Seismic Reservoir Modeling. Dario Grana

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Seismic Reservoir Modeling - Dario Grana


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href="#ulink_2da3342f-5d70-5881-9d5a-f691908f9b50">Figure 1.2. The random variable in Figure 1.2 could represent, for example, the P‐wave velocity of a porous rock, expressed in km/s.

      The triangular function fX(x) is a non‐negative function with integral equal to 1 (i.e. the area of the triangle with base of length 4 and height 0.5); hence, it satisfies the conditions in Eqs. (1.12)(1.14), and it is a valid PDF. We now compute the probability P(2 < X ≤ 3) using the definition in Eq. (1.15):

      The probability P(2 < X ≤ 3) represents the area of the region highlighted in Figure 1.2 and can also be computed as the area of a right trapezoid, rotated by 90°.

      In many applications, it is useful to describe the distribution of the continuous random variable using the concept of cumulative distribution function (CDF), FX : → [0, 1]. We assume that a random variable X has a PDF fX(x); then, the CDF FX(x) is defined as the integral of fX(x) in the interval (−∞, x]:

      (1.16)

.

      The CDF takes values in the interval [0, 1] and it is non‐decreasing, i.e. if a < b, then FX(a) ≤ FX(b). When FX(x) is strictly increasing, i.e. if a < b, then FX(a) < FX(b), we can define its inverse function, namely the quantile (or percentile) function. Given a CDF FX(x) = p, the quantile function is

and it associates p ∈ [0, 1] to the corresponding value x of the random variable X. The definition of percentile is similar to the definition of quantiles, but using percentages rather than fractions in [0, 1]. For example, the 10th percentile (or P10) corresponds to the quantile 0.10. In many practical applications, we often use quantiles and percentiles. The 25th percentile is the value x of the random variable X such that P(Xx) = 0.25, which means that the probability of taking a value lower than or equal to x is 0.25. The 25th percentile is also called the lower quartile. Similarly, the 75th percentile (or P75) is the upper quartile. The 50th percentile (or P50) is called the median of the distribution, since it represents the value x of the random variable X such that there is an equal probability of observing values lower than or equal to x and observing values greater than x, i.e. P(Xx) = 0.50 = P(X > x).

      A continuous random variable is completely defined by its PDF (or by its CDF); however, in some special cases, the probability distribution of the continuous random variable can be described by a finite number of parameters. In this case, the PDF is said to be parametric, because it is defined by its parameters. Examples of parameters are the mean and the variance.

      The mean is the most common measure used to describe the expected value that a random variable can take. The mean μX of a continuous random variable is defined as:

      (1.17)

      The mean μX is also called the expected value and it is often indicated as E[X]. However, the mean itself is not sufficient to represent the full probability distribution, because it does not contain any measure of the uncertainty of the outcomes of the random variable. A common measure for the uncertainty is the variance

:

      (1.18)

      The variance

describes the spread of the distribution around the mean. The standard deviation σX is the square root of the variance,
. Other parameters are the skewness coefficient and the kurtosis coefficient (Papoulis and Pillai 2002). The skewness coefficient is a measure of the asymmetry of the probability distribution with respect to the mean, whereas the kurtosis coefficient is a measure of the weight of the tails of the probability distribution.


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