Seismic Reservoir Modeling. Dario Grana
Читать онлайн книгу.of the favorable event, generally called the probability of success, is a real number p such that 0 ≤ p ≤ 1. The probability mass function pX(x) is then:
(1.29)
The mean of the Bernoulli distribution is then μX = p and the variance is
The Bernoulli distribution has several applications in earth sciences. In reservoir modeling, for example, we can use the Bernoulli distribution for the occurrence of a given facies or rock type. For instance, we define a successful event as finding a high‐porosity sand rather than impermeable shale. The probability of success is generally unknown and it depends on the overall proportions of the two facies.
1.4.2 Uniform Distribution
A common distribution for discrete and continuous properties is the uniform distribution on a given interval. According to a uniform distribution, a random variable is equally likely to take any value in the assigned interval. Hence, the PDF is constant within the interval, and 0 elsewhere. If a random variable X is distributed according to a uniform distribution U([a, b]) in the interval [a, b], then its PDF fX(x) can be written as:
(1.30)
The mean μX of a uniform distribution in the interval [a, b] is:
(1.31)
and it coincides with the median, whereas the variance
(1.32)
An example of uniform distribution in the interval [1, 3] is shown in Figure 1.7. The uniform distribution is sometimes called non‐informative because it does not provide any additional knowledge other than the interval boundaries.
1.4.3 Gaussian Distribution
Most of the random variables of interest in reservoir modeling are continuous. The most common PDF for continuous variables is the Gaussian distribution, commonly called normal distribution. We say that a random variable X is distributed according to a Gaussian distribution
A Gaussian distribution
Figure 1.7 Uniform probability density function in the interval [1, 3].
Figure 1.8 Standard Gaussian probability density function with 0 mean and variance equal to 1.
As shown in Eq. (1.15), in order to compute any probability associated with the random variable X, we must compute an integral. For a Gaussian distribution, the integral of Eq. (1.33) has no analytical form; therefore, we must use numerical tables of the cumulative density function of the standard Gaussian distribution (Papoulis and Pillai 2002). For example, if X is a random variable distributed according to a standard Gaussian distribution
To compute a probability associated with a non‐standard Gaussian distribution