trans Pattern Anal Mach Intell 22(6):601–609CrossRefGoogle Scholar
  25. Tenenbaum J, De Silva V, Langford J (2000) A global geometric framework for nonlinear dimension reduction. Science 290:2319–2323CrossRefGoogle Scholar
  26. Wahba G (1990) Spline models for observational data. SIAM, PhiladelphiaCrossRefGoogle Scholar
  27. Wang W, Carreira-Perpiñán MÁ (2010) Manifold blurring mean shift algorithms for manifold denoising. In: Proceedings of 2010 IEEE conference on computer vision and pattern recognition. San Francisco, pp 1759–1766Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2016

Authors and Affiliations

  1. 1.Institute of ArchitectureUniversity of VeniceVeneziaItaly

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