Seismic Reservoir Modeling. Dario Grana

coincide. For example, in a skewed distribution the larger the skewness, the larger the difference between the mode and the median. For a multimodal distribution, the mode can be a more representative statistical estimator, because the mean might fall in a low probability region. A comparison of these three statistical estimators is given in Figure 1.3, for symmetric unimodal, skewed unimodal, and multimodal distributions.

      The parameters of a probability distribution can be estimated from a set of n observations {x_i}_i=1,…,n. The sample mean

is an estimate of the mean and it is computed as the average of the data:

      (1.19)

Figure 1.3 Comparison of statistical estimators: mean (square), median (circle), and mode (diamond) for different distributions (symmetric unimodal, skewed unimodal, and multimodal distributions).

whereas the sample variance

is estimated as:

      (1.20)

      where the constant 1/(n − 1) makes the estimator unbiased (Papoulis and Pillai 2002).

      

      1.3.2 Multivariate Distributions

      In many practical applications, we are interested in multiple random variables. For example, in reservoir modeling, we are often interested in porosity and fluid saturation, or P‐wave and S‐wave velocity. To represent multiple random variables and measure their interdependent behavior, we introduce the concept of joint probability distribution. The joint PDF of two random variables X and Y is a function f_X,Y : ℝ × ℝ → [0, +∞] such that 0 ≤ f_X,Y(x, y) ≤ +∞ and

.

      The probability P(a < X ≤ b, c < Y ≤ d) of X and Y being in the domain (a, b] × (c, d] is defined as the double integral of the joint PDF:

      (1.21)

      Given the joint distribution of X and Y, we can compute the marginal distributions of X and Y, respectively, as:

      (1.22)

      (1.23)

      In the multivariate setting, we can also introduce the definition of conditional probability distribution. For continuous random variables, the conditional PDF of X∣Y is:

(1.24)

      where the joint distribution f_X,Y(x, y) is normalized by the marginal distribution f_Y(y) of the conditioning variable. An analogous definition can be derived for the conditional distribution f_Y∣X(y) of Y∣X. All the definitions in this section can be extended to any finite number of random variables.

An example of joint and conditional distributions in a bivariate domain is shown in Figure 1.4. The surface plot in Figure 1.4 shows the bivariate joint distribution f_X,Y(x, y) of two random variables X and Y centered at (1, −1). The contour plot in Figure 1.4 shows the probability density contours of the bivariate joint distribution as well as the conditional distribution f_Y∣X(y) for the conditioning value x = 1 and the marginal distributions f_X(x) and f_Y(y).

The conditional probability distribution in Eq. (1.24) can also be computed using Bayes' theorem (Eq. 1.8) as:

(1.25)

Figure 1.4 Multivariate probability density functions: bivariate joint distribution (surface and contour plots), conditional distribution for x = 1, and marginal distributions.

Figure 1.5 shows an example where the uncertainty in the prior probability distribution of a property X is relatively large, and it is reduced in the posterior probability distribution of the property X conditioned on the property Y, by integrating the information from the data contained in the likelihood function. In seismic reservoir