Читать книгу Seismic Reservoir Modeling - Dario Grana - Страница 24

We then introduce the log‐Gaussian distribution (or log‐normal distribution) that is strictly related to the Gaussian distribution. Log‐Gaussian distributions are commonly used for positive random variables. For example, suppose that we are interested in the probability distribution of P‐wave velocity. To avoid positive values of the PDF for negative (hence non‐physical) P‐wave velocity values, we can take the logarithm of P‐wave velocity and assume a Gaussian distribution in the logarithmic domain. In this case, the distribution of P‐wave velocity is said to be log‐Gaussian.

We say that a random variable Y is distributed according to a log‐Gaussian distribution with mean μ_Y and variance , if X = log(Y) is distributed according to a Gaussian distribution . The PDF f_Y(y) can be written as:

Figure 1.9 Log‐Gaussian probability density function associated with the standard Gaussian distribution in Figure 1.8.

(1.34)

where μ_X and are the mean and the variance of the random variable in the logarithmic domain.

The mean μ_Y and the variance of the log‐Gaussian distribution are related to the mean μ_X and variance of the associated Gaussian distribution, according to the following transformations:

(1.35)

(1.36)

(1.37)

(1.38)

Figure 1.9 shows the log‐Gaussian distribution associated with the standard Gaussian distribution shown in Figure 1.8.

A log‐Gaussian distributed random variable takes only positive real values. Its distribution is unimodal but it is not symmetric since the PDF is skewed toward 0. The skewness s of a log‐Gaussian distribution is always positive and is given by:

(1.39)

For 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.