Seismic Reservoir Modeling. Dario Grana. Читать онлайн. Newlib. NEWLIB.NET

Автор: Dario Grana
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: География
Год издания: 0
isbn: 9781119086192
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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 {xi}i=1,…,n. The sample mean

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

      (1.19)

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 fX,Y : × → [0, +∞] such that 0 ≤ fX,Y(x, y) ≤ +∞ and

.

      The probability P(a < Xb, c < Yd) 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 XY is:

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

Graphs depict multivariate probability density functions describing bivariate joint distribution, conditional distribution for x = 1, and marginal distributions.