Non-parametric Covariance Modeling Using Fast Fourier Transform

  • Tingting Yao
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 80)


Geostatistics is used in the petroleum industry to model spatial distributions of reservoir properties from limited well samples. Covariance models provide the basic measure of spatial continuity used to weight the information available at different sample locations, as in kriging. Traditionally, a closed-form analytical model defined by a few parameters is fitted to allow for interpolation of sample covariance values while ensuring the positive-definite condition. For cokriging, the modeling task is made even more difficult because of the restrictions imposed by the linear coregionalization model. Some non-parametric modeling methods have been proposed to model covariance values to ensure the positive-definiteness constraints based on eigenvalue correction or Fourier series fitting. These methods are not widely used due to the associated intensive CPU time. Bochner’s theorem maps the positivedefinite constraints into much simpler constraints on the Fourier transform of the covariance, that is, the density spectrum. Accordingly, it is proposed to transform the experimental (cross) covariance tables into quasi-density spectrum tables using the Fast Fourier Transform (FFT). These quasi-density spectrum tables are then smoothed under some constraints. An inverse FFT yields permissible (jointly) positive-definite (cross) covariance tables. The extention of Bochner’s theorem allows several (cross) covariance tables to be modeled simultaneously with much less effort than the linear model of coregionalization. A case study using actual petroleum reservoir data (porosity and seismic reflection energy) illustrate the application of the proposed algorithm.


Sample Covariance Density Spectrum Smoothing Window Kriging System Covariance Table 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Tingting Yao
    • 1
  1. 1.ExxonMobil Upstream Research CompanyHoustonUSA

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