Mathematical Geosciences

, Volume 47, Issue 2, pp 149–171 | Cite as

Wavelet-Based Semiparametric Estimation of Ocean Surface Temperature



Fractional-order pseudodifferential equations are considered to represent ocean climate variability when anomalous diffusion processes affect heat transfer in ocean surface. The driven process of these equations is assumed to be a regular spatiotemporal Gaussian random field representing normal conditions in the ocean. Linear regression in the log-wavelet domain is applied for the estimation of the parameters characterizing the pseudodifferential equation defining the anomalous diffusion process. The non-parametric framework is adopted in the estimation of the probability distribution of the driven spatiotemporal random field. Finally, ocean surface temperature values are approximated by plug-in least-square estimation from the computed parameter estimates, the estimated distribution of the driven process, and the integral version of the fractional-order pseudodifferential equation. The ability of the approach presented to process strong spatial-correlated ocean surface temperature curve data is illustrated with a real-data example, where sample information from weather stations in Hawaii ocean is analyzed.


Fractional-order pseudodifferential equations Global warming Log-wavelet regression Ocean surface temperature anomalies Spatial functional statistical estimation 



This work has been supported in part by project MTM2012-32674 (co-funded FEDER) of the DGI, MEC, and P09-FQM-5052 of the Andalousian CICE, Spain.

Supplementary material

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Supplementary material 1 (pdf 39 KB)
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Supplementary material 2 (pdf 39 KB)
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Supplementary material 3 (pdf 31 KB)
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Supplementary material 4 (pdf 31 KB)


  1. Akhiezer NI, Glazman IM (1961) Theory of linear operators in Hilbert spaces. Ungar, New YorkGoogle Scholar
  2. Angelini C, De Canditiis D, Leblanc F (2003) Wavelet regression estimation in nonparametric mixed effect models. J Multivar Anal 85:267–291CrossRefGoogle Scholar
  3. Biswas A, Si BCh (2011) Application of continuous wavelet transform in examining soil spatial variation: a review. Math Geosci 43:379–396CrossRefGoogle Scholar
  4. Bosch EH, González AP, Vivas JG, Easley GR (2009) Directional wavelets and a wavelet variogram for two-dimensional data. Math Geosci 41:611–641CrossRefGoogle Scholar
  5. Bosch EH, Oliver MA, Webster R (2004) Wavelets and the generalization of the variogram. Math Geosci 36:147–186Google Scholar
  6. Chandrasekhar E, Rao VE (2012) Wavelet analysis of geophysical well-log data of Bombay Offshore Basin, India. Math Geosci 44:901–928CrossRefGoogle Scholar
  7. Chatterjee S, Dimitrakopoulos R, Mustapha H (2012) Dimensional reduction of pattern-based simulation using wavelet analysis. Math Geosci 44:343–374CrossRefGoogle Scholar
  8. Christakos G (1992) Random field models in earth science. Dover Publications, New YorkGoogle Scholar
  9. Christakos G (2000) Modern spatiotemporal geostatistics. Oxford University Press, New YorkGoogle Scholar
  10. Chyzak F, Paule P, Scherzer O, Schoisswohl A, Zimmermann B (2001) The construction of orthonormal wavelets using symbolic methods and a matrix analytical approach for wavelets on the interval. Exp Math 10:66–86CrossRefGoogle Scholar
  11. Cohen A, Daubechies I, Vial P (1994) Wavelets on the interval and fast wavelet transforms. J Appl Comput Harmon Anal 1:54–81CrossRefGoogle Scholar
  12. Gloaguen E, Dimitrakopoulos R (2009) Two-dimensional conditional simulations based on the wavelet decomposition of training images. Math Geosci 41:679–701CrossRefGoogle Scholar
  13. Härdle W, Marron JS (1985) Optimal bandwidth selection in nonparametric regression function estimation. Ann Stat 13:1465–1481CrossRefGoogle Scholar
  14. Jia R-Q, Wang J, Zhou D-X (2003) Compactly supported wavelet bases for Sobolev spaces. J Appl Comput Harmon Anal 15:224–241CrossRefGoogle Scholar
  15. Kelbert M, Leonenko NN, Ruiz-Medina MD (2005) Fractional random fields associated with stochastic fractional heat equations. Adv Appl Probab 108:108–133CrossRefGoogle Scholar
  16. Frías MP, Ruiz-Medina MD (2011) Computing functional estimators of spatiotemporal long-range dependence parameters in the spectral-wavelet domain. J Stat Plan Infer 141:2417–2427CrossRefGoogle Scholar
  17. Frías MP, Ruiz-Medina MD (2012) Filtering and functional parameter estimation of spatiotemporal strong-dependence models. J Environ Stat 3:2Google Scholar
  18. Frías MP, Ruiz-Medina MD, Anh VV (2013) Wavelet-based estimation of anisotropic spatiotemporal long-range dependence. Stoch Anal Appl 31:359–380CrossRefGoogle Scholar
  19. Hruska M, Corea W, Seeburger D, Schweller W, Crane WH (2009) Automated segmentation of resistivity image logs using wavelet transform. Math Geosci 41:703–716CrossRefGoogle Scholar
  20. Leonenko NN, Ruiz-Medina MD (2006) Scaling laws for the multidimensional Burgers equation with quadratic external potentials. J Stat Phys 124:191–205CrossRefGoogle Scholar
  21. Meyer Y (1991) Ondelettes sur intervalle. Rev Mat Iberoam 7:115–133CrossRefGoogle Scholar
  22. Milne AE, Lark RM (2009) Wavelet transforms applied to irregularly sampled soil data. Math Geosci 41:661–678CrossRefGoogle Scholar
  23. Prokoph A, Bilali HEl (2008) Cross-wavelet analysis: a tool for detection of relationships between paleoclimate proxy records. Math Geosci 40:575–586Google Scholar
  24. Ruiz-Medina MD (2012) Spatial functional prediction from spatial autoregressive Hilbertian processes. Environmetrics 23:119–128CrossRefGoogle Scholar
  25. Ruiz-Medina MD, Espejo RM (2012) Spatial autoregressive functional plug-in prediction of ocean surface temperature. Stoch Environ Res Risk Assess 26:335–344CrossRefGoogle Scholar
  26. Ruiz-Medina MD, Espejo RM (2013) Integration of spatial functional interaction in the extrapolation of ocean surface temperature anomalies due to global warming. Int J Appl Earth Obs 22:27–39CrossRefGoogle Scholar
  27. Stein EM (1970) Singular integrals and differential properties of functions. Princeton University Press, New JerseyGoogle Scholar
  28. Stein ML (1999) Interpolation of spatial data: some theory for kriging. Springer, New YorkCrossRefGoogle Scholar
  29. Stein ML (2009) Spatial interpolation of high-frequency monitoring data. Ann Appl Stat 3:272–291CrossRefGoogle Scholar
  30. Watkins L, Neupauer RM, Compo GP (2009) Wavelet analysis and filtering to identify dominant orientations of permeability anisotropy. Math Geosci 41:643–659CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2014

Authors and Affiliations

  1. 1.Department of Statistics and O.R, Faculty of SciencesUniversity of GranadaGranadaSpain
  2. 2.Department of Statistics and O.RUniversity of JaénJaénSpain

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