Seismic Reservoir Modeling. Dario Grana

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Seismic Reservoir Modeling - Dario Grana


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these reasons, the log‐Gaussian distribution is a convenient and useful model to describe unimodal positive random variables in earth sciences.

      An example of application of log‐Gaussian distributions in seismic reservoir characterization can be found in Section 5.3, where we assume a multivariate log‐Gaussian distribution of elastic properties in the Bayesian linearized seismic inversion.

      

      1.4.5 Gaussian Mixture Distribution

      Gaussian and log‐Gaussian distributions are unimodal parametric distributions. However, many rock properties in the subsurface, for example porosity and permeability, are multimodal. Multimodal distributions can be described by non‐parametric distributions, but these distributions require a large amount of data to be estimated. In many applications, multimodal distributions can be approximated by Gaussian mixture distributions, i.e. linear combinations of Gaussian distributions.

      We say that a random variable X is distributed according to a Gaussian mixture distribution of n components, if the PDF fX(x) can be written as:

      (1.40)f Subscript upper X Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript k equals 1 Overscript n Endscripts pi Subscript k Baseline script upper N left-parenthesis upper X semicolon mu Subscript upper X bar k Baseline comma sigma Subscript upper X bar k Superscript 2 Baseline right-parenthesis comma

      where the distributions script upper N left-parenthesis upper X semicolon mu Subscript upper X bar k Baseline comma sigma Subscript upper X bar k Superscript 2 Baseline right-parenthesis represent the Gaussian components with mean μX∣k and variance sigma Subscript upper X bar k Superscript 2, and the coefficients πk are the weights of the linear combination with the condition sigma-summation Underscript k equals 1 Overscript n Endscripts pi Subscript k Baseline equals 1. The condition on the sum of the weights guarantees that fX(x) is a valid PDF.

Graph depicts bivariate Gaussian mixture probability density function estimated from a geophysical dataset, in the porosity–velocity domain.

      

      1.4.6 Beta Distribution

      The Beta distribution is a PDF for continuous random variables defined in the interval [0, 1]. The PDF is uniquely defined by two positive parameters, namely α and β, that determine the shape of the distribution and its derivatives. The generalization to the multivariate domain is the Dirichlet distribution.

      A random variable X is distributed according to a Beta distribution B(X; α, β) with parameters α > 0 and β > 0, in the interval [0, 1], if its PDF fX(x) can be written as:

      (1.42)mu Subscript upper X Baseline equals StartFraction alpha Over alpha plus beta EndFraction comma

      whereas the variance sigma Subscript upper X Superscript 2 is:

      (1.43)sigma Subscript upper X Superscript 2 Baseline equals StartFraction italic alpha beta Over left-parenthesis alpha plus beta right-parenthesis squared left-parenthesis alpha plus beta plus 1 right-parenthesis EndFraction period


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