Seismic Reservoir Modeling. Dario Grana

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 f_X(x) can be written as:

      (1.40)

      where the distributions represent the Gaussian components with mean μ_X∣k and variance , and the coefficients πk are the weights of the linear combination with the condition . The condition on the sum of the weights guarantees that f_X(x) is a valid PDF.

Figure 1.10 shows an example of Gaussian mixture PDF with two components with weights equal to 0.4 and 0.6, means equal to −2 and 2, and equal variance of 1. The resulting PDF is multimodal owing to the presence of two components with different mean values and asymmetric due to the larger weight of the second component.

An example of bivariate Gaussian mixture distribution with two components is shown in Figure 1.11. The PDF in Figure 1.11 is estimated from a geophysical dataset of porosity and P‐wave velocity. The two components represent two different rock types, namely high‐porosity sand and impermeable shale. The mean of porosity in the sand component is 0.22, whereas the mean of porosity in shale is 0.06. Porosity and P‐wave velocity are negatively correlated in both components; however, the Gaussian mixture distribution shows a larger variance within the sand component, probably because of the rock heterogeneity and the fluid effect. The distribution is truncated for porosity less than 0, to avoid non‐physical values.

Figure 1.10 Univariate Gaussian mixture probability density function, with two components with weights equal to 0.4 and 0.6, means equal to −2 and 2, and equal variance of 1.

Figure 1.11 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 f_X(x) can be written as:

(1.41)

where the denominator is a normalizing constant to ensure that Eq. (1.41) is a valid PDF (i.e. the integral of the PDF is equal to 1). The mean μ_X of a Beta distribution is:

      (1.42)

      whereas the variance is:

      (1.43)

The Beta distribution is a suitable model for random variables describing volumetric fractions, since it is defined on the interval [0, 1]; however, it is not commonly used in practical applications owing to the limited analytical tractability. In geoscience applications, it is often convenient to use the Beta PDF as a prior distribution in Monte Carlo sampling algorithms to sample bounded random variables such as fluid saturations. Figure 1.12 shows two examples of Beta distributions with parameters α = β = 0.1 (convex curve, solid line) and α = β = 2 (concave curve, dashed line) that could be used to describe the distribution of fluid saturations and mineral fractions in a hydrocarbon reservoir, respectively. The convex distribution in Figure 1.12 shows high likelihood for values near the boundaries of the interval [0, 1], and low likelihood in the middle of the interval. This behavior is often observed in fluid saturations in the subsurface, where partial saturations are unlikely owing to the gravity effect. The concave distribution in Figure 1.12 shows low likelihood for values near the boundaries of the interval [0, 1], and high likelihood in the middle of the interval. This behavior is often observed in mineral fractions in the subsurface,