• 060 Ma
    Late Paleocene

    March 2007

  • 050 Ma
    Early Eocene

    March 2007

  • 040 Ma
    Middle Eocene

    March 2007

  • 030 Ma
    Early Ollgocene

    March 2007

  • 020 Ma
    Early Miocene

    March 2007

  • 010 Ma
    Late Miocene

    March 2007

  • 000 Ma
    Present Day

    March 2007

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IMODE 103: Higher-Order Statistics with E&P Applications

Class Information

Upcoming Class Information

Date: TBA
Location: Houston, TX
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IMODE 103 is open for registration! Attendees can register online for the class.

Detailed Overview

Mathematical models that we use in seismology to describe phenomena which occur naturally or which include interactions with nature in their realizations are incomplete because not all the complexities of these phenomena can be taken into account in our mathematical models. Moreover, our understanding of these phenomena is generally incomplete. The introduction of statistics in these models, especially higher-order statistics (HOS), allows us to include properties inherent in nature in our models while compensating for some of our ignorance, from which we expect to be relieved in the future by a better and more complete theory. Our objective in this short course is to describe the notion of HOS from its first principles and show how HOS can help us to improve some of the models in seismology and therefore to improve our E&P data-processing and interpretation tools.

The course is intended for geophysicists working in data processing and R&D, and for people with an interest in understanding current and emerging technologies for seismic data processing.

Course Outline

Seismc Data Representation as Random Variables

  • Examples of random variables
  • From seismic signals to seismic random variables
  • Probability-density functions (PDF) of seismic random variables
  • Moments and cumulants of seismic random variables
  • Negentropy: a measurement of non-Gaussianity

Uncorrelatedness and Independence

  • Joint probability and the Kullback-Leibler divergence
  • Joint moments and joint cumulants
  • Uncorrelatedness and whiteness of random variables
  • Independence of random variables
  • Analysis of uncorrelatedness and independence with scatterplots
  • Whitening

Independent Component Analysis (ICA)

  • Maximizing contrast functions
  • Cumulant-tensor diagonalization
  • Negentropy maximization

LAB 1: Wavefield Decomposition

Statistics of Complex Random Variables

  • Statistics of complex random variables
  • Statistics of complex vector variables
  • Statistical independence of complex random variables

ICA of Complex Random Variables

  • Whiteness of complex random variables
  • Negentropy maximization
  • Permutation inconsistency problems
  • Cascaded ICA

LAB 2: Wavefield Decomposition

Statistics of Signals

  • Moments and their spectra
  • Cross-moments and their spectra

Biocoherence-Correlation Applications

Imaging Beyond Seismic Bandwidth

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